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 \[\begin{align} \frac{\$5}{7} &= \frac{x}{56} \\ 7x &= \$5 \times 56 \\ x &= \$40. \end{align}\] 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

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

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. \(_{\square}\)