Is the product bigger?

The idea

If both numbers exceed 1, the product is greater than both; if both lie between 0 and 1, it is less than both; if one lies between 0 and 1 and the other exceeds 1, it lies between them.

Smaller after multiplying?

With counting numbers a product is never smaller than the numbers multiplied: 3 × 8 = 24. Now look at decimals. 2.25 × 8 = 18 is bigger than both. 0.25 × 8 = 2 is bigger than 0.25 but smaller than 8.

Where will the product 0.25 × 0.8 fall, compared with 0.25 and 0.8?

• Bigger than both
• Between the two
• Smaller than both

Stretch or shrink?

Move the sliders. The bars show the two numbers and their product on the same scale. The red dashed line marks 1. Find an example for each situation in the list.

Above 1 stretches, below 1 shrinks

Multiplying by a number more than 1 makes things bigger. Multiplying by a number between 0 and 1 takes only a part, so it makes things smaller.

If both numbers are more than 1, the product is greater than both. If both are between 0 and 1, it is less than both. If one is between 0 and 1 and the other is more than 1, the product lies between them.

• Situation | Example
• Both numbers are more than 1 | 3.4 × 6.5 = 22.1: greater than both
• Both are between 0 and 1 | 0.75 × 0.4 = 0.3: less than both
• One of each | 0.75 × 5 = 3.75: between 0.75 and 5

Notes

Both numbers more than 1: product greater than both. Both between 0 and 1: product less than both. One of each: product between them.

Check yourself

Without multiplying: 0.9 × 0.8 is ...

Without multiplying: is 12.5 × 1.2 greater than 12.5?

Radha says, “Multiplying always makes a number bigger.” Which product shows she is wrong?

What is 0.6 × 0.5? It must be less than both 0.6 and 0.5.

Answer: 0.3

6 × 5 = 30, 2 places: 0.30 = 0.3, which is less than 0.5 and 0.6.

Which of these is sure to be less than 1, without multiplying?

7.5 × 0.9 lies ...

• greater than both 0.9 and 0.8. That happens only when both numbers are more than 1.
• less than both 0.9 and 0.8 — correct. Both numbers are between 0 and 1, so the product is less than both. (It is 0.72.)
• between 0.8 and 0.9. That happens only when one number is below 1 and the other is above 1.
• Yes. Both numbers are more than 1, so the product is greater than both. — correct. It is 15, which is greater than 12.5 and 1.2.
• No. Multiplying by a decimal always makes a number smaller.. Only a decimal between 0 and 1 makes things smaller. 1.2 is more than 1.
• It cannot be said without multiplying.. We can say it: both numbers are more than 1.
• 3 × 8 = 24, which is more than 8. This fits Radha’s claim. We need an example that breaks it.
• 1.5 × 2 = 3, which is more than 2. This also fits her claim. We need a factor between 0 and 1.
• 0.5 × 8 = 4, which is less than 8 — correct. 0.5 is between 0 and 1, so taking half of 8 gives something smaller than 8.
• 0.7 × 6. One number is below 1 and the other above 1. The product is between 0.7 and 6, which does not decide it. (It is 4.2.)
• 0.7 × 0.6 — correct. Both numbers are between 0 and 1, so the product is less than both. So it is less than 1.
• 7 × 0.6. One number is below 1 and the other above 1. The product lies between 0.6 and 7: not sure to be below 1. (It is 4.2.)
• between 0.9 and 7.5 — correct. One number is below 1 and the other is above 1, so the product lies between them. (It is 6.75.)
• above 7.5. To be above both numbers, both would have to be more than 1.
• below 0.9. To be below both numbers, both would have to be between 0 and 1.