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Math foundations
SAT math fractions made simple: models, rules, and practice
Learn what a fraction means, how to compare and calculate with fractions, and how to catch the denominator mistakes that trip up beginners.
SPSAT Practice Editorial7 min read
The short version
What you'll take away
- The denominator counts equal parts in a whole; the numerator counts the parts you have.
- Add and subtract only after the fractions name parts of the same size. Multiply across; divide by multiplying by the reciprocal.
- Simplify by dividing the numerator and denominator by the same nonzero factor.
In this article
A fraction is a number that describes equal parts
A fraction has a numerator on top and a denominator on the bottom. In \(\displaystyle \frac{3}{5}\), the 5 says the whole is split into five equal parts, and the 3 says we are counting three of them. The parts must be equal: five unequal pieces do not make fifths.
Think of a paper strip cut into five equal lengths. Shade three lengths and you have \(\displaystyle \frac{3}{5}\). The fraction can also mean division: \(\displaystyle \frac{3}{5}\) is 3 divided by 5, or 0.6. Fractions are numbers, so they can sit on a number line just like whole numbers and decimals.
The study lab
Read the fraction 3/5
Each box is one equal fifth of the same whole; shaded boxes show the numerator.
3/5 of one chocolate bar
Equivalent fractions name the same amount
If you split every piece in half, the total amount does not change. That is why \(\displaystyle \frac{1}{2}\) = \(\displaystyle \frac{2}{4}\) = \(\displaystyle \frac{3}{6}\). To make an equivalent fraction, multiply or divide the numerator and denominator by the same nonzero number.
To simplify \(\displaystyle \frac{12}{18}\), divide both numbers by 6: 12 ÷ 6 over 18 ÷ 6 = \(\displaystyle \frac{2}{3}\). A quick check: multiplying \(\displaystyle \frac{2}{3}\) by \(\displaystyle \frac{6}{6}\) gets you back to \(\displaystyle \frac{12}{18}\). Dividing only the top or only the bottom changes the value.
The study lab
One-half, three ways
Chocolate bars split into different numbers of equal pieces, plus a number line marking their shared value.
1/2
2/4
3/6
Compare fractions by making their parts match
The denominator tells you the size of each part. With the same denominator, more parts means a larger fraction: \(\displaystyle \frac{5}{8}\) > \(\displaystyle \frac{3}{8}\). With different denominators, first rewrite both fractions with a common denominator.
For example, compare \(\displaystyle \frac{2}{3}\) and \(\displaystyle \frac{3}{5}\). A common denominator is 15: \(\displaystyle \frac{2}{3}\) = \(\displaystyle \frac{10}{15}\) and \(\displaystyle \frac{3}{5}\) = \(\displaystyle \frac{9}{15}\). Since \(\displaystyle \frac{10}{15}\) is greater than \(\displaystyle \frac{9}{15}\), \(\displaystyle \frac{2}{3}\) is greater. You can also use decimals when the conversion is simple, but equivalent fractions work reliably every time.
- Same denominator? Compare the numerators.
- Different denominators? Find a common denominator, then compare.
- Sanity check: a fraction with a larger numerator is not automatically larger if its denominator also changes.
Multiply and divide with a different rule
To multiply fractions, multiply the numerators together and the denominators together: \(\displaystyle \frac{2}{3}\) × \(\displaystyle \frac{3}{5}\) = \(\displaystyle \frac{6}{15}\) = \(\displaystyle \frac{2}{5}\). You may simplify before or after multiplying, as long as you divide top and bottom by the same factor.
To divide by a fraction, multiply by its reciprocal (flip the second fraction): \(\displaystyle \frac{3}{4}\) ÷ \(\displaystyle \frac{2}{5}\) = \(\displaystyle \frac{3}{4}\) × \(\displaystyle \frac{5}{2}\) = \(\displaystyle \frac{15}{8}\) = 1 \(\displaystyle \frac{7}{8}\). The order matters: keep the first fraction and flip the divisor.
Practice: try these before checking the answers
Work each one on paper. For every answer, ask whether it is simplified and whether its size makes sense. Answers with a whole-number part are written as mixed numbers where helpful.
- 1. Simplify \(\displaystyle \frac{15}{25}\).
- 2. Which is larger: \(\displaystyle \frac{4}{9}\) or \(\displaystyle \frac{5}{12}\)?
- 3. Calculate \(\displaystyle \frac{2}{5}\) + \(\displaystyle \frac{1}{3}\).
- 4. Calculate \(\displaystyle \frac{7}{8}\) - \(\displaystyle \frac{1}{6}\).
- 5. Calculate \(\displaystyle \frac{3}{4}\) × \(\displaystyle \frac{2}{9}\).
- 6. Calculate \(\displaystyle \frac{2}{3}\) ÷ \(\displaystyle \frac{4}{5}\).
A quick fraction check for SAT practice
When a fraction appears in an SAT Math problem, identify what the fraction represents before calculating. It might be a part of a whole, a ratio, a rate, or the result of division. That meaning can help you choose a useful equation and catch an answer that is too large or too small.
Then check the mechanics: for addition or subtraction, did you create equal-sized parts? For multiplication, did you multiply both tops and both bottoms? For division, did you flip only the second fraction? Finally, simplify and compare your result with the situation in the question.
- Write the operation before doing the arithmetic.
- Estimate the result: multiplying two positive fractions less than 1 gives a smaller positive number.
- If your result seems off, check the common denominator or reciprocal step first.
Sources & further reading
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