Ratio, Rate, and Proportion

Definition

A ratio is a relationship between two numbers (or two objects) that defines the quantity of the first in comparison to the second. For example, for most mammals the ratio of legs to noses is 4:1 4:1 , but for humans, the ratio of legs to noses is 2:1 2:1 . Ratios can also be written in fractional form, so comparing three boys with five girls could be written 3:5 3:5 or 35 \frac{3}{5} .

Be careful, however; we often think of fractions in the form partwhole \frac{\text{part}}{\text{whole}} , but a ratio is not necessarily comparing a part to a whole. In the example above, the denominator, 5, represents girls, and if we wanted to compare boys to the total number of people, we would need the ratio 3:8=38 3:8 = \frac{3}{8} .

When two objects are proportional, it means that their ratios are equal. Specifically, two ratios, a:b a:b and c:d c:d , are "in the same proportion" or "proportional" if and only if:

a:b::c:d    ab=cd    ad=bc a:b :: c:d \implies \frac{a}{b}=\frac{c}{d} \implies ad = bc

Technique

The ratio of Alice's pay to Bob's pay is 54 \frac{5}{4} . The ratio of Bob's pay to Charlie's pay is 10:9 10:9 . If Alice is paid $75, how much is Charlie paid?

Since the ratio of Alice's pay to Bob's pay is 5:4 5:4 , Bob's pay must be b b , where 54=75b \frac{5}{4}=\frac{75}{b} . Cross-multiplying by the denominators, 5b=4(75) 5b = 4(75) , so b=60 b = 60 .

Continuing in the same way, we compare Bob to Charlie: 109=60c    10c=9(60)    c=54 \frac{10}{9}=\frac{60}{c} \implies 10c = 9(60) \implies c = 54 . Thus, Charlie is paid $54. _\square

 

The total number of vegetables is 158. If the ratio of cucumbers to carrots is 4:7 4:7 , and the ratio of cucumbers to radishes is 1635 \frac{16}{35} , how many more radishes are there than carrots?

The ratio of cucumbers to carrots is 4:7=16:28 4:7 = 16:28 . The ratio of cucumbers to radishes is 16:35 16:35 . Thus the ratio of cucumbers to carrots to radishes is 16:28:35 16: 28: 35 .

Let 16x 16x , 28x 28x , and 35x 35x be the number of cucumbers, carrots, and radishes, respectively. Then, since the total is 158:

16x+28x+35x=79x=158 16x + 28x + 35x = 79x = 158

Thus, x=2 x = 2 and there are 2(35)=70 2(35) = 70 radishes and 2(28)=56 2(28) = 56 carrots. Therefore, there are 14 more radishes than carrots. _\square

Note by Arron Kau
5 years, 5 months ago

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