Math Lecture Notes

Arrange the following numbers in order from least to greatest. 2 cubed, 4 squared, 6 to the 0 power, 9, and 10 to the first. The first thing we need to do is simplify these numbers. 2 cubed means 2 times itself 3 times. So that's 2 times 2 times 2. 2 times 2 is 4, times 2 is 8. So that is 8. 4 squared is 4 times itself 2 times. 4 times 4 is 16. Now this next one is very interesting because it doesn't matter what this number is. The base number doesn't matter. Any number raised to the 0 power is always 1. 9 of course is simply 9. And 10 to the first is 10 times itself one time, or not really 10 times itself at all, it's just 10. You're not multiplying 10 by anything else because you just have 10 written one time. And now they want these numbers in order from least to greatest. So the smallest number would be 1, followed by 8. Then 9, 10, and 16. But of course, those are not in our answer choices, because the numbers are written as they originally were. So now I'm going to go back and write them the way they were written to begin with. 1 was 6 to the 0. 8 was 2 cubed. 9 was always 9, 10 was from 10 to the first, and 16 was 4 squared. And now we just need to find which answer choice has this as the answer. So of course, we should be starting with 6 to the 0, so that takes it down to either b or d. But then the next should be 2 cubed, which is not the case for b, so our answer is d. The Charleston Recycling Company collects 50,000 tons of recyclable material every month. The chart shows the kinds of materials that are collected by the company's five trucks. What is the second most common material that is recycled? This is a pie chart, and a pie chart is a visual representation of data. The data represented in the chart will add up to a total of 100%. 40% plus 25% plus 15% plus 5% plus 15% is a total of 100%. The larger the slice, the larger the percent. The larger the slice and percent means the more common that piece of data is, or the more times that data occurs. So in this case, they're asking what the second most common common material is. We can see that paper is the most commonly recycled material. It has the largest slice of the pie and thus the largest percentage. So the second most common recycled would be glass at 25%. B. The Charleston Recycling Company collects 50,000 tons of recyclable material every month. The chart shows the kinds of materials that are collected by the company's five trucks. Approximately how much paper is recycled every month? This pie chart. is a total of 100 percent. It's a total 100 percent of 50,000 tons. This 40 percent means 40 percent of the total. Paper is 40 percent, so 40 percent of the total amount of recyclable material collected, or 50,000 tons, 40% of that is paper. And that's what we want to find is how much is that? Well, of means to multiply. And I'm going to multiply 40% and 50,000, which means I need to change 40%. to a decimal. So move the decimal two places to the left. So I'm actually multiplying 4 tenths or 40 hundredths, it's the same thing, times 50,000. So we have 50,000 times 4 tenths. 4 times 0 is 0, 4 times 0 is 0, 4 times 0 is 0, 4 times 0 is 0, and 4 times 5 is 20. We have one number behind the decimal, so our answer should have one number behind the decimal, and we get 20,000 tons. So 20,000 tons of the total 50,000 tons of recyclable material is paper. Dorothy is half her sister's age. She will be three-fourths of her sister's age in 20 years. How many years old is she? We have two unknowns here. We don't know Dorothy's age, and we don't know her sister's age. So I'm going to make up two variables for this. I'm going to use D for Dorothy's. Current age. And I'll use S for the sister's current age. And I say current because they're talking about their ages now and then also their ages in 20 years. But what we want to know is how many years old she is now. So we're going to deal with... their current ages. So first we have that Dorothy is, and is in math means equals. It's one of my favorite words. I love the equal sign. So Dorothy is, Dorothy equals half, her sister's age, so half s. So it's kind of like translating. You're translating from words into mathematical equations. The next sentence says, she will be, so Dorothy. will be three-fourths of her sister's age in 20 years. So Dorothy, in 20 years, will be three-fourths of her sister's age. sister's age in 20 years. Remember that D and S were Dorothy and her sister's current ages. So when they talk about in 20 years, that means we have to add 20 to their current age. So Dorothy's age in 20 years, so plus 20, will be or equals 3 fourths of her sister's age. Also, in 20 years. So we want to solve for Dorothy's age, obviously, and we can do this in lots of different ways. But the first thing I notice is that since I know that D is 1 half S, I can replace D in this equation with 1 half S, and that's called substitution. So that's the method I'm going to use. I'm going to replace D with 1 half S plus 20 equals 3 fourths times the quantity S plus 20. Well, from here we have lots of options for solving. And the first thing I think about is, really I just want to get rid of this fraction, this 3 fourths. So, I think I'll start by distributing the 3 fourths to what's inside my parentheses. So, I have 1 half s plus 20 is equal to. 3 fourths s plus, and then with the 3 fourths and the 20, I can put 20 over 1, and I can simplify multiplying these numbers by cross-canceling 4 and 20. I'm simply multiplying two fractions together. so I can use this shortcut. I'm going to divide both 4 and 20 by 4. 4 divided by 4 is 1, and 20 divided by 4 is 5. So, 3 times 5 is 15. 3 fourths of 20 is 15. That makes sense. Okay, so we need to get all of our variables on the same side. So to do that, I'm going to subtract 1 half s from both sides. 1 half s minus 1 half s is 0. I'm going to bring down 20, and that equals... Now, to subtract fractions, I have to have common denominators. So 1 half... is also 2 fourths, so I'm going to change 1 half to 2 fourths so that I can subtract these. Then when I subtract, you just subtract the numerators. 3 minus 2 is 1, and your denominators stay the same, 1 fourth s plus 15. Next, I need to subtract 15 from both sides. 20 minus 15 is 5, and 5 is of the sister's age. So finally, I need to multiply both sides by the multiplicative inverse of which is 4 over 1, or just 4. So multiply both sides by 4. 4 times 5 is 20 and 20 is her sister's age. A fourth of 4 is 1, so we're left with S, the sister's age. Well, remember that Dorothy is half of her sister's age, so since we now know her sister's age is 20, we know that Dorothy is half of that and half of 20 is 10. So that means Dorothy is 10 years old. You could have solved this also by just kind of playing with the numbers, playing with the answers, and seeing which ones would work with the information you were given, but I always love to solve a good algebra problem. Chan receives a bonus from his job. He pays 30% in taxes, gives 30% to charity, and uses another 25% to pay off an old debt. He has $600 remaining from his bonus. That was the total amount of Chan's bonus. First I want to start with totaling the percentage that he spent or used, the percentage that is not left. So he pays 30% in taxes, gives 30% to change. charity, and uses the 25% to pay off an old debt. That's a total of 85% that he has used, which means he had 15% left. and the $600 is the 15% of his total. $600 is 15% of the total amount of his bonus. $600 is as equals... Change the percent to a decimal. Of means to multiply. And we don't know the total, so use a variable, like t. And now we just need to solve for t. So divide both sides by 15 hundredths. So, 600 divided by 15 hundredths. We need to move the decimal two places to the right. So, we do the same to 600, add two zeros, and 15 goes into 60 four times. 15 times 4 is 60. You subtract and get 0. When you bring down 0, 15 goes into 0, 0 times, and that's just going to keep happening two more times until we get 4,000. So, what was the total amount of Chan's bonus? $4,000. A tire on a car rotates 500 rpm or revolutions per minute when the car is traveling at 50 kilometers per hour. What is the circumference of the tire in meters? Well, the first thing I want to do is take this 50 kilometers per hour. and convert it into meters, since that's what they want the circumference in. So to convert kilometers to meters, I multiply it by the conversion factor of 1000 meters is 1 kilometer. And when I multiply these fractions, I can cross cancel these units kilometers, and I'll be left with meters per hour. And I'll go ahead and do that first. So our kilometers cross cancel, 50 times 1000 is 50,000 meters per 1 hour. So every hour this car goes 50,000 meters. and we know that it goes 500 revolutions every minute. So then I want to convert my 50,000 meters per hour into meters per minute so I can more easily compare it to my revolutions per minute. So I'll multiply it by the conversion factor of 1 hour is 60 minutes. So again, my hours cross cancel since I'm multiplying fractions and I get 50,000 meters every 60 minutes. What I want to find though is what one revolution is. How many meters is one revolution? Because one revolution is one time all the way around the tire. That would be the circumference, which is what they're asking me for. I'm going to take this 50,000 meters per 60 minutes and multiply it times my 500 revolutions per minute. Every 1 minute, the car goes 500 revolutions. so that again my minutes, these units, minutes cross cancel. I can cross cancel my 50,000 and my 500 by dividing them both by 500. 500 divided by 500 is 1, and 50,000 divided by 500 is 100. So then 100 meters times 1 is 100 meters divided by 60 revolutions. So now I found that every 100 meters, this car, it's going 60 revolutions every 100 meters, which I can then simplify by canceling out the zeros and I get 10 sixths meters per revolution. So it goes 10 sixths of a meter every revolution. And again, a revolution is one time all the way around the wheel, which is the circumference. So our circumference is 10 sixths. A combination lock uses a three digit code. Each digit can be any of the ten available integers 0 through 9. How many different combinations are possible? In this probability problem, there are three independent events. Independent meaning they don't affect each other. So to find the possible outcomes, we would have to find the product of the possible outcomes for these three independent events. So, the possible outcomes would be first the possible outcomes for the first digit, 10, since there are 10 available numbers, times the possible outcomes for the second digit, which again is 10, 10 integers, times the possible outcomes for the third digit, which is which is again 10. So the probability is 10, or the possible outcomes is 10 times 10, which is 100, times 10, which is 1000 possible outcomes. This is the quicker way to do it. Of course, you can always see by actually writing the combinations. 0 0 0 would be the first, then 0 0 1, 0 0 2, 0 0 3. 0, 0, 4, 5, 6, etc., all the way up to 9, 9, 9, which makes it seem like there are 999 possible outcomes. But really, we have to count this first one, the 0, 0, 0, which gives us the total of 1,000 possible outcomes. Which of the following expressions is equivalent to the quantity a plus b times the quantity a minus b? Well, this is actually a rule that you would memorize, but you don't have to memorize it. You can always just take these binomials and multiply them together using FOIL. The F stands for first, meaning to multiply the first terms in each set of parentheses. So A times A. And A times A is A squared. Then, O stands for outer or outside, so you multiply the two terms on the outsides, or A and negative B. So, A times negative B is negative A. I stands for inner or inside, so you multiply the two terms on the inside, B and A. B times A is BA or A, so plus A. And finally, the L stands for last, so you multiply the two terms that are in the last positions of each set of parentheses. Positive B times negative B, which is negative B squared. Then to simplify, we combine like terms. Negative a b and positive a b are like terms, and they cancel each other out because they're additive inverses of each other, or opposites. So we're left with a squared minus b squared, or answer a. Which of the following expressions represents the ratio of the area of a circle to its circumference? We've got some pretty key words here. The first one I see is ratio. A ratio is a comparison of two numbers, like a fraction. And what we're comparing is the area of a circle to its circumference. Well, the area of a circle is pi times radius squared, while the circumference of a circle is 2 times pi times radius. So, the area is going to be our numerator. Since it was listed first, area, 2 circumference means that the area should be in the numerator, divided by this 2 as your division bar, the circumference, 2 times pi times radius. And then we need to simplify. First I see that pi divided by pi cancels. Pi divided by pi is 1. Anything divided by itself is 1. So we can cancel it. And then I can simplify r squared divided by r. r. When you divide, you subtract the exponents. Or another way to think about it is r squared is r times r. And I'm dividing that by r. So again, you have something divided by itself. itself is 1 or it cancels. So I'm left with just r in the numerator divided by 2 in the denominator since those r's cancel. So we just have r divided by 2, which is answer e. Kyle bats third in the batting order for the Badgers baseball team. The table shows the number of hits that Kyle had in each of seven consecutive games played during one week in July. What is the mode of the numbers shown in the table? Mode means the number that occurs the most. So out of these numbers, 1 occurs 1, 2, 3 times, 2 occurs twice, 3 occurs once, and 4 occurs once. So the number that occurs the most, or the mode, is the number 1. Kyle bats third in the batting order for the Badgers baseball team. The table shows the number of hits that Kyle had in each of seven consecutive games played during one week in July. What is the mean? of the numbers in the distribution shown in the table? Well, the mean of a set of numbers is the average. And to find the average or the mean, you must add all of the numbers. of the numbers together and then divide them by the number of numbers there are. So first we'll add all the numbers together. 1 plus 2 plus 3 plus 1 plus 1 plus 4 plus 2. Then we divide by the number of numbers there are, which is 1, 2, 3, 4, 5, 6, 7. So we do this. we divide by 7. And again, this is how we find the mean. add the numbers together, and then divide by the number of numbers there are. So that is 1 plus 2 is 3, 3 plus 3 is 6, plus 1 is 7, plus 1 is 8, plus 4 is 12, plus 2 is 14. So 14 divided by 7. And 14 divided by 7 is 2. Therefore, the mean or the average... The average of these numbers is 2. Jamie had $6.50 in his wallet when he left home. He spent $4.25 on drinks and $2 on a magazine. Later his friend repaid him $2.50 that he had borrowed the previous day. How much money does Jamie have in his wallet? The first thing I'd like to do is find the total amount of money that Jamie spent. I see that he spent $4.25 and $2. So to find the total amount of money he spent, I need to add $4.25 plus $2. And it's very important that whenever you add numbers with decimals or or subtract numbers with decimals that you line up the decimals. So now I'm ready to add, and I can just bring my decimal down before I even start. 5 plus 0 is 5, 2 plus 0 is 2, 4 plus 2 is 6. So now I know he spent a total of $6.25. And if he spent that much money, then that's money that is taken away from what he spent. what he had. So he started with $6.50, but he spent $6.25. So I have to subtract what he spent from what he had before. And again, we need to line up our decimals. I can't take 5 from 0, so I'm going to borrow from this 5. It becomes a 4. And now I have 10 minus 5 is 5. And 4 minus 2 is 2. I can bring down the decimals. down that decimal, 6 minus 6 is 0. So that means after he spent the $4.25 on the drinks and $2 on the magazine, he only had 25 cents left. But his friend repaid him $2.50, which means his friend gave him $2.50. So that's money that's added to his total. So now I'm going to take what he had after he spent his money and add the money his friend gave him. And again, we line up our decimals. 5 plus 0 is 5, 2 plus 5 is 7, bring down the decimal, 0 plus 2 is 2. So after spending money and getting repaid money, he now has a total of $2.75, which is the answer. D A sailboat is 19 meters long. What is its length in inches? There are many ways to approach this problem and solve it. The first thing I'm going to do is convert my meters into centimeters because then it's a quick conversion into inches from there. So I know that one meter is equal to 100 centimeters. We can use a proportion to solve. We can take this conversion factor and use it as a ratio. 1 meter is 100 centimeters, which equals 19 meters is how many centimeters. Then to solve the proportion, we cross multiply. 1 times x is x, and 100 times 19 is 1900. So it's 1900 centimeters. Now, I can use another proportion to solve for or to find inches. I know that 2 and 54 hundredths centimeters is 1 inch. and I have 1900 centimeters and I need to know how many inches that is. And again, to solve a proportion, we cross multiply. So, 2 and 54 hundredths... centimeters times X, 2 and 54 hundredths X is equal to 1 times 1900, which is 1900. And I forgot to put my little line up here, so I'll do that. And now we need to solve for x, which means dividing both sides by 2 and 54 hundredths. 2 and 54 hundredths divided by 2 and 54 hundredths is 1, times x is x, and that equals 1900 divided by 2 and 54 hundredths, so we need to do some division. which means I have to move this decimal two places to the right. So, I have to take my decimal in 1900 and move it two places to the right, which means adding two zeros. So, now my number is a hundred... 90,000 being divided by 254. So 254 does not go into 1, or into 19, or into 190, but it does go into 1, or into 19, or into 190. go into 1900. We can use compatible numbers here to get a very quick estimate of how many times it goes into 1900. So 254 is very close to 250. So 250, 500, 750, 1000, 1250, 1500, 1750. 1,800. Sorry, 1,750 and then 2,000. It only goes seven times because 2,000 is too much. 2,540 only goes seven times into 1,900. We can see what that is right here. 7 times 4 is 28. 7 times 5 is 35. 36, 37. 7 times 2 is 14. 3 is 17, so it's 1778. Then we subtract those, and we have to borrow from the 9, and that becomes an 8. This is 10, and we borrow from the 10, it becomes a 9. This is 10, so 10 minus 8 is 2, 9 minus 7 is 2, and 8 minus 7 is 1, 122. Now we bring down our next 0, Again, we can use that 250 rule to see how many times it goes into 1,220. So 250, 500, 750, 1,000, and then it would be 1,250, but 1,250 is too much, so it only goes four times. So let's see, 254 times 4, 4 times 4 is 16, carry the 1, 4 times 5 is 20. plus 1 is 21. 4 times 2 is 8, plus 2 is 10. So that's 1,016. And then we subtract. So we need to borrow from this 2, that becomes a 1. So this is 10 minus 6 is 4. 1 minus 1 is 0. And 2 minus 0 is 2. So that's 204. And then we bring down our 0 here. So again, how many times does 250 go into 2040? So we have 250, 500, 750, 1000, so 4 more times for 2000, so it looks like a total of 8 times. Let's check that. Let's see where I have some room. We'll just do it up here. 254 times 8. 8 times 4 is 32. 8 times 5 is 40, plus 3. 43. 8 times 2 is 16. 16 plus 4 is 20. So it's 8 times is correct, and it's 2,032. We subtract and we get 8. And that would be our remainder. It would be 8 out of 254. Or we could keep going to find our decimal. But if we look at our answers, we don't need to keep going because none of our answers have the decimal as part of it. So our answer is D, 748 inches. Mrs. Patterson's classroom has 16 empty chairs. All the chairs are occupied when every student is present. If two-fifths of the students are absent, how many students make up her entire class? This is a great problem to use a proportion to solve. We can take this fraction, two-fifths, and use that as our first ratio, two being the amount of students who are absent out of the five total. That equals, again, number of students who are absent divided by the total. It says in the problem that there are 16 empty chairs, which means there are 16 students missing since usually all the chairs are occupied when every student is present. Every empty chair is a student who's absent. So we have 16 absent students out of... And this is what we don't know. What's the total? How many students make up her entire class? And to solve a proportion, we simply need to cross multiply. 2 times x is 2x, and that equals 16 times 5, which is 80. And then we solve for x by dividing both sides by 2. And x equals 5. forty. So that means there are forty students in her entire class. Rachel spent twenty-four dollars and fifteen cents on vegetables. She bought two pounds of onions, three pounds of carrots, and one and a half pounds of mushrooms. If the onions cost three dollars and sixty-nine cents per pound, she would have to pay the cost of the onions. The carrots cost $4.29 per pound. What is the price per pound of mushrooms? First, I want to figure out how much money was spent on the onions that were $3.69 per pound and how much was spent on the carrots that were $4.29 per pound. Let's start with the onions. Those onions cost $3.69 for every pound that's purchased, and she purchased 2 pounds. So we need to multiply 369 times 2 to find the total amount of money spent on onions. So 2 times 9 is 18, carry the 1. 2 times 6 is 12, plus 1 is 13, write the 3, carry the 1. 2 times 3 is 6, plus 1 is 7. Two numbers behind the decimal, two numbers behind the decimal. So that means she spent $7.38 out of her total on onions. We're going to do something similar to find the total amount of money spent on carrots. So next we'll deal with our carrots. The carrots were $4.29 per pound, so $4.29 times 3, since she bought 3 pounds of carrots. 3 times 9 is 27, write the 7, carry the 2. 3 times 2 is 6, plus 2 is 8, and 3 times 4 is 12. Two numbers behind the decimal. two numbers behind the decimal. So she spent a total of $12.87 on carrots. And so now I'm going to find the total spent on onions and carrots. So that means I'm just going to add these two values together. So $12.87 plus $7.38. Make sure that when you add or subtract decimals that you line those decimals up. So we can go ahead and bring that down right now. 7 plus 8 is 15. 8, 9, 10, 11, 12, 7, 9, 10. That means she spent $20.25 on the onions and carrots. Knowing that will help us find out what amount was left to be spent on the mushrooms. Since I know her total spent on all of the vegetables is $24.15, I can take that total and subtract $20.25. which was spent on onions and carrots. What I'll have is the amount of money she spent on the mushrooms. Again, we line up our decimals. 5 minus 5 is 0. I can't take 2 from 1, so I borrow from the 4. It becomes a 3. Then I have 11 minus 2, which is 9. Bring that decimal down. 3 minus 0 is 3. She had $3.90 to spend on mushrooms. mushrooms, but what we need to know is what the price per pound was, which we can find using this information. She bought one and a half pounds of mushrooms. We need to take the total amount spent on mushrooms and divide that by how many pounds of mushrooms she purchased, which was one and a half or one and five tenths. Now we need to move the decimal one place in both places. both of our numbers, so that's where it is now. 15 goes into 39, 2 times. 15 times 2 is 30. Subtract and you get 9. Bring down the 0. 15 goes into 90, 6 times. 15 times 6 is 90. Subtract and you get 0. So no remainder, but we can add a 0. zero here since we're talking about money. That means that the mushrooms were $2.60 per pound, which is answer A. In the figure, A, B, and C are points on the number line, where O is the origin. What is the ratio of the distance BC to the distance A? First, we need to find the distance between points B and C. The distance between two points is the absolute value of the difference of the coordinates. That is the absolute value of B is 5 minus C is 8. which is the absolute value of negative 3, and that is 3. So the distance between points B and C is 3. Now, we're going to do the same thing to find the distance between points A and B. Again, it's the absolute value of the difference of the coordinates of point A and point B. That's the absolute value of A is negative 6 minus B is 5, which is the absolute value of negative 6 minus 5 is negative 11, and that is 11. So the distance from point A to point B is 11. Now they've asked us what the ratio of those distances is, and it gives you the order that it wants the ratio written in. So the distance BC should be the first number in the ratio, 3, 2, and we use a colon or you can write 2, it's fine too, the distance from points A to B, which is... 11. So our ratio is 3 to 11. We'll answer D Jesse invests $7,000 in a certificate of deposit that pays interest at the rate of 7.5% annually. How much interest in dollars does Jesse gain from this investment during the first year that he holds the certificate? This is a great problem to use this formula on. And some people call it IPERT, just as a way to remember it. The I stands for the interest, the amount of interest in dollars, like what we're trying to find. P stands for principal. or the amount that's being invested initially. That would be our $7,000. The R is the rate or the percentage, but written as a decimal. So we're going to take 7 and... and write it as a decimal. That will be our rate. Then T stands for time. You may want to make a note of that. This I is the interest, P is the principle, R is the rate as a decimal, and the T is the time. So we're going to take our information from our problem and plug it in. So our amount of interest is equal to the principal, $7,000. times the rate as a decimal again, so we're taking 7.5% and changing it to a decimal, which means taking the decimal and moving it two places to the left. So, you add that 0, so it's 75 thousandths and then times the year or the time and we're just trying to find out about the first year. So that means it's been in there for one year and then we just need to multiply. those things together. So 1 times anything is just that, so I can really kind of ignore that. So I just need to multiply these two numbers together. So I have this 75 thousandths and I'm going to... to put it on top because it's very easy to multiply times 7,000 since the first thing I'm going to do is just write down those three zeros. The only number I really have to multiply by is the 7. So 7 times 5 is 35, write the 5, carry the 3. 7 times 7 is 49, plus 3 is 52. And then we have three numbers behind the decimal. so three numbers behind the decimal, and that means that he's going to make $525 off of his investment. So that's the number you would need to bubble in, is 525. In an election in Kimball County, candidate A obtained 36,800 votes. His opponent, candidate B, obtained 32,100 votes. 2,100 votes went to write-in candidates. What percentage of the votes went to candidate A? To find a percent, we need the part over the whole. We know the part. The part is the number of votes candidate A received, which is 36,800. What I need to know is the whole, or the total, number of votes. To find the total number of votes, we need to add up 36,800. 32,100 and 2,100 so that we can find the total. We're adding. Those are 0 and again we get 0. Then we have 8, 9, 10, so that's a 0. Carry the 1. We have 6, 7, 8, 9, 10, 11. Carry the 1. We have 3, 6, 7, 8, 9, 10, 11. So that means there were a total of 71,000 votes. Now to find our percent, we need to divide. That'll give us our decimal, which will then change into a percent. But before I divide, I'm going to first simplify this fraction by canceling two zeros. And now my division will be a lot easier. So we need to divide 368 by 710. Well, 710 doesn't go into 368. at all, which means I'm going to have to add a decimal and a zero. Now, I can divide 710 into 3680. So, I can use compatible numbers here, and 710 is very close to 700. 700 times 5 would be 3500, so I'm thinking 710 is going to go about 5 times into 3,680. So let me multiply this, we get 0, and that's 5, and that's 35. So 3,550. Yep, looks good. So that's 5, we have 3,550. Now we need to subtract, and that's 0, 3, 1, so 130. Then we need to add a 0 and bring it down, 1300. So, 710 and it's only going to go into 1300 one time because 700 plus 700 would be 1400 and that's too much. So, it's just one time. 710 times 1 is 710. Then we need to subtract. So, I'm going to borrow from the 1, that's a 0. That's 13, borrow from the 13, that's 12, which makes that 10. We'll borrow, well we don't need to borrow that 0, so 10 minus 1 is 9, and 12 minus 7 is 5, 590. Which makes sense, if you add 90 back to 710, you get 800. If you add 500 to 800, you get the 1300. We need to add another 0 and bring it down. So now we need to know how many times 710 goes into 5,900. So I'm again going to use that compatible number of 700. So let's see, 700 times 8 would be 5,600. So I'm thinking it goes in there about 8 times. So that's 0 and that's 8 and that's 56. So yes, that's very close, so that's 8. And we have 5680. We can subtract, and that's 0. Borrow from the 9, that's an 8, so that's 10. 10 minus 8 is 2. 8 minus 6 is 2, so we get 220. And we could keep going, but really, that's all I need to know, because as I look at my answers, it's very clear which one of these is my answer. This is, well, it's about 518 thousandths, and then when we change it into our percent, we move our decimal two places to the right, we get 51 and 8 tenths percent, which is answer A. Francine can ride 16 miles on her bicycle in 45 minutes. At this speed, how many minutes would it take Francine to ride 60 miles? This problem is a good candidate for using a proportion to solve. There's more than one way to solve it, but I like proportions, so I'm going to use a proportion. First, I know that Francine can go 16 miles in 45 minutes. That's going to be my first ratio. So I have 16 miles in 45 minutes. And the key to setting up a successful proportion is to be consistent. So since our first ratio we have miles to minutes, then our second ratio will also be miles to minutes. So they want to know how many minutes it would take Francine to ride 60 miles. So my miles is 60, and my minutes is what I'm trying to find, so that's where I'll put a variable. To solve a proportion, we cross multiply. 16 times X is 16X, and that equals 60 times 45, 0. 6 times 5 is 30, carry the 3. 6 times 4 is 24, plus 3 is 27, so 2,700. And then we solve for x by dividing both sides by 16. So x equals, and now I need to find out what 2,700 divided by 16 is. 16 goes into 2,700 one time, and that's 16, and we subtract and get 11, and we bring down our 0. We could use 15 as a compatible number to see how many times 15 would go into 110. So we have 15, 30, 45, 60, 75, 90, 105. So 7 times. So let's see if 16 will go into 110 7 times. 7 times 6 is 42. 7 times 1 is 7, plus 4 is 11. And that's a little too big, which means I know 16 goes into 110 six times. So 16 times 6, 6 times 6 is 36, 6 times 1 is 6, plus 3 is 9. And that works. So we need to subtract, and we get 14. Then we bring down our next 0, so we have 140. So I know 16 is going to go into 140 at least 7 times, but it should go even more than that. So let's see what 16 times 8 is. 8 times 6 is 48, 8 times 1 is 8, plus 4 is 12, and that's all we're going to be able to do, so 8 times and we get 128. When we subtract those numbers, we get 12. So that means we need to add a decimal and a zero and bring it down. 16 goes into 120 seven times. 16 times 7, we found earlier, is 112. We subtract and we get 8. We add another 0 and bring it down. 16 goes into 80 less than 6 times, and actually exactly 5 times. 16 times 5 is 80. We subtract and we get a remainder of 0. That means if it took her 45 minutes to go 16 miles, It should take her 168 and 75 hundredths minutes to go 60 miles. And this is the answer you would grid in. Figure 9 shows two quarter circles centered on the origin of the Cartesian coordinate plane. The inner circle has a radius of 2 units. The outer circle has a radius of 3 units. What is the area of the shaded region? So that's right in here. So if you can just imagine it, this circle would continue going. as with this larger circle. So, since it's been split into quarters, all we're finding the area of is a quarter of those circles. So, first let's start with the formula for the area of a circle. is. It's pi times radius squared. As we discussed though, this is only a quarter of these circles, so our area is a quarter of the area of these circles. And then we need to find the area of just the part between those two circles. So that would be the area of the large circle minus the area of the small circle. minus the area of the small circle. The area of the large circle would be pi times the radius, which is 3 squared, minus the area of the small circle, which would be pi times 2 squared. The 3 and the 2 came from the radius. The radius of the small circle is 2, and the radius of the outer circle has a radius of 5. 3. So that's where these numbers came from. And now let's simplify that. So we get area is 1 4th of 3 squared is 9, so 9 pi minus 2 squared is 4, so 4 pi. Now since each one of these terms has a pi in it, we can factor the pi out. So we get area equals 1 4th pi times the quantity 9. 9 minus 4. Then we can simplify 9 minus 4 so we have area equals a fourth pi times 5. And a fourth times 5 is 5 fourths, so that's 5 fourths pi. But of course, if you're needing to bubble this in on an answer sheet, then you don't have the option of bubbling pi. So, we're going to have to multiply 5 fourths times pi. And 5 fourths times pi is 1 and 25 hundredths pi. And for pi, we'll just use 3 and 14 hundredths. So we're doing 1 and 25 hundredths times 3 and 14 hundredths. And that I'm going to do over here. So 3 and 14 hundredths times 1 and 25 hundredths. And that's my pi. 5 times 4 is 20. 5 times 1 is 5. That's 7. 5 times 3 is 15. Put a zero placeholder. Two times four is eight. Two times one is two. Two times three is six. Put two zero placeholders. 1 times 4 is 4, 1 times 1 is 1, 1 times 3 is 3. Add that together. That's 0, 7 plus 8 is 15, carry the 1, 5, 9, 10, 12, carry the 1, 6, 7, 8, 9, 3. And then we have four numbers behind the decimal. One, two, three, four numbers behind the decimal. So, and then we could round this to three and ninety-three hundredths would be the area of just the shaded region. The figure shows an irregular quadrilateral and the length of its individual sides. Which of the following equations best represents the perimeter of the quadrilateral? The perimeter is the distance around a figure. To find the perimeter of any shape, you can simply add all of the sides. So the perimeter is m plus 3 plus m plus 2 plus m plus 2m. And now we just need to combine like terms. So the perimeter is... m plus m plus m plus 2m, so that's 1, 2, 3m's plus 2m's is a total of 5m's, plus 3 plus 2 is 5. So the perimeter is 5m plus 5. Answer D David bought 200 shares of Oracle stock yesterday and sold it today. His profit was $22. At what price did he buy the stock yesterday? We have this table of information. to use. So we're only focusing on Oracle stock, which presently is being sold for $19.11 per share. It tells us that he bought 200 shares. and made a profit of $22. So the first thing we want to do is find out his profit per share, which means we need to divide his profit by his number of shares. 200 does not go into 22, so I have to add a decimal and a 0. Now, 200 will divide into 220. 200 goes into 220 one time, and 200 times 1 is 200. We subtract and we get 20. So, we add another 0 and bring it down. 200 goes into 200 one time. 200 times 1 is 200. We subtract and we get 0. Now we know that he made 11 cents per share. If today's price per share is $19.11, then we have to subtract. his profit to find out how much he bought them for the day before. So $19.11 minus 11 cents would just be $19. Answer B. Marjorie buys a package of stocks consisting of 100 shares each of Microsoft and Apple, as well as 200 shares of Garmin at today's closing prices as shown in the table. What is the average price per share that she pays for these stocks? To find this average price per share, first I need to find the total price. Then I have to find the total shares and divide because this is a ratio, price per share, price divided by shares. So first I'm going to figure out how much money she spent on these shares. So she got 100 shares of Microsoft. Let's start with Microsoft. She bought 100 shares for $45.14 each, so times 45, 14. Multiplying by 100 just means you move that decimal two places to the right. It's $4,514 that she spent on her Microsoft shares. She also bought 100 shares of apple, so 100 times an apple is $16.90. Again, multiplying by 100 just means to move that decimal two places to the right. So she spent $1,690 on her apple shares. Then she got 200 shares of Garmin. So, Garmin 200 times, and Garmin is $29.30. So, multiplying by 100 means we move it two places to the right, but multiplying by 200 means not only do we multiply our, or move our decimal two places to the right, but we're also going to double that number. So, if you double 0, you get 0. If you double 3, you get 6. If you double 9, you get 18, and then we carry the 1 over here. 2 times 2 is 4, plus 1 is 5. So, all I did was multiply 200 times $29.30. So, I can use this then to find my total amount of money. I'm just going to move this up a little so I have some more room. So I'm going to add these three costs together to find my total cost or my total price. So 4 plus 0 plus 0 is 4. 10 plus 9 is 10 plus 6 is 16. Then we have 8 plus 6 is 14, plus 5 is 19, plus 1 is 20. 5 plus 1 is 6, plus 4 is 10, plus 2 is 12. That means she spent a total of $12,064 on all of these stocks. That's our price. Now we need to find our shares, and that total can be found right here. Just add these up. 200 plus 100 plus 100, that's a total of 400 shares. The average price per share would be $12,064 divided by 400 shares. All we have to do is divide those two numbers. 400 goes into 1,206 three times. 400 times 3 is 1,200. Subtract and you get 6. Bring down the 4. 400 does not go into 64 at all, so that's a 0, which means we add a decimal and a 0 and bring it down. 400 goes into 640 one time, so bring that decimal up. 1 times 400 is 400. Then you subtract and you get 240. Add another 0, bring it down, and 400 goes into 2,400 six times. 400 times 6 is 2,400. You subtract and you get a remainder of 0, which means the average price per share is $30.16. Pradeep decides to invest $4,500 in Cisco Systems stock and buys it at the price shown in the table. At what price should he sell it to obtain a profit of $10,000? 10%. So he's buying Cisco Systems stock at $3.50 per share. If he wants to make a profit of 10%, then he needs to to sell his stock when the price per share is 10% higher. There are several ways to solve this problem. I'll show you one. First, I want to know what is 10% of $3.50. Well, of means to multiply. So I'm multiplying times 1 tenth, or times 10 hundredths. And multiplying times 10 hundredths means just moving that decimal one place to the left. So that is 35 hundredths, or 35 cents. So that means that his stock needs to be 35 cents higher. The price per share needs to be 35 cents. greater. So $3.50 plus 35 cents is $3.85. So in order to make a profit, a 10% profit, then he needs to sell his stock at $3.85 per share. Morris started work today at 7 a.m. and worked until 4.30 p.m. He earns $12 per hour for his regular shift, which is 8 hours, and 50% more per hour for overtime. How much did Morris make today? The first thing we want to do is find out how long Morris worked. So if he worked from 7 a.m. to 4.30 p.m. Then from 7 a.m. to noon, that's 5 hours. And from noon to 4.30 p.m. is 4.5 hours. which means he worked for a total of 9.5 hours. Now they told us that he makes $12 an hour for just his regular 8 hour shift. So for 8 of these hours, he made $12 an hour. And the rest, the left over, so 1.5 hours, would be his overtime. And for overtime, he makes 50% more per hour. So his regular pay would be 8 hours times $12 an hour, which is $96. But then for overtime, he makes 1 times as much as he made. For his regular shift, so 12 times 50% more, which would be 1 and a half, times, he worked that for 1 and a half hours. So first I'm going to multiply 12 times 1 and a half. 5 times 2 is 10, carry the 1, 5 times 1 is 5, plus 1 is 6. Put a 0 placeholder, 1 times 2 is 2, 1 times 1 is 1. So we add this together and we get 180, but there's one number behind the decimal. So that means that he was making $18 an hour for that 1 hours that he worked overtime. So now we multiply 18 times 1 and 5 times 8 is 40, carry the 4, 5 times 1 is 5 plus 4 is 9, 0 placeholder, 1 times 8 is 8, 1 times 1 is 1. We add those together, 17, carry the 1, 1 number behind the decimal, 1 number behind the decimal. So that means he made $27 in overtime. His total amount that he made that day would be the sum of the amount of money he made for his regular shift and the amount he made for overtime. We need to add $96.27. 7 plus 6 is 13, carry the 1, 9, 10, 11, 12. That means he made $123 that day.

Arrange the following numbers in order from least to greatest. 2 cubed, 4 squared, 6 to the 0 power, 9, and 10 to the first. The first thing we need to do is simplify these numbers.

2 cubed means 2 times itself 3 times. So that's 2 times 2 times 2. 2 times 2 is 4, times 2 is 8. So that is 8. 4 squared is 4 times itself 2 times. 4 times 4 is 16. Now this next one is very interesting because it doesn't matter what this number is.

The base number doesn't matter. Any number raised to the 0 power is always 1. 9 of course is simply 9. And 10 to the first is 10 times itself one time, or not really 10 times itself at all, it's just 10. You're not multiplying 10 by anything else because you just have 10 written one time. And now they want these numbers in order from least to greatest.

So the smallest number would be 1, followed by 8. Then 9, 10, and 16. But of course, those are not in our answer choices, because the numbers are written as they originally were. So now I'm going to go back and write them the way they were written to begin with. 1 was 6 to the 0. 8 was 2 cubed. 9 was always 9, 10 was from 10 to the first, and 16 was 4 squared. And now we just need to find which answer choice has this as the answer.

So of course, we should be starting with 6 to the 0, so that takes it down to either b or d. But then the next should be 2 cubed, which is not the case for b, so our answer is d. The Charleston Recycling Company collects 50,000 tons of recyclable material every month.

The chart shows the kinds of materials that are collected by the company's five trucks. What is the second most common material that is recycled? This is a pie chart, and a pie chart is a visual representation of data.

The data represented in the chart will add up to a total of 100%. 40% plus 25% plus 15% plus 5% plus 15% is a total of 100%. The larger the slice, the larger the percent. The larger the slice and percent means the more common that piece of data is, or the more times that data occurs.

So in this case, they're asking what the second most common common material is. We can see that paper is the most commonly recycled material. It has the largest slice of the pie and thus the largest percentage. So the second most common recycled would be glass at 25%. B.

The Charleston Recycling Company collects 50,000 tons of recyclable material every month. The chart shows the kinds of materials that are collected by the company's five trucks. Approximately how much paper is recycled every month?

This pie chart. is a total of 100 percent. It's a total 100 percent of 50,000 tons.

This 40 percent means 40 percent of the total. Paper is 40 percent, so 40 percent of the total amount of recyclable material collected, or 50,000 tons, 40% of that is paper. And that's what we want to find is how much is that? Well, of means to multiply. And I'm going to multiply 40% and 50,000, which means I need to change 40%.

to a decimal. So move the decimal two places to the left. So I'm actually multiplying 4 tenths or 40 hundredths, it's the same thing, times 50,000. So we have 50,000 times 4 tenths. 4 times 0 is 0, 4 times 0 is 0, 4 times 0 is 0, 4 times 0 is 0, and 4 times 5 is 20. We have one number behind the decimal, so our answer should have one number behind the decimal, and we get 20,000 tons.

So 20,000 tons of the total 50,000 tons of recyclable material is paper. Dorothy is half her sister's age. She will be three-fourths of her sister's age in 20 years.

How many years old is she? We have two unknowns here. We don't know Dorothy's age, and we don't know her sister's age. So I'm going to make up two variables for this. I'm going to use D for Dorothy's.

Current age. And I'll use S for the sister's current age. And I say current because they're talking about their ages now and then also their ages in 20 years.

But what we want to know is how many years old she is now. So we're going to deal with... their current ages. So first we have that Dorothy is, and is in math means equals. It's one of my favorite words.

I love the equal sign. So Dorothy is, Dorothy equals half, her sister's age, so half s. So it's kind of like translating.

You're translating from words into mathematical equations. The next sentence says, she will be, so Dorothy. will be three-fourths of her sister's age in 20 years. So Dorothy, in 20 years, will be three-fourths of her sister's age.

sister's age in 20 years. Remember that D and S were Dorothy and her sister's current ages. So when they talk about in 20 years, that means we have to add 20 to their current age. So Dorothy's age in 20 years, so plus 20, will be or equals 3 fourths of her sister's age. Also, in 20 years.

So we want to solve for Dorothy's age, obviously, and we can do this in lots of different ways. But the first thing I notice is that since I know that D is 1 half S, I can replace D in this equation with 1 half S, and that's called substitution. So that's the method I'm going to use.

I'm going to replace D with 1 half S plus 20 equals 3 fourths times the quantity S plus 20. Well, from here we have lots of options for solving. And the first thing I think about is, really I just want to get rid of this fraction, this 3 fourths. So, I think I'll start by distributing the 3 fourths to what's inside my parentheses.

So, I have 1 half s plus 20 is equal to. 3 fourths s plus, and then with the 3 fourths and the 20, I can put 20 over 1, and I can simplify multiplying these numbers by cross-canceling 4 and 20. I'm simply multiplying two fractions together. so I can use this shortcut. I'm going to divide both 4 and 20 by 4. 4 divided by 4 is 1, and 20 divided by 4 is 5. So, 3 times 5 is 15. 3 fourths of 20 is 15. That makes sense.

Okay, so we need to get all of our variables on the same side. So to do that, I'm going to subtract 1 half s from both sides. 1 half s minus 1 half s is 0. I'm going to bring down 20, and that equals... Now, to subtract fractions, I have to have common denominators.

So 1 half... is also 2 fourths, so I'm going to change 1 half to 2 fourths so that I can subtract these. Then when I subtract, you just subtract the numerators. 3 minus 2 is 1, and your denominators stay the same, 1 fourth s plus 15. Next, I need to subtract 15 from both sides. 20 minus 15 is 5, and 5 is of the sister's age.

So finally, I need to multiply both sides by the multiplicative inverse of which is 4 over 1, or just 4. So multiply both sides by 4. 4 times 5 is 20 and 20 is her sister's age. A fourth of 4 is 1, so we're left with S, the sister's age. Well, remember that Dorothy is half of her sister's age, so since we now know her sister's age is 20, we know that Dorothy is half of that and half of 20 is 10. So that means Dorothy is 10 years old. You could have solved this also by just kind of playing with the numbers, playing with the answers, and seeing which ones would work with the information you were given, but I always love to solve a good algebra problem.

Chan receives a bonus from his job. He pays 30% in taxes, gives 30% to charity, and uses another 25% to pay off an old debt. He has $600 remaining from his bonus.

That was the total amount of Chan's bonus. First I want to start with totaling the percentage that he spent or used, the percentage that is not left. So he pays 30% in taxes, gives 30% to change. charity, and uses the 25% to pay off an old debt.

That's a total of 85% that he has used, which means he had 15% left. and the $600 is the 15% of his total. $600 is 15% of the total amount of his bonus. $600 is as equals... Change the percent to a decimal.

Of means to multiply. And we don't know the total, so use a variable, like t. And now we just need to solve for t.

So divide both sides by 15 hundredths. So, 600 divided by 15 hundredths. We need to move the decimal two places to the right.

So, we do the same to 600, add two zeros, and 15 goes into 60 four times. 15 times 4 is 60. You subtract and get 0. When you bring down 0, 15 goes into 0, 0 times, and that's just going to keep happening two more times until we get 4,000. So, what was the total amount of Chan's bonus? $4,000.

A tire on a car rotates 500 rpm or revolutions per minute when the car is traveling at 50 kilometers per hour. What is the circumference of the tire in meters? Well, the first thing I want to do is take this 50 kilometers per hour. and convert it into meters, since that's what they want the circumference in.

So to convert kilometers to meters, I multiply it by the conversion factor of 1000 meters is 1 kilometer. And when I multiply these fractions, I can cross cancel these units kilometers, and I'll be left with meters per hour. And I'll go ahead and do that first. So our kilometers cross cancel, 50 times 1000 is 50,000 meters per 1 hour. So every hour this car goes 50,000 meters.

and we know that it goes 500 revolutions every minute. So then I want to convert my 50,000 meters per hour into meters per minute so I can more easily compare it to my revolutions per minute. So I'll multiply it by the conversion factor of 1 hour is 60 minutes.

So again, my hours cross cancel since I'm multiplying fractions and I get 50,000 meters every 60 minutes. What I want to find though is what one revolution is. How many meters is one revolution? Because one revolution is one time all the way around the tire. That would be the circumference, which is what they're asking me for.

I'm going to take this 50,000 meters per 60 minutes and multiply it times my 500 revolutions per minute. Every 1 minute, the car goes 500 revolutions. so that again my minutes, these units, minutes cross cancel.

I can cross cancel my 50,000 and my 500 by dividing them both by 500. 500 divided by 500 is 1, and 50,000 divided by 500 is 100. So then 100 meters times 1 is 100 meters divided by 60 revolutions. So now I found that every 100 meters, this car, it's going 60 revolutions every 100 meters, which I can then simplify by canceling out the zeros and I get 10 sixths meters per revolution. So it goes 10 sixths of a meter every revolution. And again, a revolution is one time all the way around the wheel, which is the circumference. So our circumference is 10 sixths.

A combination lock uses a three digit code. Each digit can be any of the ten available integers 0 through 9. How many different combinations are possible? In this probability problem, there are three independent events. Independent meaning they don't affect each other. So to find the possible outcomes, we would have to find the product of the possible outcomes for these three independent events.

So, the possible outcomes would be first the possible outcomes for the first digit, 10, since there are 10 available numbers, times the possible outcomes for the second digit, which again is 10, 10 integers, times the possible outcomes for the third digit, which is which is again 10. So the probability is 10, or the possible outcomes is 10 times 10, which is 100, times 10, which is 1000 possible outcomes. This is the quicker way to do it. Of course, you can always see by actually writing the combinations.

0 0 0 would be the first, then 0 0 1, 0 0 2, 0 0 3. 0, 0, 4, 5, 6, etc., all the way up to 9, 9, 9, which makes it seem like there are 999 possible outcomes. But really, we have to count this first one, the 0, 0, 0, which gives us the total of 1,000 possible outcomes. Which of the following expressions is equivalent to the quantity a plus b times the quantity a minus b?

Well, this is actually a rule that you would memorize, but you don't have to memorize it. You can always just take these binomials and multiply them together using FOIL. The F stands for first, meaning to multiply the first terms in each set of parentheses. So A times A. And A times A is A squared.

Then, O stands for outer or outside, so you multiply the two terms on the outsides, or A and negative B. So, A times negative B is negative A. I stands for inner or inside, so you multiply the two terms on the inside, B and A.

B times A is BA or A, so plus A. And finally, the L stands for last, so you multiply the two terms that are in the last positions of each set of parentheses. Positive B times negative B, which is negative B squared. Then to simplify, we combine like terms. Negative a b and positive a b are like terms, and they cancel each other out because they're additive inverses of each other, or opposites.

So we're left with a squared minus b squared, or answer a. Which of the following expressions represents the ratio of the area of a circle to its circumference? We've got some pretty key words here.

The first one I see is ratio. A ratio is a comparison of two numbers, like a fraction. And what we're comparing is the area of a circle to its circumference. Well, the area of a circle is pi times radius squared, while the circumference of a circle is 2 times pi times radius.

So, the area is going to be our numerator. Since it was listed first, area, 2 circumference means that the area should be in the numerator, divided by this 2 as your division bar, the circumference, 2 times pi times radius. And then we need to simplify.

First I see that pi divided by pi cancels. Pi divided by pi is 1. Anything divided by itself is 1. So we can cancel it. And then I can simplify r squared divided by r.

r. When you divide, you subtract the exponents. Or another way to think about it is r squared is r times r. And I'm dividing that by r. So again, you have something divided by itself.

itself is 1 or it cancels. So I'm left with just r in the numerator divided by 2 in the denominator since those r's cancel. So we just have r divided by 2, which is answer e. Kyle bats third in the batting order for the Badgers baseball team. The table shows the number of hits that Kyle had in each of seven consecutive games played during one week in July.

What is the mode of the numbers shown in the table? Mode means the number that occurs the most. So out of these numbers, 1 occurs 1, 2, 3 times, 2 occurs twice, 3 occurs once, and 4 occurs once.

So the number that occurs the most, or the mode, is the number 1. Kyle bats third in the batting order for the Badgers baseball team. The table shows the number of hits that Kyle had in each of seven consecutive games played during one week in July. What is the mean? of the numbers in the distribution shown in the table?

Well, the mean of a set of numbers is the average. And to find the average or the mean, you must add all of the numbers. of the numbers together and then divide them by the number of numbers there are. So first we'll add all the numbers together. 1 plus 2 plus 3 plus 1 plus 1 plus 4 plus 2. Then we divide by the number of numbers there are, which is 1, 2, 3, 4, 5, 6, 7. So we do this.

we divide by 7. And again, this is how we find the mean. add the numbers together, and then divide by the number of numbers there are. So that is 1 plus 2 is 3, 3 plus 3 is 6, plus 1 is 7, plus 1 is 8, plus 4 is 12, plus 2 is 14. So 14 divided by 7. And 14 divided by 7 is 2. Therefore, the mean or the average... The average of these numbers is 2. Jamie had $6.50 in his wallet when he left home. He spent $4.25 on drinks and $2 on a magazine.

Later his friend repaid him $2.50 that he had borrowed the previous day. How much money does Jamie have in his wallet? The first thing I'd like to do is find the total amount of money that Jamie spent. I see that he spent $4.25 and $2. So to find the total amount of money he spent, I need to add $4.25 plus $2.

And it's very important that whenever you add numbers with decimals or or subtract numbers with decimals that you line up the decimals. So now I'm ready to add, and I can just bring my decimal down before I even start. 5 plus 0 is 5, 2 plus 0 is 2, 4 plus 2 is 6. So now I know he spent a total of $6.25. And if he spent that much money, then that's money that is taken away from what he spent.

what he had. So he started with $6.50, but he spent $6.25. So I have to subtract what he spent from what he had before. And again, we need to line up our decimals.

I can't take 5 from 0, so I'm going to borrow from this 5. It becomes a 4. And now I have 10 minus 5 is 5. And 4 minus 2 is 2. I can bring down the decimals. down that decimal, 6 minus 6 is 0. So that means after he spent the $4.25 on the drinks and $2 on the magazine, he only had 25 cents left. But his friend repaid him $2.50, which means his friend gave him $2.50. So that's money that's added to his total.

So now I'm going to take what he had after he spent his money and add the money his friend gave him. And again, we line up our decimals. 5 plus 0 is 5, 2 plus 5 is 7, bring down the decimal, 0 plus 2 is 2. So after spending money and getting repaid money, he now has a total of $2.75, which is the answer.

D A sailboat is 19 meters long. What is its length in inches? There are many ways to approach this problem and solve it. The first thing I'm going to do is convert my meters into centimeters because then it's a quick conversion into inches from there. So I know that one meter is equal to 100 centimeters.

We can use a proportion to solve. We can take this conversion factor and use it as a ratio. 1 meter is 100 centimeters, which equals 19 meters is how many centimeters.

Then to solve the proportion, we cross multiply. 1 times x is x, and 100 times 19 is 1900. So it's 1900 centimeters. Now, I can use another proportion to solve for or to find inches. I know that 2 and 54 hundredths centimeters is 1 inch.

and I have 1900 centimeters and I need to know how many inches that is. And again, to solve a proportion, we cross multiply. So, 2 and 54 hundredths... centimeters times X, 2 and 54 hundredths X is equal to 1 times 1900, which is 1900. And I forgot to put my little line up here, so I'll do that.

And now we need to solve for x, which means dividing both sides by 2 and 54 hundredths. 2 and 54 hundredths divided by 2 and 54 hundredths is 1, times x is x, and that equals 1900 divided by 2 and 54 hundredths, so we need to do some division. which means I have to move this decimal two places to the right.

So, I have to take my decimal in 1900 and move it two places to the right, which means adding two zeros. So, now my number is a hundred... 90,000 being divided by 254. So 254 does not go into 1, or into 19, or into 190, but it does go into 1, or into 19, or into 190. go into 1900. We can use compatible numbers here to get a very quick estimate of how many times it goes into 1900. So 254 is very close to 250. So 250, 500, 750, 1000, 1250, 1500, 1750. 1,800. Sorry, 1,750 and then 2,000.

It only goes seven times because 2,000 is too much. 2,540 only goes seven times into 1,900. We can see what that is right here. 7 times 4 is 28. 7 times 5 is 35. 36, 37. 7 times 2 is 14. 3 is 17, so it's 1778. Then we subtract those, and we have to borrow from the 9, and that becomes an 8. This is 10, and we borrow from the 10, it becomes a 9. This is 10, so 10 minus 8 is 2, 9 minus 7 is 2, and 8 minus 7 is 1, 122. Now we bring down our next 0, Again, we can use that 250 rule to see how many times it goes into 1,220.

So 250, 500, 750, 1,000, and then it would be 1,250, but 1,250 is too much, so it only goes four times. So let's see, 254 times 4, 4 times 4 is 16, carry the 1, 4 times 5 is 20. plus 1 is 21. 4 times 2 is 8, plus 2 is 10. So that's 1,016. And then we subtract. So we need to borrow from this 2, that becomes a 1. So this is 10 minus 6 is 4. 1 minus 1 is 0. And 2 minus 0 is 2. So that's 204. And then we bring down our 0 here.

So again, how many times does 250 go into 2040? So we have 250, 500, 750, 1000, so 4 more times for 2000, so it looks like a total of 8 times. Let's check that.

Let's see where I have some room. We'll just do it up here. 254 times 8. 8 times 4 is 32. 8 times 5 is 40, plus 3. 43. 8 times 2 is 16. 16 plus 4 is 20. So it's 8 times is correct, and it's 2,032. We subtract and we get 8. And that would be our remainder.

It would be 8 out of 254. Or we could keep going to find our decimal. But if we look at our answers, we don't need to keep going because none of our answers have the decimal as part of it. So our answer is D, 748 inches. Mrs. Patterson's classroom has 16 empty chairs. All the chairs are occupied when every student is present.

If two-fifths of the students are absent, how many students make up her entire class? This is a great problem to use a proportion to solve. We can take this fraction, two-fifths, and use that as our first ratio, two being the amount of students who are absent out of the five total. That equals, again, number of students who are absent divided by the total. It says in the problem that there are 16 empty chairs, which means there are 16 students missing since usually all the chairs are occupied when every student is present.

Every empty chair is a student who's absent. So we have 16 absent students out of... And this is what we don't know. What's the total? How many students make up her entire class?

And to solve a proportion, we simply need to cross multiply. 2 times x is 2x, and that equals 16 times 5, which is 80. And then we solve for x by dividing both sides by 2. And x equals 5. forty. So that means there are forty students in her entire class.

Rachel spent twenty-four dollars and fifteen cents on vegetables. She bought two pounds of onions, three pounds of carrots, and one and a half pounds of mushrooms. If the onions cost three dollars and sixty-nine cents per pound, she would have to pay the cost of the onions.

The carrots cost $4.29 per pound. What is the price per pound of mushrooms? First, I want to figure out how much money was spent on the onions that were $3.69 per pound and how much was spent on the carrots that were $4.29 per pound. Let's start with the onions.

Those onions cost $3.69 for every pound that's purchased, and she purchased 2 pounds. So we need to multiply 369 times 2 to find the total amount of money spent on onions. So 2 times 9 is 18, carry the 1. 2 times 6 is 12, plus 1 is 13, write the 3, carry the 1. 2 times 3 is 6, plus 1 is 7. Two numbers behind the decimal, two numbers behind the decimal.

So that means she spent $7.38 out of her total on onions. We're going to do something similar to find the total amount of money spent on carrots. So next we'll deal with our carrots. The carrots were $4.29 per pound, so $4.29 times 3, since she bought 3 pounds of carrots.

3 times 9 is 27, write the 7, carry the 2. 3 times 2 is 6, plus 2 is 8, and 3 times 4 is 12. Two numbers behind the decimal. two numbers behind the decimal. So she spent a total of $12.87 on carrots. And so now I'm going to find the total spent on onions and carrots.

So that means I'm just going to add these two values together. So $12.87 plus $7.38. Make sure that when you add or subtract decimals that you line those decimals up. So we can go ahead and bring that down right now.

7 plus 8 is 15. 8, 9, 10, 11, 12, 7, 9, 10. That means she spent $20.25 on the onions and carrots. Knowing that will help us find out what amount was left to be spent on the mushrooms. Since I know her total spent on all of the vegetables is $24.15, I can take that total and subtract $20.25. which was spent on onions and carrots. What I'll have is the amount of money she spent on the mushrooms.

Again, we line up our decimals. 5 minus 5 is 0. I can't take 2 from 1, so I borrow from the 4. It becomes a 3. Then I have 11 minus 2, which is 9. Bring that decimal down. 3 minus 0 is 3. She had $3.90 to spend on mushrooms.

mushrooms, but what we need to know is what the price per pound was, which we can find using this information. She bought one and a half pounds of mushrooms. We need to take the total amount spent on mushrooms and divide that by how many pounds of mushrooms she purchased, which was one and a half or one and five tenths.

Now we need to move the decimal one place in both places. both of our numbers, so that's where it is now. 15 goes into 39, 2 times.

15 times 2 is 30. Subtract and you get 9. Bring down the 0. 15 goes into 90, 6 times. 15 times 6 is 90. Subtract and you get 0. So no remainder, but we can add a 0. zero here since we're talking about money. That means that the mushrooms were $2.60 per pound, which is answer A. In the figure, A, B, and C are points on the number line, where O is the origin. What is the ratio of the distance BC to the distance A?

First, we need to find the distance between points B and C. The distance between two points is the absolute value of the difference of the coordinates. That is the absolute value of B is 5 minus C is 8. which is the absolute value of negative 3, and that is 3. So the distance between points B and C is 3. Now, we're going to do the same thing to find the distance between points A and B.

Again, it's the absolute value of the difference of the coordinates of point A and point B. That's the absolute value of A is negative 6 minus B is 5, which is the absolute value of negative 6 minus 5 is negative 11, and that is 11. So the distance from point A to point B is 11. Now they've asked us what the ratio of those distances is, and it gives you the order that it wants the ratio written in. So the distance BC should be the first number in the ratio, 3, 2, and we use a colon or you can write 2, it's fine too, the distance from points A to B, which is... 11. So our ratio is 3 to 11. We'll answer D Jesse invests $7,000 in a certificate of deposit that pays interest at the rate of 7.5% annually. How much interest in dollars does Jesse gain from this investment during the first year that he holds the certificate?

This is a great problem to use this formula on. And some people call it IPERT, just as a way to remember it. The I stands for the interest, the amount of interest in dollars, like what we're trying to find. P stands for principal.

or the amount that's being invested initially. That would be our $7,000. The R is the rate or the percentage, but written as a decimal. So we're going to take 7 and... and write it as a decimal.

That will be our rate. Then T stands for time. You may want to make a note of that. This I is the interest, P is the principle, R is the rate as a decimal, and the T is the time.

So we're going to take our information from our problem and plug it in. So our amount of interest is equal to the principal, $7,000. times the rate as a decimal again, so we're taking 7.5% and changing it to a decimal, which means taking the decimal and moving it two places to the left. So, you add that 0, so it's 75 thousandths and then times the year or the time and we're just trying to find out about the first year. So that means it's been in there for one year and then we just need to multiply.

those things together. So 1 times anything is just that, so I can really kind of ignore that. So I just need to multiply these two numbers together.

So I have this 75 thousandths and I'm going to... to put it on top because it's very easy to multiply times 7,000 since the first thing I'm going to do is just write down those three zeros. The only number I really have to multiply by is the 7. So 7 times 5 is 35, write the 5, carry the 3. 7 times 7 is 49, plus 3 is 52. And then we have three numbers behind the decimal.

so three numbers behind the decimal, and that means that he's going to make $525 off of his investment. So that's the number you would need to bubble in, is 525. In an election in Kimball County, candidate A obtained 36,800 votes. His opponent, candidate B, obtained 32,100 votes.

2,100 votes went to write-in candidates. What percentage of the votes went to candidate A? To find a percent, we need the part over the whole.

We know the part. The part is the number of votes candidate A received, which is 36,800. What I need to know is the whole, or the total, number of votes. To find the total number of votes, we need to add up 36,800. 32,100 and 2,100 so that we can find the total.

We're adding. Those are 0 and again we get 0. Then we have 8, 9, 10, so that's a 0. Carry the 1. We have 6, 7, 8, 9, 10, 11. Carry the 1. We have 3, 6, 7, 8, 9, 10, 11. So that means there were a total of 71,000 votes. Now to find our percent, we need to divide.

That'll give us our decimal, which will then change into a percent. But before I divide, I'm going to first simplify this fraction by canceling two zeros. And now my division will be a lot easier. So we need to divide 368 by 710. Well, 710 doesn't go into 368. at all, which means I'm going to have to add a decimal and a zero.

Now, I can divide 710 into 3680. So, I can use compatible numbers here, and 710 is very close to 700. 700 times 5 would be 3500, so I'm thinking 710 is going to go about 5 times into 3,680. So let me multiply this, we get 0, and that's 5, and that's 35. So 3,550. Yep, looks good. So that's 5, we have 3,550. Now we need to subtract, and that's 0, 3, 1, so 130. Then we need to add a 0 and bring it down, 1300. So, 710 and it's only going to go into 1300 one time because 700 plus 700 would be 1400 and that's too much.

So, it's just one time. 710 times 1 is 710. Then we need to subtract. So, I'm going to borrow from the 1, that's a 0. That's 13, borrow from the 13, that's 12, which makes that 10. We'll borrow, well we don't need to borrow that 0, so 10 minus 1 is 9, and 12 minus 7 is 5, 590. Which makes sense, if you add 90 back to 710, you get 800. If you add 500 to 800, you get the 1300. We need to add another 0 and bring it down.

So now we need to know how many times 710 goes into 5,900. So I'm again going to use that compatible number of 700. So let's see, 700 times 8 would be 5,600. So I'm thinking it goes in there about 8 times. So that's 0 and that's 8 and that's 56. So yes, that's very close, so that's 8. And we have 5680. We can subtract, and that's 0. Borrow from the 9, that's an 8, so that's 10. 10 minus 8 is 2. 8 minus 6 is 2, so we get 220. And we could keep going, but really, that's all I need to know, because as I look at my answers, it's very clear which one of these is my answer. This is, well, it's about 518 thousandths, and then when we change it into our percent, we move our decimal two places to the right, we get 51 and 8 tenths percent, which is answer A.

Francine can ride 16 miles on her bicycle in 45 minutes. At this speed, how many minutes would it take Francine to ride 60 miles? This problem is a good candidate for using a proportion to solve. There's more than one way to solve it, but I like proportions, so I'm going to use a proportion.

First, I know that Francine can go 16 miles in 45 minutes. That's going to be my first ratio. So I have 16 miles in 45 minutes.

And the key to setting up a successful proportion is to be consistent. So since our first ratio we have miles to minutes, then our second ratio will also be miles to minutes. So they want to know how many minutes it would take Francine to ride 60 miles.

So my miles is 60, and my minutes is what I'm trying to find, so that's where I'll put a variable. To solve a proportion, we cross multiply. 16 times X is 16X, and that equals 60 times 45, 0. 6 times 5 is 30, carry the 3. 6 times 4 is 24, plus 3 is 27, so 2,700.

And then we solve for x by dividing both sides by 16. So x equals, and now I need to find out what 2,700 divided by 16 is. 16 goes into 2,700 one time, and that's 16, and we subtract and get 11, and we bring down our 0. We could use 15 as a compatible number to see how many times 15 would go into 110. So we have 15, 30, 45, 60, 75, 90, 105. So 7 times. So let's see if 16 will go into 110 7 times. 7 times 6 is 42. 7 times 1 is 7, plus 4 is 11. And that's a little too big, which means I know 16 goes into 110 six times. So 16 times 6, 6 times 6 is 36, 6 times 1 is 6, plus 3 is 9. And that works. So we need to subtract, and we get 14. Then we bring down our next 0, so we have 140. So I know 16 is going to go into 140 at least 7 times, but it should go even more than that.

So let's see what 16 times 8 is. 8 times 6 is 48, 8 times 1 is 8, plus 4 is 12, and that's all we're going to be able to do, so 8 times and we get 128. When we subtract those numbers, we get 12. So that means we need to add a decimal and a zero and bring it down. 16 goes into 120 seven times. 16 times 7, we found earlier, is 112. We subtract and we get 8. We add another 0 and bring it down. 16 goes into 80 less than 6 times, and actually exactly 5 times. 16 times 5 is 80. We subtract and we get a remainder of 0. That means if it took her 45 minutes to go 16 miles, It should take her 168 and 75 hundredths minutes to go 60 miles.

And this is the answer you would grid in. Figure 9 shows two quarter circles centered on the origin of the Cartesian coordinate plane. The inner circle has a radius of 2 units. The outer circle has a radius of 3 units. What is the area of the shaded region?

So that's right in here. So if you can just imagine it, this circle would continue going. as with this larger circle.

So, since it's been split into quarters, all we're finding the area of is a quarter of those circles. So, first let's start with the formula for the area of a circle. is. It's pi times radius squared. As we discussed though, this is only a quarter of these circles, so our area is a quarter of the area of these circles.

And then we need to find the area of just the part between those two circles. So that would be the area of the large circle minus the area of the small circle. minus the area of the small circle. The area of the large circle would be pi times the radius, which is 3 squared, minus the area of the small circle, which would be pi times 2 squared.

The 3 and the 2 came from the radius. The radius of the small circle is 2, and the radius of the outer circle has a radius of 5. 3. So that's where these numbers came from. And now let's simplify that.

So we get area is 1 4th of 3 squared is 9, so 9 pi minus 2 squared is 4, so 4 pi. Now since each one of these terms has a pi in it, we can factor the pi out. So we get area equals 1 4th pi times the quantity 9. 9 minus 4. Then we can simplify 9 minus 4 so we have area equals a fourth pi times 5. And a fourth times 5 is 5 fourths, so that's 5 fourths pi.

But of course, if you're needing to bubble this in on an answer sheet, then you don't have the option of bubbling pi. So, we're going to have to multiply 5 fourths times pi. And 5 fourths times pi is 1 and 25 hundredths pi.

And for pi, we'll just use 3 and 14 hundredths. So we're doing 1 and 25 hundredths times 3 and 14 hundredths. And that I'm going to do over here.

So 3 and 14 hundredths times 1 and 25 hundredths. And that's my pi. 5 times 4 is 20. 5 times 1 is 5. That's 7. 5 times 3 is 15. Put a zero placeholder. Two times four is eight. Two times one is two.

Two times three is six. Put two zero placeholders. 1 times 4 is 4, 1 times 1 is 1, 1 times 3 is 3. Add that together.

That's 0, 7 plus 8 is 15, carry the 1, 5, 9, 10, 12, carry the 1, 6, 7, 8, 9, 3. And then we have four numbers behind the decimal. One, two, three, four numbers behind the decimal. So, and then we could round this to three and ninety-three hundredths would be the area of just the shaded region.

The figure shows an irregular quadrilateral and the length of its individual sides. Which of the following equations best represents the perimeter of the quadrilateral? The perimeter is the distance around a figure. To find the perimeter of any shape, you can simply add all of the sides. So the perimeter is m plus 3 plus m plus 2 plus m plus 2m.

And now we just need to combine like terms. So the perimeter is... m plus m plus m plus 2m, so that's 1, 2, 3m's plus 2m's is a total of 5m's, plus 3 plus 2 is 5. So the perimeter is 5m plus 5. Answer D David bought 200 shares of Oracle stock yesterday and sold it today.

His profit was $22. At what price did he buy the stock yesterday? We have this table of information. to use. So we're only focusing on Oracle stock, which presently is being sold for $19.11 per share.

It tells us that he bought 200 shares. and made a profit of $22. So the first thing we want to do is find out his profit per share, which means we need to divide his profit by his number of shares. 200 does not go into 22, so I have to add a decimal and a 0. Now, 200 will divide into 220. 200 goes into 220 one time, and 200 times 1 is 200. We subtract and we get 20. So, we add another 0 and bring it down.

200 goes into 200 one time. 200 times 1 is 200. We subtract and we get 0. Now we know that he made 11 cents per share. If today's price per share is $19.11, then we have to subtract.

his profit to find out how much he bought them for the day before. So $19.11 minus 11 cents would just be $19. Answer B. Marjorie buys a package of stocks consisting of 100 shares each of Microsoft and Apple, as well as 200 shares of Garmin at today's closing prices as shown in the table. What is the average price per share that she pays for these stocks?

To find this average price per share, first I need to find the total price. Then I have to find the total shares and divide because this is a ratio, price per share, price divided by shares. So first I'm going to figure out how much money she spent on these shares. So she got 100 shares of Microsoft.

Let's start with Microsoft. She bought 100 shares for $45.14 each, so times 45, 14. Multiplying by 100 just means you move that decimal two places to the right. It's $4,514 that she spent on her Microsoft shares. She also bought 100 shares of apple, so 100 times an apple is $16.90. Again, multiplying by 100 just means to move that decimal two places to the right.

So she spent $1,690 on her apple shares. Then she got 200 shares of Garmin. So, Garmin 200 times, and Garmin is $29.30. So, multiplying by 100 means we move it two places to the right, but multiplying by 200 means not only do we multiply our, or move our decimal two places to the right, but we're also going to double that number. So, if you double 0, you get 0. If you double 3, you get 6. If you double 9, you get 18, and then we carry the 1 over here.

2 times 2 is 4, plus 1 is 5. So, all I did was multiply 200 times $29.30. So, I can use this then to find my total amount of money. I'm just going to move this up a little so I have some more room.

So I'm going to add these three costs together to find my total cost or my total price. So 4 plus 0 plus 0 is 4. 10 plus 9 is 10 plus 6 is 16. Then we have 8 plus 6 is 14, plus 5 is 19, plus 1 is 20. 5 plus 1 is 6, plus 4 is 10, plus 2 is 12. That means she spent a total of $12,064 on all of these stocks. That's our price.

Now we need to find our shares, and that total can be found right here. Just add these up. 200 plus 100 plus 100, that's a total of 400 shares.

The average price per share would be $12,064 divided by 400 shares. All we have to do is divide those two numbers. 400 goes into 1,206 three times. 400 times 3 is 1,200.

Subtract and you get 6. Bring down the 4. 400 does not go into 64 at all, so that's a 0, which means we add a decimal and a 0 and bring it down. 400 goes into 640 one time, so bring that decimal up. 1 times 400 is 400. Then you subtract and you get 240. Add another 0, bring it down, and 400 goes into 2,400 six times.

400 times 6 is 2,400. You subtract and you get a remainder of 0, which means the average price per share is $30.16. Pradeep decides to invest $4,500 in Cisco Systems stock and buys it at the price shown in the table. At what price should he sell it to obtain a profit of $10,000?

10%. So he's buying Cisco Systems stock at $3.50 per share. If he wants to make a profit of 10%, then he needs to to sell his stock when the price per share is 10% higher.

There are several ways to solve this problem. I'll show you one. First, I want to know what is 10% of $3.50.

Well, of means to multiply. So I'm multiplying times 1 tenth, or times 10 hundredths. And multiplying times 10 hundredths means just moving that decimal one place to the left. So that is 35 hundredths, or 35 cents.

So that means that his stock needs to be 35 cents higher. The price per share needs to be 35 cents. greater.

So $3.50 plus 35 cents is $3.85. So in order to make a profit, a 10% profit, then he needs to sell his stock at $3.85 per share. Morris started work today at 7 a.m. and worked until 4.30 p.m. He earns $12 per hour for his regular shift, which is 8 hours, and 50% more per hour for overtime.

How much did Morris make today? The first thing we want to do is find out how long Morris worked. So if he worked from 7 a.m. to 4.30 p.m.

Then from 7 a.m. to noon, that's 5 hours. And from noon to 4.30 p.m. is 4.5 hours. which means he worked for a total of 9.5 hours.

Now they told us that he makes $12 an hour for just his regular 8 hour shift. So for 8 of these hours, he made $12 an hour. And the rest, the left over, so 1.5 hours, would be his overtime. And for overtime, he makes 50% more per hour. So his regular pay would be 8 hours times $12 an hour, which is $96.

But then for overtime, he makes 1 times as much as he made. For his regular shift, so 12 times 50% more, which would be 1 and a half, times, he worked that for 1 and a half hours. So first I'm going to multiply 12 times 1 and a half.

5 times 2 is 10, carry the 1, 5 times 1 is 5, plus 1 is 6. Put a 0 placeholder, 1 times 2 is 2, 1 times 1 is 1. So we add this together and we get 180, but there's one number behind the decimal. So that means that he was making $18 an hour for that 1 hours that he worked overtime. So now we multiply 18 times 1 and 5 times 8 is 40, carry the 4, 5 times 1 is 5 plus 4 is 9, 0 placeholder, 1 times 8 is 8, 1 times 1 is 1. We add those together, 17, carry the 1, 1 number behind the decimal, 1 number behind the decimal. So that means he made $27 in overtime. His total amount that he made that day would be the sum of the amount of money he made for his regular shift and the amount he made for overtime.

We need to add $96.27. 7 plus 6 is 13, carry the 1, 9, 10, 11, 12. That means he made $123 that day.

Transcript for:Math Lecture Notes

Transcript for:
Math Lecture Notes