Okay, so we're going to see some applications of first-degree equations. So this is a finance-related question. A financial manager has $15,000 to invest for her company. She plans to invest part of the money in tax-free bonds at 5% annual interest and the remainder in taxable bonds at 8% annual interest. She wants to earn $1,020 per year in interest from the investments.
find out how much she should invest in each bond in order to earn this amount of interest per year. Okay, so the total amount of money she has is $15,000. So let's write down all the information we have.
So this is the total amount. Out of which she's going to invest X amount at 5%. and the remainder which is 15,000 minus x at 8%.
Now what is the interest you're going to get? If you invest x dollars then your interest will be 0.05x because it's 5%. That's 0.05.
What is the amount of interest you gain on the other part? It's 0.08x and we know that, oh sorry, 0.08, sorry not times x, but times 15,000 minus x. And we know that the sum of this total interest that she wants is going to be this $1,020. So 0.05x plus 0.08 times 15,000. minus x must equal 1, 0, 2, 0. Okay?
And essentially then we simply solve for x. Alright? So for example in this case, 0.05x minus 0.08x is equal to, Well, let's do it stage by stage here. So this is what we got for the excess parts from the distributing here.
There's also plus 0.08 times 15,000 is equal to 1,020. So we want to isolate x. So here we get negative 0.03x is equal to 1,020 minus 0.08 times 15,000.
Okay, so we're going to solve for x. x is going to be 1, 0, 2, 0, minus. And this is 8, 100. So we can write this as 8 times 15,000 divided by 100. over negative 0.03. So this is 1, 0, 2, 0 minus, since two zeros go away here, We're left with 8 times 150. So that's 8 times 5 is 40. Carry the 4. 8 times 1 is 8, plus 4 is 12. So minus 1,200 divided by negative.03, or simply negative 180 divided by negative.03. We're doing this out just to remind us with a little bit of arithmetic here.
So negative $180, this is divided by negative. divided by 100, which is negative 180 times 100 over negative 3. We can cancel the negative 3 and the negative 18 here. This is negative 60. So we get negative times negative is a positive 6,000.
So x is $6,000. So $6,000 needs to go. at 5%, and the remaining $15,000 minus X, $15,000 minus $6,000, that's $9,000. So this is how much you should put in the 5% bond and $9,000 in the 8% bond.
Okay, so that is an example of a financial application. Another application is, and let's just erase this. We're going to redraw this picture in just a second here. So this second problem is all about cars.
And one car is behind another car, and it's trying to catch up to the other car. So that's the whole story. And, all right, I think that's good enough.
Got enough space here. Okay, so let's read this question. Two cars are traveling on a straight road, the first at 60 miles an hour and the second at 45 miles an hour.
If the second car is 90 miles ahead of the first, how long will it take for the first car to catch up to the second car? Okay, so let's draw a picture. So here is the road. And we're traveling, say, in this direction. And we're going to draw a black dot for the second car.
Somewhere here, this is the second car. And the first car is somewhere here, this is the first car. The first car is going at 60 miles per hour. And from here to here...
is 90 miles. That's how far the second car, which is going 45 miles per hour, is behind. The first car is behind the second car.
Well, we know that eventually, because this guy is going at 60 miles an hour, this guy is going at 45, the 60 mile an hour car will eventually, at some point, let's make that a third dot here. Oops. Eventually, let's put a dot somewhere right here.
They will meet at some point. the first car will catch up to the second car. The question is, well, how long will it take?
Or you can also ask, how much more distance will they end up traveling? So there's some amount of distance that the second car is going to go to meet by the time the first car meets the second car. So this distance from the second dot to where they meet, we're going to call that D. And the question is, how much time will it take?
Well, we know that whatever the time is, let's label the time as t. The time is the same traveled for both vehicles. So the time elapsed for the first car is the same as the time elapsed for the second car. They are the same. That's the key observation that we have to notice.
Okay, so the time is the same. Well, how much distance is the first car going to travel? The first car will travel...
90 plus D, the second car will travel just D, right? The distance of D. All right.
Well, if we're traveling at 45 miles an hour, and we go time T hours, we're going to reach D. We're going to travel at D. And at the same time, at 60 miles an hour, If we multiply by time, we're going to get a distance, but we're not going to get d. We're going to get 90 plus d, right? So we have two equations now.
There's 45t is equal to d, because in the time elapsed, the car going 45 miles an hour will travel d. In the time elapsed t, the second time elapsed t is equal to d. car which is traveling 60 miles an hour if you multiply by time you'll get 90 plus D so what we do now is we substitute D is equal to 45 T and to here so what we get is 60 T is 90 plus 45t we subtract 45t from both sides we get 15t is equal to 90 and t is in hours here so t is 90 over 15 and what is that 6. So t is 6 hours.
Because time is measured in hours. The unit of time is hours. So t is 6 hours. So let's check.
Does this make sense? How much, how for the first car? How far is that one going to travel?
45 times 6. 6 times that's 30. So 270 miles is how far. This is D. What about 60 times T?
times six this is 360 miles is that 90 plus 270 yep that is 90 plus 270 so so it turns out that this d right here is 270 miles so the first car travels a grand total of 360 miles the second car travels a total of 270 miles they meet after six hours you Alright, so those are two applications. There are more, but these are the two we're going to talk about, and we're going to move to a different topic in the next video.