Operations on Fractions

Adding and subtracting fractions needs a common denominator; multiplying and dividing does not. This asymmetry is the source of nearly every fraction error in Classes 4 and 5, and CTET tests both the procedure and the reasoning behind it.

Addition and subtraction

Like fractions: add or subtract numerators, keep the denominator

2/7 + 3/7 = 5/7. For unlike fractions, first make the denominators the same using the LCM: 1/4 + 1/6, LCM of 4 and 6 is 12, so 3/12 + 2/12 = 5/12.

The reason is physical: you can only add pieces of the same size. Quarters and sixths must first be renamed as twelfths.

Multiplication

a/b x c/d = (a x c) / (b x d)

• 2/3 x 4/5 = 8/15. No common denominator is needed.
• 'Of' means multiply: 1/2 of 8 is 1/2 x 8 = 4.
• Multiplying by a proper fraction makes a number smaller, which contradicts what children learnt about whole numbers.
• Simplify before multiplying where possible: 3/4 x 8/9 becomes 1/1 x 2/3 = 2/3.

Division

a/b ÷ c/d = a/b x d/c (multiply by the reciprocal)

3/4 ÷ 1/2 = 3/4 x 2/1 = 6/4 = 3/2. Ask the meaning aloud: how many halves are there in three-quarters? One and a half. Dividing by a proper fraction makes a number larger, the second shock for a child.

The number line check

Every fraction operation can be checked on a number line, and this is the fastest way to catch a wrong answer. 1/2 + 1/3 must land between 1/2 and 1, so an answer of 2/5, which is less than a half, is visibly wrong. Teaching children to estimate the position before computing prevents most of the errors listed below.

Mixed numbers

Convert to improper fractions before any operation. 2 1/3 becomes (2 x 3 + 1)/3 = 7/3. Convert back at the end.

✗ 1/2 + 1/3 = 2/5, adding numerators and denominators.  |  ✓ 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

✗ Multiplication always makes a number bigger.  |  ✓ Multiplying by a fraction less than 1 makes it smaller.

Frequently asked questions

Why do we need a common denominator to add fractions?

Because only parts of the same size can be added. Quarters and sixths are different-sized pieces, so both must first be renamed as twelfths before the numerators can be combined.

Why does multiplying by a fraction make a number smaller?

Because multiplying by a proper fraction means taking a part of the number. Half of eight is four, so the result is smaller. Only multiplication by a number greater than one increases the value.

How do you divide one fraction by another?

Multiply the first fraction by the reciprocal of the second, that is, turn the second fraction upside down and multiply. Three-quarters divided by one-half becomes three-quarters times two, which is one and a half.