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

$\begin{aligned} 2x &= (10)(5) \\ 2x &= 50 \\ x &= 25.\end{aligned}$

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 $N = \pm 6$.