Pieces of the same size

The idea

To add or subtract two rational numbers, first write them with the same denominator b, then use a/b + c/b = (a + c)/b and a/b – c/b = (a – c)/b.

Different pieces

Brahmagupta also gave rules for adding and subtracting fractions. Look at a half and a third. They are made of different size pieces.

What is 1/2 + 1/3?

• 2/5
• 5/6
• 1/6

Same size first

First cut both bars into pieces of the same size: choose a common denominator b. Then add or subtract the tops.

Same denominator

To add or subtract two rational numbers, first write them with the same denominator b. Then a/b + c/b = (a + c)/b and a/b − c/b = (a − c)/b.

The pieces are now the same size, so you only count them: the tops add or subtract and the bottom stays b.

Notes

To add or subtract rational numbers, write them with the same denominator b. Then a/b + c/b = (a + c)/b and a/b − c/b = (a − c)/b.

Check yourself

What is 2/7 + 3/7? The bottom stays 7. What is the top?

Answer: 5

2 + 3 = 5, so 2/7 + 3/7 = 5/7.

What is the correct first step for 1/4 + 1/6?

Subtract 5/6 − 1/3. Write 1/3 as 2/6. What is the top of 5/6 − 2/6?

Answer: 3

5 − 2 = 3, so 5/6 − 1/3 = 3/6 = 1/2.

The rules work for negative fractions too. What is −2/7 + 5/7?

• Add the tops and add the bottoms: 2/10. 2/10 is smaller than 1/4 itself. A sum must be bigger than each part.
• Write both with the same denominator, such as 12 — correct. Yes. 1/4 = 3/12 and 1/6 = 2/12, so the sum is 5/12.
• Multiply the two fractions. Multiplying is a different operation. To add, make the pieces the same size.
• −3/7. The tops are −2 and 5. Add them: −2 + 5 = 3, not −3.
• 3/7 — correct. Yes. The tops are −2 and 5, and −2 + 5 = 3. The bottom stays 7.
• 7/14. The bottom does not change when the pieces are the same size.