To estimate a standard-form answer, round each number to one significant figure, then deal with the digits and the powers of ten separately. The digits give you the front of the answer, and the index laws give you the power. It is quicker than keying the whole calculation in twice.
This lesson uses skills from applying index laws and comparing quantities with different powers of ten.
How do I estimate step by step?
- Round each decimal part to 1 significant figure. Leave the powers of ten alone.
- Combine the digits using the operation in the question.
- Combine the powers of ten using the index laws: add for multiplication, subtract for division.
- Put the two parts together and convert to standard form if needed.
- Compare with the exact answer. The power of ten should match, and the first digits should be close.
Worked example
Estimate (4.8 × 106) × (2.1 × 10−3) ÷ (5.9 × 102). Then find the exact value.
Step 1, round. 4.8 → 5, 2.1 → 2, 5.9 → 6. The estimate is (5 × 106) × (2 × 10−3) ÷ (6 × 102).
Step 2, digits. 5 × 2 ÷ 6 = 10/6 ≈ 1.7.
Step 3, powers. 106 × 10−3 ÷ 102 = 106 − 3 − 2 = 101.
Step 4, estimate. About 1.7 × 101 = 17.
Step 5, exact value. 4.8 × 2.1 = 10.08, and 10.08 ÷ 5.9 = 1.7085… The powers give 106−3−2 = 101. So the exact answer is 1.7085… × 101 ≈ 1.71 × 101, which is 17.1 to 3 significant figures. It agrees with the estimate of 17.
The mistake to watch for
The usual slip is a keying error that the student cannot see on the display.
Mistaken calculator entry: 4.8 × 106 × 2.1 × 10−3 ÷ 5.9 × 102
Without brackets around the last term, the calculator divides by 5.9 and then multiplies by 102. The display shows about 1.71 × 105.
The correction: compare with the estimate of about 17. A result of 1.71 × 105 is ten thousand times too big, so something is wrong with the entry. Put brackets around the whole denominator, or use the fraction key, and the display gives 17.1 (3 s.f.), which fits.
Check yourself
Try these without a calculator, then open each answer.
1. Estimate (3.9 × 105) × (2.1 × 104) in standard form.
Show answer
3.9 → 4 and 2.1 → 2, so 4 × 2 = 8. Powers: 105+4 = 109. Estimate: 8 × 109. The exact value is 8.19 × 109.
2. Estimate (9.7 × 10−3) ÷ (2.1 × 10−6).
Show answer
9.7 → 10 and 2.1 → 2, so 10 ÷ 2 = 5. Powers: 10−3−(−6) = 103. Estimate: 5 × 103. The exact value is about 4.62 × 103.
3. A student says (6.2 × 108) × (3.1 × 10−5) = 1.922 × 1013. Use an estimate to show it is wrong, and give the correct value.
Show answer
Estimate: 6 × 3 = 18, and 108−5 = 103, so about 18 × 103 = 1.8 × 104. The student’s power of 13 is far too big. The exact working is 6.2 × 3.1 = 19.22, so 1.922 × 104. The student added 8 and 5 instead of combining 8 and −5.
Where this leads next
Estimating links directly to precision, bounds and measurement, where you decide how accurate an answer needs to be. To practise the whole module together, use the mixed practice set. The non-calculator working trainer supports the same habit of checking by hand.
If a calculator slip keeps costing you marks and you only notice afterwards, online one-to-one Mathematics tuition gives you a teacher who can build the checking routine with you.