Key takeaway: Equivalent ratios are ratios that describe the same relationship, found by multiplying or dividing both terms of a ratio by the same nonzero number, such as 4:7 and 20:35.
Grade 66.RP.A.3aRatios, Rates and Percents
Definition of Equivalent Ratios
Equivalent ratios are different-looking ratios that express the same comparison. If a smoothie uses 4 scoops of yogurt for every 7 scoops of fruit, a bigger batch with 20 scoops of yogurt and 35 scoops of fruit tastes exactly the same.
Two ratios are equivalent when one can be turned into the other by multiplying (or dividing) both terms by the same nonzero number. Equivalent ratios always simplify to the same simplest form.
Formula and notation
For any nonzero number \(k\):
\[a : b \;=\; (a \times k) : (b \times k)\]
Here \(a\) and \(b\) are the terms of the original ratio and \(k\) is the scale factor. In fraction form, \(\frac{4}{7} = \frac{20}{35}\) because both sides were scaled by \(k = 5\).
Key facts
- Multiply or divide both terms by one number; adding the same amount to both terms breaks the ratio.
- Equivalent ratios appear as rows of a ratio table and as points on a straight line through the origin.
- To test two ratios, simplify each one and compare the simplest forms.
- The scale factor can be a fraction, which is how you shrink a recipe.
Equivalent Ratios example
A paint mix uses 9 cups of blue for every 12 cups of white. Is the mix of 33 cups blue and 44 cups white equivalent? Find the missing term of \(9 : 12 = 15 : \square\).
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Simplify \(9 : 12\) by dividing by 3: \(3 : 4\).
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Simplify \(33 : 44\) by dividing by 11: \(3 : 4\). The simplest forms match, so the mixes are equivalent.
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For \(9 : 12 = 15 : \square\), the scale factor on the first term is \(15 : 9 = \frac{5}{3}\).
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Apply it to the second term: \(12 \times \frac{5}{3} = 20\).
Yes, \(33 : 44\) is equivalent to \(9 : 12\), and the missing term is 20.
Common mistake
Watch out
Leo changes \(5 : 8\) into \(8 : 11\) by adding 3 to each term.
Why it is wrong
Adding the same number keeps the difference, not the ratio. Doubling \(5 : 8\) gives \(10 : 16\), but \(8 : 11\) makes the first quantity a much larger share of the second. Multiply both terms by one number: \(5 : 8 = 15 : 24\).
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Ratios, Rates and Percents