What are equivalent fractions?

Equivalent fractions are different fractions that name the same amount or value. For example:

• 1/2, 2/4, and 3/6 are equivalent because each represents one-half.

Although the numerators and denominators look different, the actual size (value) of the fraction is the same.

Visual idea (easy picture you can draw)

• Draw a circle (like a pizza) and shade half of it — that's 1/2.
• Draw another circle divided into 4 equal slices and shade 2 slices — that's 2/4. It looks the same amount shaded.

This shows visually that 1/2 = 2/4.

How to make equivalent fractions (the rule)

Multiply or divide both the numerator (top number) and denominator (bottom number) by the same nonzero number.

• Multiplying by the same number gives a new fraction equal to the original.
• Dividing by the same number (a common factor) simplifies a fraction to an equivalent one.

Important: You may never multiply or divide only the numerator or only the denominator if you want the value to stay the same.

Examples — creating equivalent fractions

1) Start with 3/5. Multiply numerator and denominator by 2:

• (3 × 2) / (5 × 2) = 6/10. So 3/5 = 6/10.

2) Start with 4/6. Divide numerator and denominator by their common factor 2:

• (4 ÷ 2) / (6 ÷ 2) = 2/3. So 4/6 = 2/3.

Finding an equivalent fraction with a specific denominator

To write an equivalent fraction with a given denominator:

1. Find what number you multiply the original denominator by to get the new denominator.
2. Multiply the numerator by the same number.

Example: Convert 2/3 to a fraction with denominator 12.

• 3 × 4 = 12, so multiply numerator by 4: 2 × 4 = 8. Answer: 8/12.

How to check if two fractions are equivalent (cross-multiplication)

For fractions a/b and c/d, they are equivalent if and only if a × d = b × c.

Example: Are 2/3 and 8/12 equivalent?

• 2 × 12 = 24 and 3 × 8 = 24. Because the products are equal, the fractions are equivalent.

Number line idea

Place fractions on a number line between 0 and 1. If two fractions land on the same point, they are equivalent.

Example: Mark 1/2 and 2/4 on a number line divided into 4 equal parts — both fall at the same point in the middle.

Common mistakes to avoid

• Multiplying only the numerator or only the denominator. (Wrong: 1/2 → 2/2 is not equivalent to 1/2.)
• Using 0 as a multiplier or making a denominator 0 (denominator can never be 0).
• Adding or subtracting numbers to numerator and denominator to try to make equivalents (e.g., 1/2 → (1+1)/(2+1) = 2/3 is not equivalent).

Practice problems

1. Write two fractions equivalent to 1/3.
2. Write a fraction equivalent to 5/8 with denominator 32.
3. Simplify 9/12 to lowest terms.
4. Are 3/4 and 6/8 equivalent? Use cross-multiplication to show it.
5. Fill in the blank: 7/? is equivalent to 21/9.
6. True or False: 4/5 = 8/15.
7. Find an equivalent fraction to 2/9 by multiplying numerator and denominator by 3.
8. Simplify 14/21.

Answers

1. Two examples: 2/6 and 3/9 (also 4/12, etc.).
2. 5/8 → denominator 32: 8 × 4 = 32 so 5 × 4 = 20. Answer: 20/32.
3. 9/12. Divide numerator and denominator by 3 → (9 ÷ 3)/(12 ÷ 3) = 3/4.
4. Check 3/4 and 6/8 by cross-multiplying: 3 × 8 = 24 and 4 × 6 = 24. Because they match, they are equivalent.
5. 7/? = 21/9. Solve: multiply 7 by what gives 21? 7 × 3 = 21, so denominator must be 3 × ? = 9 → ? = 3. Answer: 7/3 = 21/9 (so blank is 3).
6. False. Check with cross-multiplication: 4 × 15 = 60, 5 × 8 = 40. Not equal, so not equivalent.
7. Multiply numerator and denominator of 2/9 by 3: (2 × 3)/(9 × 3) = 6/27.
8. Simplify 14/21. Divide both by 7 → (14 ÷ 7)/(21 ÷ 7) = 2/3.

Quick rules to remember

• To get equivalent fractions: multiply or divide numerator and denominator by the same nonzero number.
• To check equivalence: use cross-multiplication (a × d ?= b × c).
• To simplify: divide numerator and denominator by their greatest common factor.

Helpful tips:

• Always do the same operation (× or ÷) to both numerator and denominator.
• If you want a specific denominator, find the factor that changes the old denominator to the new one, then use that factor on the numerator.
• Use pictures (circles, bars, or number lines) when fractions feel confusing — they make equivalence obvious.
• Practice writing several equivalents for the same fraction to see patterns (e.g., 1/2 → 2/4 → 3/6 → 4/8).
• Keep an eye out for simplifying: if both numerator and denominator are even, you can divide both by 2.
• When in doubt, cross-multiply to test if two fractions are the same.

Good work learning equivalent fractions! Keep practicing with drawings and number lines — they make the idea stick.