Comparing a product with the numbers multiplied

The idea

When one of the numbers being multiplied is between 0 and 1, the product is less than the other number; when one of them is greater than 1, the product is greater than the other number.

Is the product always bigger?

Multiply 3 × 5 = 15. The product is bigger than both 3 and 5. Now try 1/4 × 8 = 2.

Compared with the two numbers 1/4 and 8, the product 2 is…

• bigger than both
• smaller than both
• between them

Move the numbers, fill the table

Drag x and y (or tap ◀ ▶). The green marker is the product. Find out what happens in each box of the table.

Below 1 shrinks, above 1 grows

A number between 0 and 1 takes only a part of the other number, so the product is smaller than it. A number greater than 1 takes more than the whole, so the product is bigger.

When one of the numbers being multiplied is between 0 and 1, the product is less than the other number. When one of them is greater than 1, the product is greater than the other number.

• Multiply | The product is
• 4/3 × 4 | 16/3: bigger than both (both above 1)
• 3/4 × 2/5 | 3/10: smaller than both (both below 1)
• 3/4 × 5 | 15/4: between 3/4 and 5 (one of each)

Notes

When one of the numbers being multiplied is between 0 and 1, the product is less than the other number. When one of them is greater than 1, the product is greater than the other number.

Check yourself

Without working it out: 5/6 × 12 is…

2/3 × 5/4: one number is below 1 and one is above 1. The product is…

Find 3/5 × 20. Your answer must be less than 20. Type a whole number.

Answer: 12

20 ÷ 5 = 4 and 3 × 4 = 12. It is less than 20, as expected.

Mina says: “Multiplication always makes a number bigger.” Which example shows she is wrong?

• more than 12. Multiplication does not always make numbers bigger. A number below 1 makes the other number smaller.
• less than 12 — correct. Yes! 5/6 is between 0 and 1, so the product is less than the other number, 12.
• equal to 12. Only multiplying by 1 leaves a number unchanged. 5/6 is less than 1.
• bigger than both. That happens only when both numbers are above 1.
• smaller than both. That happens only when both numbers are below 1.
• between 2/3 and 5/4 — correct. Yes! It is less than 5/4 (because of 2/3) and more than 2/3 (because of 5/4). It is 5/6.
• 3 × 5 = 15. Here the product is bigger than both numbers, which fits what Mina says.
• 1/4 × 8 = 2 — correct. Yes! 2 is smaller than 8: 1/4 is below 1.
• 5/4 × 4 = 5. Here the product is bigger than both numbers (both are above 1). It fits what Mina says.