Finding Missing Values in Ratios

A ratio is a comparison between two or more quantities. 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 \( \frac{3}{5} \).

Ratios are equivalent when they simplify to the same ratio. For example, the ratios \(4:10\) and \(14:35\) are equivalent because they both simplify to \(2:5.\)

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 use some fraction sense or cross-multiply to find the unknown value.

Given the equivalent ratios below, what is the value of \(x\,?\)

\[ 6: 15 = 10 : x \].

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ANSWER

Expressing them as fractions, we get \( \frac{6}{15} = \frac{10}{x}, \) or in simplified form \( \frac{2}{5} = \frac{10}{x}. \)

The values in the right fraction are five times greater than the corresponding values in the left fraction, so \(x= 5\times 5 = 25.\)

Solving by cross-multiplication, we get

\[\begin{align} 2x &= (10)(5) \\ 2x &= 50 \\ x &= 25.\end{align}\]

Every 3 shelves require 18 screws. How many screws are needed for 4 shelves? How many shelves can we build with 42 screws?

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ANSWER

We can use equivalent ratios to find the missing values. Using the first two rows of the table, we know that \(18:3\) is equivalent to \(? : 4.\) The fraction \(18:3\) simplifies to \(6:1\) so every shelf requires 6 screws. Therefore, four shelves require \(4 \times 6 = 24\) screws.

Since every shelf requires 6 screws, we can build \(42 \div 6 = 7\) shelves with 42 screws.

For what value of \(N\) does \( 4 : N = N : 9 \).

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ANSWER

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

Hence, this has solutions \( N = \pm 6 \).