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 $4:1$, but for humans, the ratio of legs to noses is $2:1$.

Ratios can be written in the fractional form, so comparing three boys with five girls could be written $3:5$ or $\frac{3}{5}$.

When quantities are proportional, their ratios are equal. For example, the ratios $\frac{2}{5}$ and $\frac{8}{20}$ are proportional. Note that $\frac{4}{10}$ and $\frac{12}{30}$ are equivalent fractions because they both simplify to $\frac{2}{5}.$

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 $25 : 30$.

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ANSWER

Since the greatest common factor of 25 and 30 is 5, the simplified ratio is $\frac{25}{5} : \frac{30}{5} = 5 : 6$. $_\square$

Are $4:6$ and $6 : 9$ equivalent?

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ANSWER

The simplified ratio of $4:6$ is $\frac{4}{2} : \frac{6}{2} = 2 : 3$.
The simplified ratio of $6:9$ is $\frac{6}{3} : \frac{ 9}{3} = 2 : 3$.
Since they have the same simplified ratio, these ratios are equivalent. $_\square$

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 $x$ in the proportion : $6: 15 = 10 : x$.

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Expressing them as fractions, we get $\frac{6}{15} = \frac{10}{x}$.

Cross-multiplying, we get $6x = 10(15)$.

Finally, solving for $x$ gives us $x = 25$. $_\square$

Solve for $x$ in the proportion: $4 : x = x : 9$.

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ANSWER

Expressing them as fractions, we get $\frac{4}{x} = \frac{x}{9}$.
Cross-multiplying, we get $4 \times 9 = x \times x$, or $36 = x^2$.

Hence, this has solutions $x = \pm 6$. $_\square$

If Calvin paid $5 for 7 pencils, how much would he pay for 56 pencils?

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ANSWER

Let $x$ be the price of $56$ pencils. Since the price of a single pencil does not change, we have $$\begin{aligned} \frac{\$5}{7} &= \frac{x}{56} \\ 7x &= \$5 \times 56 \\ x &= \$40. \end{aligned}$$ Hence, Calvin would pay $\$40$ for $56$ pencils.

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

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ANSWER

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

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

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

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ANSWER

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

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

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

Thus, $x = 2$ and there are $2(35) = 70$ radishes and $2(28) = 56$ carrots.

Therefore, there are 14 more radishes than carrots. $_\square$

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?

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ANSWER

Let the distances covered by Alice, Bob, and Cathy at the same time be $a, b,$ and $c,$ respectively. Then

$$\begin{aligned} a:b = 100:80 &\implies a:b = 100 \times 5 : 80 \times 5 = 500:400\\ b:c = 100:80 &\implies b:c = 100 \times 4 : 80 \times 4 = 400:320, \end{aligned}$$

since we can multiply both terms of the ratio be the same integer, without changing the ratio.

Now since $b$ is equal in both ratios, we can equate both ratios:

$$a:b:c = 500:400:320 \implies a:c = 500:320=100:64.$$

Thus, when Alice covers 100 m, Cathy covers 64 m, implying Alice beats Cathy by 36 m in a 100 m race. $_{\square}$