Multiply fractions as parts of parts

A fraction of a fraction

Draw a rectangle and divide it into four equal columns. Shade three columns to show 3/4. Divide the same rectangle into three equal rows. Take two of those rows within the shaded part. Six of the twelve equal small rectangles are selected:

The twelve parts come from dividing each of four columns into three. The six selected parts come from selecting two rows in each of three columns. This explains why both numerators and both denominators multiply.

Use the same rule for more than one whole

3/5 × 4 = 12/5 = 22/5: four lots of three fifths contain twelve fifths. The denominator stays five because each part is still one fifth of the same whole.

5/4 × 3 = 15/4 = 33/4: three lots of five quarters contain fifteen quarters.

You can simplify factors first: 3/8 × 4/9 = 1/6 because 3/9 = 1/3 and 4/8 = 1/2. This divides a numerator and a denominator by the same non-zero factor; it does not cancel arbitrary digits.

Check your understanding 1 Must multiplication make a number larger?

Find 3/5 × 5/6. Compare the result with 3/5 and explain without relying only on a rule.

Your product, comparison and explanation

Show answer

The product is 15/30 = 1/2. Taking five sixths of a positive amount keeps less than all of it, so the result is less than 3/5. Multiplying by a fraction below one can make an amount smaller.

A proper and an improper factor

The product need not be below one. Half of 3/2 is 3/4, half of 2/1 is 1, and half of 5/2 is 5/4 = 11/4.

In each case taking half reduces the starting positive amount. Whether the result is below, equal to or above one also depends on how large that starting amount is.

Transfer to a whole-number amount

Check your understanding 2 Keep the whole unit with each answer

A ribbon is 7/4 m long. Find the total length of four such ribbons, then find half the length of one ribbon.

Your total length, half-length and checks

Show answer

Four lengths total 7/4 × 4 = 7 m. Half of one is 1/2 × 7/4 = 7/8 m. Check 4 × 13/4 = 7 m and 2 × 7/8 = 7/4 m. The second operation scales down rather than up; it uses one ribbon, not the four-ribbon total.