Ratio and Proportion
A ratio is a relationship between two numbers that defines the quantity of the first in comparison to the second. For example, for most mammals, the ratio of legs to noses is , but for humans, the ratio of legs to noses is .
Ratios can be written in the fractional form, so comparing three boys with five girls could be written or .
When quantities are proportional, their ratios are equal. For example, the ratios and are proportional. Note that and are equivalent fractions because they both simplify to
Simplify Ratios
We can simplify a ratio by dividing both sides by the greatest common factor. Note that a ratio and its simplified form are equivalent fractions.
Simplify .
Since the greatest common factor of 25 and 30 is 5, the simplified ratio is .
Are and equivalent?
The simplified ratio of is .
The simplified ratio of is .
Since they have the same simplified ratio, these ratios are equivalent.
Finding Unknowns In Ratios
To find the unknown term in a ratio, we can write the ratios as fractions, and then cross-multiply and solve the resulting equation.
Solve for in the proportion : .
Expressing them as fractions, we get .
Cross-multiplying, we get .
Finally, solving for gives us .
Solve for in the proportion: .
Expressing them as fractions, we get .
Cross-multiplying, we get , or .Hence, this has solutions .
Ratio and Proportion Word Problems
If Calvin paid $5 for 7 pencils, how much would he pay for 56 pencils?
Let be the price of pencils. Since the price of a single pencil does not change, we have Hence, Calvin would pay for pencils.
The ratio of Alice's pay to Bob's pay is . The ratio of Bob's pay to Charlie's pay is . If Alice is paid $75, how much is Charlie paid?
Since the ratio of Alice's pay to Bob's pay is , Bob's pay must be , where . Cross-multiplying by the denominators, we get , so .
Continuing in the same way, we compare Bob to Charlie: Thus, Charlie is paid $54.
The total number of vegetables is 158. If the ratio of cucumbers to carrots is , and the ratio of cucumbers to radishes is , how many more radishes are there than carrots?
The ratio of cucumbers to carrots is . The ratio of cucumbers to radishes is . Thus, the ratio of cucumbers to carrots to radishes is .
Let , , and be the number of cucumbers, carrots, and radishes, respectively. Then, since the total is 158,
Thus, and there are radishes and carrots.
Therefore, there are 14 more radishes than carrots.
If Alice beats Bob by 20 meters in a 100-meter race, and Bob beats Cathy by 20 meters in a 100-meter race, by what distance does Alice beat Cathy?
Let the distances covered by Alice, Bob, and Cathy at the same time be and respectively. Then
since we can multiply both terms of the ratio be the same integer, without changing the ratio.
Now since is equal in both ratios, we can equate both ratios:
Thus, when Alice covers 100 m, Cathy covers 64 m, implying Alice beats Cathy by 36 m in a 100 m race.