Equivalent Fractions: Definition, Examples And Easy Methods

What Are Equivalent Fractions?

Equivalent fractions are different ways of writing the same part of a whole. The numbers change, but the amount does not.

Picture a pizza cut into 2 slices. Eat 1 slice and you have eaten 1/2. Now picture the same pizza cut into 8 slices. Eat 4 and you have eaten 4/8. The slices are smaller, but you ate exactly the same amount of pizza. So 1/2 = 4/8.

Every fraction has two parts:

In equivalent fractions, both parts grow or shrink together. That keeps the size of the fraction the same.

Why Multiplying By The Same Number Works

Multiplying the top and bottom by the same number is the same as multiplying by 1. Any fraction with the same number on top and bottom, like 2/2, 3/3 or 10/10, equals 1.

Multiplying a number by 1 never changes its value. So:

• 1/2 × 2/2 = 2/4
• 1/2 × 3/3 = 3/6
• 1/2 × 5/5 = 5/10

Each answer is still one half, just cut into more pieces. Dividing works the same way, because dividing the top and bottom by 4 is the same as dividing by 4/4, which is also 1.

This is why the number you use can be anything except zero. 0/0 is not equal to 1. It is undefined, so it breaks the rule.

Picturing Equivalent Fractions

Drawings make equivalent fractions easy to see. Two models work best.

Fraction bars. Draw three bars of the same length. Split the first into 3 parts and shade 2. Split the second into 6 parts and shade 4. Split the third into 12 parts and shade 8. All three shaded areas line up at the same point, so 2/3 = 4/6 = 8/12.

Number line. Mark 0 and 1 on a line. Split it into fourths above the line and eighths below it. The mark for 3/4 lands on exactly the same spot as 6/8. Same point on the line means the same value.

These models show something students often miss. The equivalent fraction has more parts, but each part is smaller by the same factor. Twice as many pieces, each half as big, gives the same total.

How To Find Equivalent Fractions

There are two ways to make an equivalent fraction: multiply the top and bottom by the same number, or divide them by the same number. A third skill, finding a missing number, uses both.

Method 1: Multiply The Numerator And Denominator

Pick any whole number greater than 0 and multiply both parts of the fraction by it. This makes a fraction with bigger numbers.

Example: find three fractions equivalent to 3/5

You can keep going forever. 3/5 = 6/10 = 9/15 = 12/20 = 15/25 = 18/30, and so on.

Method 2: Divide The Numerator And Denominator

Find a number that divides evenly into both the top and the bottom, then divide both by it. This makes a fraction with smaller numbers.

Example: find fractions equivalent to 24/36

• Divide by 2: 12/18
• Divide by 3: 8/12
• Divide by 6: 4/6
• Divide by 12: 2/3

Dividing only works when the number goes into both parts evenly. You cannot divide 5/7 by 2, because 5 and 7 are not even. When the only shared factor left is 1, the fraction is in simplest form. More on that below.

Method 3: Find A Missing Number

Many homework problems give you one full fraction and one with a blank. Find what the known number was multiplied or divided by, then do the same to the other number.

Example: 3/4 = ?/20

1. Compare the denominators: 4 became 20.
2. 4 × 5 = 20, so the factor is 5.
3. Multiply the top by 5 too: 3 × 5 = 15.
4. Answer: 3/4 = 15/20

Example: 5/? = 15/24

1. Compare the numerators: 5 became 15.
2. 5 × 3 = 15, so the factor is 3.
3. Divide the other denominator by 3: 24 ÷ 3 = 8.
4. Answer: 5/8 = 15/24

Example: 18/30 = 3/?

1. 18 became 3, so the top was divided by 6.
2. Divide the bottom by 6: 30 ÷ 6 = 5.
3. Answer: 18/30 = 3/5

When the factor is not a whole number, cross multiply instead. For 4/6 = 10/?, write 4 × ? = 6 × 10, so 4 × ? = 60 and ? = 15. Check: 4/6 and 10/15 both simplify to 2/3.

How To Check If Two Fractions Are Equivalent

To test whether two fractions are equivalent, you can cross multiply, simplify both, give them the same denominator, or turn them into decimals. All four give the same answer, so pick the one that fits the numbers.

Cross Multiplication

Multiply the top of each fraction by the bottom of the other. If the two products match, the fractions are equivalent.

Is 4/10 equal to 6/15?

• 4 × 15 = 60
• 10 × 6 = 60
• 60 = 60, so yes, they are equivalent.

Is 3/7 equal to 5/12?

• 3 × 12 = 36
• 7 × 5 = 35
• 36 ≠ 35, so no.

Cross multiplication is the fastest test for most pairs, especially when the numbers do not share an obvious factor.

Simplify Both Fractions

Reduce each fraction to simplest form. If they end up as the same fraction, they are equivalent.

• 12/16 simplifies to 3/4
• 15/20 simplifies to 3/4
• Same result, so 12/16 = 15/20.

A fraction simplifier does this in one step if the numbers are large.

Give Them The Same Denominator

Rewrite both fractions with a common denominator, then compare the numerators.

Is 2/3 equal to 8/12?

• 12 is a multiple of 3, so rewrite 2/3 with a denominator of 12: 2/3 = 8/12.
• The numerators match, so yes.

For awkward pairs, use the least common multiple of the two denominators. An LCM calculator finds it quickly.

Convert To Decimals

Divide each numerator by its denominator. Equal decimals mean equal fractions.

• 3/8 = 0.375
• 9/24 = 0.375
• Same decimal, so 3/8 = 9/24.

This works well with a calculator. Be careful with repeating decimals, though. 1/3 shows as 0.3333333 on most screens, so tiny rounding differences can make two equal fractions look slightly different.

Equivalent Fractions Chart

This chart lists the first four equivalent fractions for the fractions you meet most often. Each row is made by multiplying the first fraction by 2, 3, 4 and 5.

The decimal column is a handy shortcut. Any fraction that matches one of those decimals belongs in that row.

Fractions That Equal Whole Numbers

A fraction equals a whole number when its numerator is a multiple of its denominator. These are equivalent fractions too, and US standards teach them in grade 3.

Any whole number can be written over 1, so 6 = 6/1. This trick matters when you multiply a whole number by a fraction. 4 × 2/3 is easier to see as 4/1 × 2/3 = 8/3.

The fraction 0/5 equals 0, and so do 0/2, 0/8 and 0/100. They are all equivalent to each other.

Equivalent Mixed Numbers And Improper Fractions

A mixed number such as 2 1/2 has equivalent improper fractions too. Convert it first, then multiply or divide as usual.

Example: find fractions equivalent to 2 1/2

1. Convert to an improper fraction: (2 × 2 + 1)/2 = 5/2
2. Multiply by 2/2: 10/4
3. Multiply by 3/3: 15/6

So 2 1/2 = 5/2 = 10/4 = 15/6. You can also write the mixed form with an equivalent fraction part: 2 1/2 = 2 2/4 = 2 3/6. The whole number part stays the same, and only the fraction part changes.

Simplest Form: The Smallest Equivalent Fraction

The simplest form of a fraction is the equivalent fraction with the smallest possible whole numbers. You get it by dividing the top and bottom by their greatest common factor (GCF).

Example: simplify 42/56

1. Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
2. Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
3. The greatest common factor is 14.
4. 42 ÷ 14 = 3 and 56 ÷ 14 = 4
5. Simplest form: 3/4

Every fraction has exactly one simplest form. That is why teachers usually ask for answers "in lowest terms." It gives everyone the same final answer. For big numbers, a GCF calculator saves listing every factor.

Where You Use Equivalent Fractions

Equivalent fractions are not just a worksheet topic. They are the tool behind adding fractions, comparing sizes, scaling recipes and reading measurements.

Adding and subtracting fractions with different denominators is where most students first need this skill for real. The fraction calculator shows each common-denominator step if you want to check your work.

When Students Learn Equivalent Fractions

In schools that follow the Common Core State Standards, equivalent fractions start in grade 3 and are extended in grade 4. Later grades rely on the skill for adding, subtracting and comparing fractions.

The grade 3 standards limit denominators to 2, 3, 4, 6 and 8. You can read the full wording in the Common Core grade 3 fraction standards. Individual states may word or number their standards differently.

Tips For Teaching Equivalent Fractions At Home

Kids understand equivalent fractions fastest when they can see and touch the pieces before they learn the multiplying rule. A few low-cost activities help.

• Fold paper. Fold a sheet in half and shade one half. Fold it again. The shaded part is now 2/4. Fold once more and it is 4/8. Nothing new was shaded.
• Use measuring cups. Fill a 1/2 cup using a 1/4 cup twice. Kids see that 2/4 = 1/2 with real food.
• Cut snacks. Cut one sandwich into 2 pieces and another into 4. Two of the four pieces match one of the halves.
• Build fraction strips. Cut paper strips of equal length into halves, thirds, fourths, sixths and eighths. Line them up to find matches.
• Move to numbers last. Once the pictures make sense, show that the top and bottom were both multiplied by the same number.

Ask "how do you know?" often. A child who can explain why 3/6 = 1/2 with a drawing understands more than one who only memorizes the rule.

Common Mistakes To Avoid

Most errors come from changing only one part of the fraction or from adding instead of multiplying. Here is how to spot and fix them.

A quick self-check catches most of these. After finding an equivalent fraction, simplify it. If you do not get your starting fraction back, something went wrong.

Practice Problems With Answers

Try these, then check your answers below. They get a little harder as you go.

1. Write two fractions equivalent to 2/7.
2. Fill in the blank: 5/6 = ?/36
3. Fill in the blank: 7/? = 28/32
4. Is 6/9 equal to 10/15?
5. Is 4/5 equal to 15/20?
6. Write 36/48 in simplest form.
7. Write 3 1/3 as two equivalent improper fractions.
8. Which is bigger, 2/3 or 7/10?

Answers

1. 4/14 and 6/21 (any pair made by multiplying by the same number works)
2. 30 (6 × 6 = 36, so 5 × 6 = 30)
3. 8 (7 × 4 = 28, so 32 ÷ 4 = 8)
4. Yes: 6 × 15 = 90 and 9 × 10 = 90
5. No: 4 × 20 = 80, but 5 × 15 = 75
6. 3/4 (divide both by 12)
7. 10/3 and 20/6
8. 7/10, because 2/3 = 20/30 and 7/10 = 21/30

Summary

Equivalent fractions name the same amount with different numbers, like 1/2 = 2/4 = 4/8. You make them by multiplying or dividing the top and bottom by the same nonzero number, which works because you are really multiplying by 1.

To test two fractions, cross multiply, simplify both, use a common denominator or compare decimals. The simplest form is the one equivalent fraction with the smallest numbers. The skill comes up whenever you add, compare or convert fractions.

Next step: work through the practice problems above, then try adding 2/3 + 1/4 by turning both into twelfths. Use the fraction simplifier to check that your answers are in lowest terms.

Frequently Asked Questions