Brahmagupta’s formula for multiplying fractions

The idea

The product of two fractions is the product of the numerators over the product of the denominators: a/b × c/d = (a × c)/(b × d); a whole number is first written as a fraction with denominator 1.

Five twelfths of seven eighteenths

You know that 1/12 × 1/18 = 1/216: the whole is cut into 12 × 18 = 216 equal parts. Now we want 5/12 × 7/18.

How many of the 216 parts are shaded?

• 5 × 7 = 35
• 5 + 7 = 12
• 7

Hunt for pairs with the same picture

Make the same picture with different pairs of fractions. Watch the two lines under the picture: gold and all.

Tops multiply, bottoms multiply

Every pair you found has the same counts: the numerators multiply to the gold squares, and the denominators multiply to all the squares. For 5/12 × 7/18 that is 5 × 7 = 35 out of 12 × 18 = 216.

A whole number is first written as a fraction with denominator 1: 3 = 3/1. 3 × 3/4 = 3/1 × 3/4 = 9/4 3/5 × 4 = 3/5 × 4/1 = 12/5

The product of two fractions is the product of the numerators over the product of the denominators: a/b × c/d = (a × c)/(b × d) A whole number is first written as a fraction with denominator 1.

• Product | Numerators | Denominators
• 2/3 × 3/8 | 2 × 3 = 6 | 3 × 8 = 24
• 5/12 × 7/18 | 5 × 7 = 35 | 12 × 18 = 216
• 3 × 3/4 | 3 × 3 = 9 | 1 × 4 = 4

Notes

The product of two fractions is the product of the numerators over the product of the denominators: a/b × c/d = (a × c)/(b × d) A whole number is first written as a fraction with denominator 1.

Check yourself

Find 3/5 × 4/7. Type a fraction like 2/9.

Answer: 12/35

3 × 4 = 12 and 5 × 7 = 35. 3/5 × 4/7 = 12/35

Find 3 × 3/4. Write 3 as a fraction first. Type a fraction or mixed number, like 2 1/10.

Answer: 2 1/4

3 = 3/1, so 3/1 × 3/4 = 9/4 = 2 1/4

Which is 7/9 × 2/3?

Which line works out 4/5 × 3 correctly?

• 9/12. That adds the tops (7 + 2) and the bottoms (9 + 3). They must be multiplied.
• 14/27 — correct. Yes! 7 × 2 = 14 and 9 × 3 = 27.
• 14/9. The numerators were multiplied, but the denominators were not: 9 × 3 = 27.
• 4/5 × 3 = (4 × 3)/(5 × 3) = 12/15. The 3 was multiplied into the denominator too. That gives 12/15 = 4/5, the original fraction!
• 4/5 × 3 = 4/(5 × 3) = 4/15. The whole number 3 multiplies the numerator, not the denominator: 3 = 3/1.
• 4/5 × 3/1 = (4 × 3)/(5 × 1) = 12/5 — correct. Yes! The whole number 3 becomes 3/1, so the denominator stays 5.