Solve Problems with Ratios and Proportions

I can multiply both the denominator and numerator by any number to find equivalent fractions. Examples are [ggfrac]6/14[/ggfrac] , [ggfrac]9/21[/ggfrac] , and [ggfrac]12/28[/ggfrac].

I can again multiply both the numerator and denominator by any number, like 2, to get [ggfrac]24/32[/ggfrac]. Because 12 and 16 have common factors, I can also divide them by the same number to find equivalent fractions. Dividing 12 and 16 by 2, I get the equivalent fraction [ggfrac]6/8[/ggfrac] . Or, dividing 12 and 16 by 4, I get the equivalent fraction [ggfrac]3/4[/ggfrac].

The ratio of blue balloons to green balloons is 4:8. It can also be reduced by dividing both 4 and 8 by 4 to get 1:2.

Not necessarily. It means that for every 5 yellow balloons, there are 7 pink balloons. So, if there are 10 yellow balloons, then there must be 14 pink balloons. If there are 20 yellow balloons, then there are 28 pink balloons.

He is not right! When 34 of the balloons are red, 3 out of every 4 balloons are red. However, the ratio 3:4 compares the part of the balloons that are red to the part of the balloons that are purple. That means that 3 out of every 7 balloons are red.

If two ratios are equivalent, that means that I can get one ratio by multiplying or dividing both parts of the other ratio by the same number. Because 2 times 3 is 6, and 3 times 3 is 9, I know that the ratios are equivalent, since I multiplied both parts by 3.

Write the ratio equation 5:3 = 10:x, where x represents the number of grade 9 students. These ratios are proportional. Since I multiply 5 by 2 to get 10, I must also multiply 3 by 2 to get the unknown value. 3 × 2 = 6, so there are 6 grade 9 students.

I can write the ratio of brown to white as 2:3, or I can write the ratio of white to brown as 3:2. Both are correct, but I have to use the correct ratio when I want to find an equivalent ratio. Suppose you want to make a bigger recipe and you are using 12 tablespoons of brown sugar. Then your ratio would either be brown to white 2:3 = 12:x OR white to brown 3:2 = [ggfrac] x/12[/ggfrac]. Either brown sugar is always the first number in the ratio, or it is always the second number.

Yes, they are! There is an extra step that helps me find out. I can reduce both ratios to [ggfrac]4/3[/ggfrac]. Since both ratios are equivalent to [ggfrac]4/3[/ggfrac], then they are also equivalent to each other!

Josh is not correct! The top numbers compare oranges to oranges, which makes sense, but the bottom numbers compare apples to total fruit. The two ratios do not compare the same things. I can change the first ratio so that it also compares the ratio of oranges to total fruit. Add the parts of the ratio to get 3 + 7 = 10. Then the ratio of oranges to total fruit is [ggfrac]3/10[/ggfrac] . The correct equation can be [ggfrac]3/10[/ggfrac] =[ggfrac]12/x[/ggfrac] , which I can solve to find that there are 40 fruits in all.