now let's work on problems involving ratios and proportions let's start with this one a certain island contains 540 cats and 675 dogs what is the ratio of cats to dogs on this island one way in which we could do so is by setting up a fraction since cats came first we're going to put it on the top of the fraction and dogs came second so we're going to put that on the bottom so there's 540 cats and 675 dogs ratios and fractions they're into convertible you can put a ratio into a fraction or you can convert a fraction back into a ratio right now what we need to do is simplify this fraction now because the last number of 540 and 675 contains either a zero or a five both numbers are divisible by 5. 540 divided by 5 is 108. 675 divided by 5 is 135. now we need to simplify this further if possible so what number goes into 108 and 135 well let's find out it turns out that both of these numbers is divisible by 9. 108 divided by 9 is 12. 135 divided by 9 is 15. now we can reduce it even further twelve is four times three fifteen is five times three and so we could cancel a three so the ratio of cats to dogs is four to five or we can write it like this using a colon so that's it for this problem now let's move on to number two the ratio of boys to girls in the class is eight to seven if there are forty boys how many girls are there in the class so what we're going to do is we're going to put the information relating to the boys on the left side and the information relating to the girls on the right side we need to set up a proportion we need to write two fractions separated by an equal sign now the ratio of boys to girls is eight to seven so that means that if there are eight boys in the class there's going to be seven girls in that same class now if there's 40 boys in the class how many girls or x will be in that same class so this is the formula that we need to solve let's cross multiply so this is going to be 8 times x and that's going to equal 40 times 7. now we can multiply 40 by 7 four times seven is twenty eight at the zero you get two eighty but it's easier if we rewrite forty as eight times five notice that we can cancel an eight so on the left side all we have is an x left over on the right side it's going to be five times seven which is thirty five so if there are 40 boys in the class there's going to be 35 girls in that same class and you could think about it conceptually so let's say we have the ratio 8 to 7 and we want to expand it to 40 to some unknown number to go from 8 to 40 we need to multiply by 5. so therefore to keep the ratio the same we need to multiply 7 by 5 which will give us 35 and that's a quick way to get the answer mentally if you see it that way try this one karen can make 14 cakes in six hours how many cakes can she make in 15 hours so what do you think we need to do to solve this problem well let's set up a proportion there's two things that we're dealing with the number of cakes that she can make and a time in which she can make them so once again let's put two fractions separated by an equal sign so she can make 14 cakes in six hours so how many cakes which means x that's what we're trying to solve for can she make in 15 hours so all of the information associated with time is on the right side and the information associated with cakes is on the left side and since 14 cakes correspond to six hours these two things are on the top of the fraction the other stuff has to be on the bottom so now let's cross multiply this is going to be 6 times x and that's going to equal 14 times 15. now fourteen times fifteen is two hundred and ten so now in order to separate six from x we need to divide them we need to divide both sides by six so it's going to be 210 divided by six which is 35. so in 15 hours she can make 35 cakes so that's the answer now let's see if we can do it the fast way so the ratio of cakes to hours is initially 14 to 6 and we want to know how many cakes or x she can make in 15 hours so we have these two numbers what do we need to multiply six by to get to fifteen six times what number is fifteen if you're not sure divide fifteen by six is 2.5 so you got to multiply 6 by 2.5 to get to 15. likewise to find x starting from 14 you must also multiply that by 2.5 fourteen times two point five will give us the same answer which is thirty five number four the left and width of a small rectangle is 9 inches and 8 inches respectively the length of the large rectangle is 24 inches if the length and the width of the two rectangles have the same ratio what is the width of the large rectangle well let's set up a proportion so we have the large and the small rectangle now let's say the top portion represents the length of the two rectangles and the bottom part of the two fractions represents the width so the left and the width of the small rectangle the length is 9 inches and the width of the small rectangle is 8 inches now for the large rectangle the left is 24. our goal is to calculate the width of the large rectangle so let's put an x now let's cross multiply so this is going to be 9x and that's equal to 8 times 24. so eight times twenty-four actually before we do that let's write nine as three times three and 24 as 8 times 3 because this will help us to give our answer as a fraction so we can get an exact answer as opposed to a decimal value so right now we have 3x on the left 8 times 8 is 64. now let's divide both sides by 3. 3 doesn't go into 64. so we can leave our answer as 64 over three so that is the exact answer of the width of the rectangle but if you want to turn that into a decimal this is about 21.3 or you could say 21.3 repeating and the units inches so that's it for this problem number five the ratio of nickels dimes and quarters in a jar is three to four to seven respectively if there are a total of 112 coins in the jar how many of them are nickels so this is a multi-step problem feel free to pause the video and try so we can set up many fractions to get the answer we're gonna have nickels quarters dimes or let me put nickels dimes quarters let me follow what we have here and then i'm going to add another category that is the total so in this case we're going to have four fractions on top i'm going to put the ratio if there are three nickels there's going to be four dimes and seven quarters in this case the total number of coins will be three plus four plus seven that's going to be 14. so this is based on the ratio on the bottom i'm going to put the actual number of coins in the jar the total number of coins in the jar is 112. now i can calculate the number of nickels dimes or quarters it doesn't matter however we want to focus on the number of nickels so we can put an x here or if you want to you can put n for nickels so all i need to do is set this fraction equal to this one so 3 over n is equal to 14 over 112. now let's say if i wanted to calculate the number of dimes i would put d for dimes and set these two equal to each other and solve for d or if i want to solve for the number of quarters i would set these two fractions equal to each other so i can calculate the number of dimes nickels or quarters um whichever i want in this problem but let's focus on n so let's cross multiply this is going to be 14 times n and that's going to equal 3 times 1 12. three times one twelve is three hundred and thirty six now let's divide both sides by fourteen three thirty six divided by fourteen is twenty four so there's 24 nickels in this jar and so that's it for this problem by the way if you want to find a rest here's what you can do at this point let's write our ratios of nickels dimes quarters and the total so it's 3 to 4 to 7 to 14. now for nickels we have 24 in the bottom to go from 3 to 24 we need to multiply by 8 and if we multiply 14 by 8 that will give us 112 the total number of coins to find the total number of dimes we need to multiply 4 by 8. 4 times 8 is 32 and to find the number of quarters it's going to be 7 times 8 which is 56 and you can check it if we add 24 32 and 56 it should give us the total number of coins in the draw which is 112. so there's 24 nickels 32 dimes and 56 quarters