# 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 \( \frac{3}{5} \).

## Equivalent Ratios

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?

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. \( _\square \)

## Finding Missing Values in Ratios

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 \(x\,?\)

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

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?

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 \( 4 : N = N : 9 \).

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 \( N = \pm 6 \).

**Cite as:**Finding Missing Values in Ratios.

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