# Ratio and Proportion

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 \( 4:1 \), but for humans, the ratio of legs to noses is \( 2:1 \). Ratios can also be written in the fractional form, so comparing three boys with five girls could be written \( 3:5 \) or \( \frac{3}{5} \).

Be careful; we often think of fractions in the form \( \frac{\text{part}}{\text{whole}} \), 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 \( 3:8 = \frac{3}{8} \).

When two objects are *proportional*, or "in the same proportion," it means that their ratios are equal. Specifically,

Two ratios, \(a:b\) and \(c:d\) are

proportional\((\text{i.e. } a:b :: c:d)\) if and only if\[ ad=bc.\]

## Simplify Ratios

A ratio is **simplified** when both numbers have no common divisor. We can simplify a ratio by dividing throughout by the greatest common divisor (highest common factor).

Simplify \( 25 : 30 \).

Since \( \gcd(25, 30 ) = 5 \), the simplified ratio is \( \frac{25}{5} : \frac{30}{5} = 5 : 6 \). \( _\square \)

Note: Two ratios are

equivalentif they both have the same simplified ratio. Another way to check if 2 ratios are equivalent is to cross-multiply and see if we get the same value.

Are \( 4:6 \) and \( 6 : 9 \) equivalent?

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. \( _\square \)

Another approach:

We are asking if \( 4 : 6 = 6 : 9 \). This is the same as saying that \( \frac{4}{6} = \frac{6}{9} \). Cross-multiplying, this is the same as saying \( 4 \times 9 = 6 \times 6 \). Since both sides of the equation are equal to 36, the two ratios are equivalent. \( _\square \)

Simplify \( 123: 321 \).

We see that 123 and 321 are both multiples of 3, so we can first divide throughout by 3 to get \( 41 : 107 \). Since 41 is a prime, and 107 is not a multiple of 41, this ratio is simplified. \(_\square\)

## Finding Unknowns In Ratios

To find the unknown term in a ratio, we state the ratios as fractions, and then cross-multiply and solve the resulting equation.

## Solve for \(x\) in the proportion : \( 6: 15 = 10 : x \).

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

Cross-multiplying, we get \( 6 \times x = 10 \times 15 \).Finally, solving for \(x\) gives us \( x = \frac{ 10 \times 15 } { 6} = \frac{ 150}{6} = 25 \). \( _\square \)

## Solve for \(x\) in the proportion: \( 4 : x = x : 9 \).

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 \( x = \pm 6 \). \( _\square \)

Find the mean proportional of \(4\) and \(36\).

Asking for the mean proportional is equivalent to asking "Solve for \(x,\) where \(4:x::x:36.\)"

Thus, \(4 \times 36 =x^2 \implies x=12.\) \(_\square\)

## Ratio and Proportion Word Problems

## 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?

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?

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?

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. \(_{\square}\)

**Cite as:**Ratio and Proportion.

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