HSPT: Comparisons: decide which fraction is greater
Learn to compare two fractions quickly and accurately by cross-multiplying or using a common denominator.
How can you tell which of two fractions is greater when they have different denominators?
Short answer
Cross-multiply: multiply each numerator by the other denominator and compare the products. Equal products mean the fractions are equal.
Know first: Equivalent fractions, Multiplication facts
Step 1 of 5
Make the fractions comparable
To compare two fractions, give them a fair match. Two fractions with the same denominator are easy: the bigger numerator wins.
When the denominators differ, you can rewrite both over a common denominator or use the shortcut of cross-multiplying. Both methods give the same order, and cross-multiplying is quicker without a calculator.
| Situation | Example | Result |
|---|---|---|
| Same denominator | 5/9 vs 4/9 | The bigger numerator wins: 5/9 |
| Same numerator | 1/3 vs 1/4 | The smaller denominator wins: 1/3 |
| One above 1/2, one below | 5/9 vs 3/7 | 5/9 is above one half, 3/7 is below |
| Simplify first | 8/10 vs 12/15 | Both are 4/5, so they are equal |
Step 2 of 5
Cross-multiply and compare
Worked example: Two close fractions
Compare Quantity A: 5/8 and Quantity B: 4/7.
- AQuantity A is greater.
- BQuantity B is greater.
- CThe two quantities are equal.
- DThe relationship cannot be determined.
- 1
Cross-multiply
A: 5 × 7 = 35. B: 4 × 8 = 32.
- 2
Compare the products
35 is greater than 32, so 5/8 is greater than 4/7.
- 3
Confirm with a common denominator
5/8 = 35/56 and 4/7 = 32/56. Same result.
- 4
See why the other choices are there
B looks greater if you only notice that 8 is bigger than 7. Equal would need equal products. Cannot be determined never fits two fixed numbers, since they always have a definite order.
Answer
Quantity A is greater.
Wrong: 8 is bigger than 7, so 4/7 is greater than 5/8.
Right: 5 × 7 = 35 and 4 × 8 = 32, so 5/8 is greater.
A bigger denominator means smaller pieces, but the numerators differ too. Only a fair comparison, by cross-multiplying or a common denominator, settles it.
Step 3 of 5
Compare fairly every time
Check yourself · Question 1
Compare Quantity A: 6/8 and Quantity B: 9/12.
Answer: C
Cross-multiply: 6 × 12 = 72 and 9 × 8 = 72. The products match, so the fractions are equal. Both simplify to 3/4.
Trap answer: Quantity B is greater.
Bigger numbers, bigger fraction
- Why it looks right
- 9 and 12 are both bigger than 6 and 8, so 9/12 feels bigger.
- Why it's wrong
- A fraction's size depends on the relationship between its top and bottom. Both fractions simplify to 3/4.
- The right answer
- The two quantities are equal.
Check yourself · Question 2
Compare Quantity A: 4/9 and Quantity B: 5/11.
Answer: A
Cross-multiply: 4 × 11 = 44 and 5 × 9 = 45. 45 is greater, so 5/11 is greater than 4/9.
Common mistake
"I can tell which fraction is bigger by looking at the denominators."
- Why it's tempting
- Comparing one pair of numbers is faster than working with both, and it is sometimes right.
- Do this instead
- Cross-multiply or rewrite over a common denominator. Then compare numerators only.
Step 4 of 5
What to remember
Remember
- Fractions can be compared directly only when they share a denominator.
- Cross-multiply: the bigger product goes with the bigger fraction, and equal products mean equal fractions.
- Two fixed numbers can always be compared, so cannot be determined is never right here.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
What two products did you compare in the practice question, and why can the answer never be *cannot be determined*?