See the idea
Which fractions are equivalent to 3/4?
6/8, 9/12, 12/16 and 15/20 are all equal to 3/4. Multiply both numbers by the same factor. The part size changes, but the amount does not.
| Start | Scale both numbers | Equivalent fraction |
|---|---|---|
| 3/4 | × 2 | 6/8 |
| 3/4 | × 3 | 9/12 |
| 3/4 | × 4 | 12/16 |
| 3/4 | × 5 | 15/20 |
Missing-number example: 3/4 = 6/□. The numerator doubled from 3 to 6, so double the denominator too: 4 × 2 = 8. The answer is 6/8.
To check 6/8 = 3/4, divide 6 and 8 by 2. Because the fractions are equal, neither is greater. For a pair that is not equal, use these fraction comparison methods.
Try it: 3/4 = □/28. What belongs in the box?
21. Since 4 × 7 = 28, multiply the numerator by 7: 3 × 7 = 21. Check by dividing 21 and 28 by 7.
Before you start
Read a fraction as a number of equal parts. You should be able to explain why the denominator names all the equal parts, including the unshaded ones.
Words used in this guide
- Numerator
- The number above the fraction bar; in 3/4 it counts three quarter-sized parts.
- Denominator
- The number below the bar; four equal parts make one whole. It cannot be zero.
- Equivalent
- Equal in value, even when the written numbers look different.
Why the method works
Multiplying a fraction by 2/2 multiplies it by 1, so its value is unchanged. The picture explains the same fact: cutting each piece into two changes how many pieces you count, but does not add or remove any of the original amount. The multiplier must be non-zero. Multiplying both numbers by zero would produce 0/0, which is undefined.
Keep the whole the same
Draw two bars of equal length. Divide one into four equal parts and the other into eight. Three quarters and six eighths cover the same length. Comparing bars of different lengths would change the whole and make the picture misleading.
Rename the parts
Multiply both the numerator and denominator by the same non-zero whole number: 3/4 = (3 × 2)/(4 × 2) = 6/8. Each quarter becomes two eighths, so the number of shaded pieces and the total number of pieces both double.
A mistake worth checking
Adding the same number to the top and bottom does not generally preserve the fraction. For example, 1/2 and 2/3 are different numbers. Use a picture or a number line to check the rule rather than memorising it on its own.
Equivalent fractions: choose a method that explains the equality
Two fractions are equivalent when they name the same number. For pictures, compare equal-sized wholes; for numbers, compare their values. Multiplying the numerator and denominator by the same non-zero number changes the size and number of parts together, leaving the amount unchanged.
A fraction chart you can rebuild
Read each row as a chain of equal values. The columns use scale factors 1, 2, 3 and 4. Make your own next column by multiplying both original numbers by 5.
| Starting fraction | × 2 / 2 | × 3 / 3 | × 4 / 4 |
|---|---|---|---|
| 1/2 | 2/4 | 3/6 | 4/8 |
| 2/3 | 4/6 | 6/9 | 8/12 |
| 3/4 | 6/8 | 9/12 | 12/16 |
| 3/5 | 6/10 | 9/15 | 12/20 |
Missing denominator: 9/12 = 3/□
The numerator changes from 9 to 3 by dividing by 3. Divide the denominator by 3 too: 12 ÷ 3 = 4. So the missing denominator is 4. Check by multiplying 3/4 by 3/3 to get back to 9/12.
When the scaling is not obvious
Compare 10/15 and 14/21. Simplify each: 10/15 = 2/3 and 14/21 = 2/3. They are equivalent. You can also check cross-products: 10 × 21 = 210 and 14 × 15 = 210. This works because rewriting both fractions with denominator 15 × 21 produces those two numerators. Cross-multiplication is a check, not a reason to add or multiply the original numerators together.
For ordinary school fraction charts, keep whole-number numerators and non-zero whole-number denominators. Do not multiply by zero: it would create a denominator of zero.
Can you explain it without a rule?
A child says 2/5 = 4/7 because both numbers increased by 2. What would you show?
Split every fifth into two equal parts. Two fifths becomes four tenths, not four sevenths: 2/5 = 4/10. Adding 2 changes the relationship between the number of shaded parts and the number of parts in the whole.
Find two fractions equivalent to 7/4. Must equivalent fractions be less than 1?
14/8 and 21/12 are equivalent to 7/4. All three equal 1¾. Equivalence describes equal values, including values greater than 1.
Worked examples, step by step
Example 1: Find a fraction equivalent to 2/3 with denominator 12.
- 3 × 4 = 12, so multiply the numerator by 4 as well.
- 2/3 = 8/12. Both fractions cover the same share of the same whole.
Example 2: Find three fractions equivalent to 4/7.
- Choose scale factors 2, 3 and 5. Apply each to both numbers, not just to the denominator.
- 4/7 = 8/14 = 12/21 = 20/35. There is no final equivalent fraction: you can keep choosing larger whole-number factors.
- Check one pair: dividing 20 and 35 by 5 returns 4 and 7.
Example 3: Are 8/14 and 12/21 equivalent?
- Both can be reduced: 8/14 = 4/7 and 12/21 = 4/7. This is often simpler than finding a large common denominator.
- A second check uses cross-products: 8 × 21 = 168 and 14 × 12 = 168. Equal cross-products confirm equality when the denominators are non-zero.
- This test works because putting a/b and c/d over denominator bd gives numerators ad and bc.
Example 4: A recipe uses 3/4 cup. Your measuring cup is marked in eighths. Which mark is needed?
- The quantity stays fixed; only the measuring unit changes. There are two eighths in each quarter.
- Three quarters contain six eighths, so use the 6/8 mark.
- Do not double the amount of an ingredient just because both written numbers doubled.
Common mistakes and how to repair them
“Adding 2 to both numbers keeps it equal.”
1/3 would become 3/5. Over fifteenths these are 5/15 and 9/15, so the values differ. Adding changes the ratio; scaling preserves it.
“Bigger numbers always mean a bigger fraction.”
Compare 3/4 and 30/40 on the same bar. Both shade the same proportion; the second description uses smaller pieces.
Check your understanding
Choose an answer for each question. You can check again after reading the explanation.
Worked answers
Which fraction equals 3/4? 9/12. Multiply numerator and denominator by 3: 3/4 = 9/12.
Complete 2/5 = ?/20. 8. There are four times as many parts, so 2 × 4 = 8 shaded parts.
Practise independently
Work these out on paper. Include a diagram or a sentence where the question asks you to explain. Open each solution after making an attempt.
3. Write 5/6 with denominator 24.
Show the reasoning
20/24. The denominator was multiplied by 4, so 5 × 4 = 20.
4. Complete 7/9 = 35/?.
Show the reasoning
45. Multiplying 7 by 5 gives 35, so multiply 9 by 5 as well.
5. Are 6/10 and 9/15 equal? Explain in two ways.
Show the reasoning
Yes. Both simplify to 3/5; the cross-products are also equal: 6 × 15 = 10 × 9 = 90.
6. Can 2/3 be written as a fraction with denominator 10 and a whole-number numerator?
Show the reasoning
No. The numerator would be 20/3, which is not a whole number. You can use denominator 30: 2/3 = 20/30.
If a method still feels uncertain, revisit the closest worked example, then try a fresh problem. A correct answer copied from a solution tells you less than an explanation you can give independently.
Learn it → Try it → Explain it
Put the idea into play
Build and compare fractions with pictures you can change.
Choose “Same amount” and build an equivalent fraction. Explain why the shaded amounts match.
Try Fraction Visuals →Free guest play has question, level and time limits. Sign in to continue or save progress. A game score is not a mastery assessment.
Questions students and parents ask
Can an improper fraction have equivalent fractions?
Yes. 7/4 = 14/8. Both are greater than one. Equivalence is about value, not whether the numerator is smaller.
Should we check with a calculator?
Exact simplification or cross-products give a reliable check. Rounded decimal displays can make different fractions appear equal.
Is the simplest form the only correct answer?
No. 4/6 and 2/3 are equal. Use simplest form when requested; another denominator may be more useful when adding or comparing.
Help at home: Fold and compare
Make two equal paper strips. Fold one into four equal sections and shade three. Split every section of the other into two and shade the same length. Ask what changed and what stayed fixed.
Then ask for 2/3 in twelfths without showing the example. If your child changes only one number, return to splitting every piece, including unshaded pieces.
Revisit with fifths and tenths another day. Look for an explanation of equal value, not just successful multiplication.
Keep the conversation focused on the reasoning you can hear or see. One short activity cannot establish lasting mastery or diagnose a learning difficulty.
How can I explain equivalent fractions to my child?
Fold two equal-length paper strips into thirds. Shade two thirds on each. On one strip, split every third in half: there are now six equal pieces and four are shaded. Nothing was added to the shaded length, so 2/3 = 4/6. Ask your child to point to what changed and what stayed the same before introducing the multiplication rule.
If your child changes 2/3 to 3/4 by adding one to both numbers, compare the shaded lengths. Three quarters is larger. Making equivalent fractions means multiplying or dividing both numbers by the same non-zero factor, not adding the same number.
Try 3/5 = ?/10 next. Listen for “each fifth becomes two tenths, so three fifths becomes six tenths.” A correct answer without that explanation is a reason to keep the model nearby.
Helping your child with fractions? Start with this parent guide →
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Choose your next learning step
- Simplify fractions — Use a common factor to rename a fraction.
- Compare fractions — Use equal denominators when the values are different.
- Help a child who is stuck on fractions — Choose a short activity based on the step they find difficult.
If your child can copy the rule but cannot explain why the value stays the same, bring that example to a free 50-minute class.
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