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 , but for humans, the ratio of legs to noses is .
Ratios can be written in the fractional form, so comparing three boys with five girls could be written or .
Equivalent Ratios
Ratios are equivalent when they simplify to the same ratio. For example, the ratios and are equivalent because they both simplify to
Are and equivalent?
The simplified ratio of is .
The simplified ratio of is .
Since they have the same simplified ratio, these ratios are equivalent.
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
.
Expressing them as fractions, we get or in simplified form
The values in the right fraction are five times greater than the corresponding values in the left fraction, so
Solving by cross-multiplication, we get
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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We can use equivalent ratios to find the missing values. Using the first two rows of the table, we know that is equivalent to The fraction simplifies to so every shelf requires 6 screws. Therefore, four shelves require screws.
Since every shelf requires 6 screws, we can build shelves with 42 screws.
For what value of does .
Expressing them as fractions, we get .
Cross-multiplying, we get , or .Hence, this has solutions .