Multiplying Decimals by Estimating
Every question with a fully worked solution.
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Theory
An estimate tells you roughly how big a product is, so it shows you where the decimal point goes. If you know a whole-number product such as \(69 \times 18 = 1242\), you can find related decimal products such as \(6.9 \times 1.8\). This Year 7 maths page shows how to estimate, use a known product and check that the point is in the right place.
To estimate a product, round each number to an easy number, often the nearest whole number, and multiply: \(4.9 \times 6.2 \approx 5 \times 6 = 30\).
If you know the answer to a whole-number multiplication, the digits of a related decimal product are the same. Only the position of the point changes.
Write each decimal using a whole number and a power of \(10\): \(0.63 = 63 \div 100\) and \(3.8 = 38 \div 10\). Then divide the known product by the same powers of \(10\).
Multiplying by a number less than 1 gives a smaller answer: \(18 \times 0.6 < 18\). Multiplying by a number greater than \(1\) gives a bigger answer.
One known product gives many answers. The digits stay the same:
| Product | Estimate | Answer |
|---|---|---|
| \(69 \times 18\) | \(70 \times 20 = 1400\) | \(1242\) (known) |
| \(0.69 \times 18\) | \(0.7 \times 20 = 14\) | \(12.42\) |
| \(6.9 \times 180\) | \(7 \times 200 = 1400\) | \(1242\) |
| \(6.9 \times 1.8\) | \(7 \times 2 = 14\) | \(12.42\) |
| \(690 \times 1.8\) | \(700 \times 2 = 1400\) | \(1242\) |
How to use a known product
- Estimate the answer by rounding each number.
- Write each decimal as the whole number \(\div 10\), \(\div 100\), \(\ldots\)
- Divide (or multiply) the known product by the same powers of \(10\).
- Compare with the estimate.
| \(7 \times 4\) | \(=\) | \(28\) |
\(7.2 \approx 7\) and \(3.9 \approx 4\). The estimate \(28\) is very close to \(28.08\).
(i) \(6.4 \times 37\)
(ii) \(0.64 \times 3.7\)
(iii) \(640 \times 0.37\)
| \(6.4 \times 37\) | \(=\) | \(2368 \div 10 = 236.8\) |
| \(0.64 \times 3.7\) | \(=\) | \(2368 \div 1000\) |
| \(\) | \(=\) | \(2.368\) |
| \(640 \times 0.37\) | \(=\) | \(2368 \div 10 = 236.8\) |
Estimates: \(6 \times 40 = 240\), \(0.6 \times 4 = 2.4\) and \(600 \times 0.4 = 240\), so all three are sensible.
| \(6.9 \times 1.8\) | \(=\) | \(1242 \div 100\) |
| \(\) | \(=\) | \(12.42\) |
| \(690 \times 1.8\) | \(=\) | \(1242 \times 10 \div 10\) |
| \(\) | \(=\) | \(1242\) |
Estimates \(7 \times 2 = 14\) and \(700 \times 2 = 1400\) agree.
| \(2.6 \times 6.9\) | \(=\) | \(1794 \div 100\) |
| \(\) | \(=\) | \(17.94\) |
Estimate \(3 \times 7 = 21\); \(17.94\) is close (not \(1.794\) or \(179.4\)). Tia pays \(\$17.94\).
(i) \(18 \times 0.45\)
(ii) \(0.18 \times 4.5\)
(iii) \(180 \times 0.045\)
(i) \(\times 10\) then \(\div 10\) cancel: \(8.1\).
(ii) Only one number is \(\div 10\): \(0.81\).
(iii) \(\times 100\) then \(\div 100\) cancel: \(8.1\).
(i) yes
(ii) no
(iii) yes
| \(5 \times 0.2\) | \(=\) | \(1\) |
| \(810 \div 1000\) | \(=\) | \(0.81\) |
The estimate is \(1\), so \(8.1\) is too big. The answer has \(1 + 2 = 3\) decimal places: \(0.81\).
Common pitfalls
Frequently asked questions
Why estimate before multiplying decimals?
The estimate shows how big the answer should be, so you can see where the decimal point goes.
How do you use 69 × 18 = 1242 to find 6.9 × 1.8?
There are two decimal places in the question, so divide by \(100\): \(12.42\).
What if my answer is far from the estimate?
The decimal point is probably in the wrong place. Count the decimal places again.
Does multiplying always make a number bigger?
No. Multiplying by a number less than \(1\) makes a positive number smaller.
How do you round for an estimate?
Round each number to one easy figure, often the nearest whole number. \(4.9 \times 6.2 \approx 5 \times 6\).