Key takeaway: A ratio pairs two quantities so that one is measured against the other, using a colon, the word to, or a fraction bar, as in 3 : 5 for 3 blue tiles against 5 red tiles.
Grade 66.RP.A.1Ratios, Rates and Percents
Definition of Ratio
A ratio sets one quantity beside another to show how they relate. It tells how much of one thing there is for a certain amount of another, such as 24 rock songs for every 40 pop songs on a playlist.
It can compare part to part (rock songs to pop songs), part to whole (rock songs to all songs) or whole to whole. The order matters: the quantity named first is written first. Both quantities must come from the same situation so the comparison makes sense.
Formula and notation
Two quantities \(a\) and \(b\) give three common written forms:
\[a:b \quad\quad a \text{ to } b \quad\quad \frac{a}{b}\]
The numbers \(a\) and \(b\) are the terms. The fraction bar here does not mean a part of a whole; it only records the comparison. Swapping the terms produces a different comparison.
Key facts
- Scaling both terms by one nonzero factor leaves the comparison unchanged and gives an equivalent ratio.
- Simplest form means the two terms share no factor larger than 1.
- Part-to-part and part-to-whole comparisons of the same group have different second terms.
- Word order sets term order, so rock to pop is not the same as pop to rock.
Ratio example
Maya’s playlist has 24 rock songs and 40 pop songs. Write rock to pop in simplest form, then rock to all songs.
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Rock to pop is \(24 : 40\). Both numbers are multiples of 8, and no larger number divides both.
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Dividing each term by 8 leaves 3 and 5, so rock to pop is \(3 : 5\).
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All songs: \(24 + 40 = 64\). Rock to all songs is \(24 : 64\), which reduces to \(3 : 8\).
Rock to pop is 3 : 5 and rock to all songs is 3 : 8.
Common mistake
Watch out
Maya reads \(3 : 5\) and says, “So \(\frac{3}{5}\) of my songs are rock.”
Why it is wrong
The comparison \(3 : 5\) sets rock against pop, which are two parts. The whole playlist is \(3 + 5 = 8\) parts, so rock is \(\frac{3}{8}\) of the songs. Add the parts before turning a part-to-part comparison into a fraction of the whole.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Ratios, Rates and Percents