Equivalent Fractions

Definition: Two or more fractions are called equivalent fractions if they represent the same part of a whole, even though their numerators and denominators are different.

For example:

• 1/2 = 2/4 = 3/6 = 4/8 = 5/10
• All these fractions represent the same value — one half.

Key idea:

• If you multiply both the numerator and denominator of a fraction by the same non-zero number, you get an equivalent fraction.
• If you divide both the numerator and denominator of a fraction by the same non-zero number, you get an equivalent fraction.
• The value of the fraction does not change.

Visual understanding:

Think of a chocolate bar. If you break it into 3 equal pieces and take 1 piece, you have 1/3 of the bar. Now break the same bar into 6 equal pieces and take 2 pieces. You still have the same amount of chocolate. So 1/3 = 2/6.

The idea behind equivalent fractions comes from a basic property of multiplication:

Multiplying by 1 does not change the value of a number.

Now, any number divided by itself equals 1. For example:

• 2/2 = 1
• 3/3 = 1
• 5/5 = 1
• 10/10 = 1

So when we multiply a fraction by 2/2 or 3/3 or 5/5, we are actually multiplying by 1. The value stays the same, but the numerator and denominator change.

Example:

• Start with 3/4
• Multiply by 2/2: (3 × 2) / (4 × 2) = 6/8
• We multiplied by 1 (since 2/2 = 1), so 3/4 = 6/8

Similarly, dividing both parts by a common factor is the reverse process. If 6/8 is given, we divide both by 2:

• (6 ÷ 2) / (8 ÷ 2) = 3/4

This is why the method works — you are always multiplying or dividing by a form of 1, so the fraction's value does not change.

Visual proof:

Draw a rectangle and shade 1 out of 3 equal parts. Now draw the same rectangle and divide each of those 3 parts into 2 equal parts. You now have 6 parts, and 2 of them are shaded. The shaded area is the same in both pictures. So 1/3 = 2/6.

There are different types of problems involving equivalent fractions:

Type 1: Find an equivalent fraction by multiplying

• Given a fraction, multiply numerator and denominator by the same number.
• Example: Find an equivalent fraction of 2/5 → Multiply by 3: (2 × 3)/(5 × 3) = 6/15

Type 2: Find an equivalent fraction by dividing (simplifying)

• Given a fraction, divide numerator and denominator by a common factor.
• Example: Simplify 12/18 → Divide by 6: (12 ÷ 6)/(18 ÷ 6) = 2/3

Type 3: Check if two fractions are equivalent

• Use cross-multiplication: if a × d = b × c, they are equivalent.
• Example: Are 3/5 and 9/15 equivalent? Check: 3 × 15 = 45, 5 × 9 = 45. Yes!

Type 4: Fill in the missing number

• Given one fraction equal to another with a blank, find the missing value.
• Example: 4/7 = __/21. Since 7 × 3 = 21, multiply numerator by 3: 4 × 3 = 12. Answer: 12/21.

Type 5: Write a family of equivalent fractions

• Write 4 or 5 equivalent fractions of a given fraction.
• Example: 1/4 → 2/8, 3/12, 4/16, 5/20

Equivalent fractions are used in many everyday situations:

• Cooking: A recipe calls for 1/2 cup of sugar. You only have a 1/4 cup measure. Since 1/2 = 2/4, you know to use two scoops of the 1/4 cup.
• Sharing equally: If you cut a cake into 6 pieces and give 2 pieces to your friend, that is 2/6 = 1/3 of the cake. Knowing equivalent fractions helps you understand the fair share.
• Comparing prices: If Shop A sells 2 kg of rice for Rs. 80 and Shop B sells 3 kg for Rs. 120, you can use equivalent fractions to compare which is cheaper.
• Measuring: A carpenter needs 3/4 of a metre of wood. His ruler marks only in eighths. Since 3/4 = 6/8, he measures 6 marks out of 8.
• Adding fractions: To add 1/3 + 1/4, you need a common denominator. You find equivalent fractions: 1/3 = 4/12 and 1/4 = 3/12. Now you can add: 4/12 + 3/12 = 7/12.
• Test scores: If you score 18/24 on a test and your friend scores 15/20, are the scores the same? Simplify: 18/24 = 3/4 and 15/20 = 3/4. Yes, they are equivalent!
• Money: Half a rupee is 50 paise. Quarter of a rupee is 25 paise. We know 50 paise = 2 × 25 paise, just like 1/2 = 2/4. Equivalent fractions help us understand these relationships.
• Map reading: If a map uses a scale where 1/4 inch represents 1 mile, then 2/8 inch also represents 1 mile. Knowing equivalent fractions helps you read maps correctly.