Fractions:
add, subtract, multiply and divide

What a fraction tells you

A fraction describes part of a whole. The denominator (bottom number) says how many equal parts the whole is cut into, and the numerator (top number) says how many of those parts you have. So 3/4 means three parts out of four.

Different fractions can describe the same amount. 1/2, 2/4 and 50/100 are equivalent because you can multiply or divide the top and bottom by the same number without changing the value. Every operation below depends on that one idea.

A fraction is also a division. 3/4 is 3 ÷ 4 = 0.75, and 3/4 of ₹2,500 is ₹1,875, the same as 75% of 2,500. The percentage-of-a-number calculator on this page is a quick way to check results like that.

Rules for adding, subtracting, multiplying and dividing fractions

After any operation, simplify by dividing the top and bottom by their greatest common factor. 18/24 becomes 3/4 once both are divided by 6.

1. 1List multiples of each denominator until one appears in both lists. For 4 and 6: 4, 8, 12 and 6, 12, so the common denominator is 12.
2. 2Multiply the top and bottom of each fraction by whatever turns its denominator into 12.
3. 3Add or subtract the new numerators and keep 12 underneath.
4. 4Simplify, and convert to a mixed number if the answer is greater than 1.

Example: 3/4 + 5/6

In decimals that is 0.75 + 0.8333… = 1.5833…, which matches 1 7/12. Checking in decimals is a good habit because it catches a wrong denominator immediately.

Estimating first helps as well. Both 3/4 and 5/6 are more than 1/2 and less than 1, so their sum must lie between 1 and 2. An answer of 8/10 fails that test at once, while 1 7/12 passes.

Notice that the denominator did not change when the fractions were added. Only the numerators were added. Adding the denominators as well, which gives 8/10, is the most common error.

All four operations side by side

For multiplication, you can cancel before you multiply: in 3/5 × 10/9, the 3 and the 9 share a factor of 3, and the 5 and the 10 share a factor of 5. Cancelling first gives 1/1 × 2/3 = 2/3 with smaller numbers.

Fractions of an amount and scaling a recipe

"Of" means multiply. To find a fraction of a quantity, multiply the quantity by the numerator and divide by the denominator, in either order.

• 2/5 of ₹3,500: 3,500 ÷ 5 = 700, then × 2 = ₹1,400.
• 3/8 of 2 kg: 2,000 g ÷ 8 = 250 g, then × 3 = 750 g.
• 7/10 of 45 minutes: 45 × 7 = 315, then ÷ 10 = 31.5 minutes.

Dividing first keeps the numbers small and often shows immediately whether the answer will be a whole number. If the fraction is bigger than 1, such as 5/4 of ₹800, the result is more than the original: 800 ÷ 4 × 5 = ₹1,000.

A laddoo recipe uses 3/4 cup of sugar for 12 pieces. To make 18 pieces, scale by 18/12 = 3/2. Multiply: 3/4 × 3/2 = 9/8 = 1 1/8 cups.

Scaling every ingredient by the same fraction keeps the recipe in balance. Work out the scale factor once, write it down, and apply it to each line. That is quicker and safer than recalculating from scratch for each ingredient.

Subtraction works the same way, as the leftover case shows. You have 2/3 of a packet of flour and use 1/4 of the full packet. What remains is 2/3 − 1/4 = 8/12 − 3/12 = 5/12 of a packet. Written as a check: 5/12 + 3/12 = 8/12 = 2/3, which returns the amount you started with.

How to compare and order fractions

To find which of two fractions is larger, cross-multiply. For 5/8 and 3/5, compute 5 × 5 = 25 and 3 × 8 = 24. The side with the larger product is the larger fraction, so 5/8 is bigger. Check: 0.625 against 0.6.

Another route is to convert both to the same denominator, or to decimals. Cross-multiplying is faster by hand because it needs no common denominator. For three or more fractions, convert them all to decimals and sort.

A benchmark trick is faster still. Compare each fraction to 1/2: a fraction is more than 1/2 when its numerator is more than half the denominator. 5/8 is above 1/2 (5 is more than 4) and 3/7 is below it (3 is less than 3.5), so 5/8 is larger without any further working.

Common fraction mistakes

• Adding numerators and denominators separately: 1/2 + 1/3 is 5/6, not 2/5.
• Flipping the first fraction when dividing instead of the second.
• Using a common denominator for multiplication. It is unnecessary and only makes the numbers larger.
• Leaving the answer unsimplified, or leaving an improper fraction when a mixed number was asked for.
• Treating 1/3 as 0.33 midway through a calculation, which accumulates error.
• Forgetting that a whole number is a fraction too: 3 is 3/1, so 3 × 2/5 is 6/5, not 6/15.

If a problem includes a mixed number such as 2 1/2, convert it to an improper fraction first. The mixed fraction guide shows how. To change a final answer into a decimal or back again, use the fraction to decimal and decimal to fraction guides.