EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Place value and estimating
Number · Lesson 2 of 7
Last Lesson and Before
Answer each one, then check.
1. Round 3.846 to 1 decimal place.
3.8
2. Work out 4500 ÷ 100.
45
3. Work out 0.3 × 0.2.
0.06
4. Work out 7 ÷ 0.1.
70 - dividing by 0.1 is the same as multiplying by 10.
5. Last lesson: list the meals from 2 starters (A, B) and 2 mains (X, Y).
AX, AY, BX, BY
Learning Objectives
1. Use a known calculation to work out related calculations.
2. Round numbers to decimal places and significant figures.
3. Estimate the answer to a calculation by rounding to 1 significant figure.
4. Estimate square roots of numbers that are not square numbers.
5. Decide whether an estimate is too big or too small.
Place Value
The value of a digit depends on its position.
▸ Multiplying by 10. Every digit moves one place to the left: 3.7 × 10 = 37.
▸ Dividing by 10. Every digit moves one place to the right: 3.7 ÷ 10 = 0.37.
▸ Multiplying by 0.1. The same as dividing by 10: 45 × 0.1 = 4.5.
▸ Dividing by 0.1. The same as multiplying by 10: 45 ÷ 0.1 = 450.
One Fact, Many Answers
Start from 38 × 142 = 5396 and use place value.
|
Calculation |
Answer |
Why |
|---|---|---|
|
38 × 142 |
5396 |
The fact we are given |
|
3.8 × 142 |
539.6 |
One number is 10 times smaller, so the answer is 10 times smaller |
|
3.8 × 1.42 |
5.396 |
10 times smaller and 100 times smaller: 1000 times smaller |
|
0.38 × 14.2 |
5.396 |
100 times smaller and 10 times smaller: 1000 times smaller |
|
5396 ÷ 142 |
38 |
Division is the inverse of multiplication |
|
5396 ÷ 14.2 |
380 |
Dividing by a number 10 times smaller gives an answer 10 times bigger |
|
53.96 ÷ 3.8 |
14.2 |
5396 ÷ 38 = 142, and both numbers are 100 times smaller |
Using a Known Fact
|
Given that 23 × 47 = 1081, work out (a) 2.3 × 0.47 and (b) 108.1 ÷ 4.7. |
1. (a) Compare with the fact: 2.3 is 23 ÷ 10 and 0.47 is 47 ÷ 100
÷ 10 and ÷ 100 make ÷ 1000
2. So divide the answer by 1000
1081 ÷ 1000 = 1.081
3. (b) From the fact, 1081 ÷ 47 = 23
division undoes multiplication
4. 108.1 is 1081 ÷ 10 and 4.7 is 47 ÷ 10
both divided by 10
5. Dividing both numbers by 10 does not change the answer
108.1 ÷ 4.7 = 23
Answer: (a) 1.081 (b) 23
PART ONE
Rounding
Decimal places and significant figures.
Significant Figures
The first significant figure is the first digit that is not zero.
▸ Counting. 4 is the first significant figure of 45 678 and of 0.004 12.
▸ Zeros in the middle count. 7.0496 has 5 significant figures; the 0 counts.
▸ Zeros at the start do not. 0.003 082 has 4 significant figures: 3, 0, 8, 2.
▸ Keep the place value. 45 678 to 2 significant figures is 46 000, not 46.
Rounding to Significant Figures
|
Number |
1 s.f. |
2 s.f. |
3 s.f. |
|---|---|---|---|
|
45 678 |
50 000 |
46 000 |
45 700 |
|
0.003 082 |
0.003 |
0.0031 |
0.003 08 |
|
7.0496 |
7 |
7.0 |
7.05 |
|
299 512 |
300 000 |
300 000 |
300 000 |
|
0.098 76 |
0.1 |
0.099 |
0.0988 |
PART TWO
Estimating
A rough answer that tells you if the real one is sensible.
How to Estimate
To estimate, round every number to 1 significant figure, then work out the calculation.
▸ Round first. Round each number to 1 s.f. before doing anything else.
▸ Never round to 0. 0.048 rounds to 0.05, not 0.
▸ Dividing by a decimal. 150 ÷ 0.5 = 300: dividing by 0.5 is the same as doubling.
▸ Show it. Write the rounded calculation down: the method marks are for the rounded numbers, not just the answer. Use ≈ for "is approximately equal to".
Estimating a Calculation
|
Work out an estimate for (48.7 × 3.14)/0.52 |
1. Round each number to 1 significant figure
48.7 ≈ 50, 3.14 ≈ 3, 0.52 ≈ 0.5
2. Write the rounded calculation
(50 × 3)/0.5
3. Work out the top
50 × 3 = 150
4. Divide by 0.5 (the same as doubling)
150 ÷ 0.5 = 300
5. Compare with the calculator
The exact answer is 294.07…, close to 300
Answer: ≈ 300
Too Big or Too Small?
Look at which way each number was rounded.
▸ Multiplying. Rounding the numbers UP gives an overestimate; rounding them DOWN gives an underestimate.
▸ Dividing by a number rounded DOWN. The answer gets bigger: an overestimate.
▸ Dividing by a number rounded UP. The answer gets smaller: an underestimate.
▸ Mixed rounding. If some numbers went up and some went down, you cannot always tell without working it out.
Estimating a Square Root
40 is not a square number, so √40 lies between two whole numbers.
√36 = 6 ——▸ √49 = 7
1. √36 = 6. The square number just below 40
2. √40 ≈ 6.3. 40 is 4 of the 13 steps from 36 to 49, so about 0.3 of the way
3. √42.25 = 6.5. Halfway between 6 and 7 - 40 is below this
4. √49 = 7. The square number just above 40
Estimating a Square Root
|
Estimate √70 to 1 decimal place without a calculator. |
1. Find the square numbers either side of 70
64 = 8² and 81 = 9²
2. So √70 is between 8 and 9
8 < √70 < 9
3. How far is 70 from 64, out of the gap from 64 to 81?
6 out of 17
4. 6 out of 17 is a bit more than a third
about 0.35
5. Check by squaring
8.4² = 70.56, which is close to 70
Answer: √70 ≈ 8.4 (the exact value is 8.366…)
Estimating a Crowd
Nobody counts a crowd one person at a time.
▸ Count a sample. Count the people in one block of seats: about 400.
▸ Count the blocks. The stadium has about 150 blocks.
▸ Multiply. 400 × 150 = 60 000.
▸ Report it sensibly. "About 60 000" - rounded to 1 significant figure, because the inputs were only estimates.
Case Study
|
CASE STUDY Enrico Fermi and "Fermi Problems" The Italian-American physicist Enrico Fermi, who won the 1938 Nobel Prize in Physics, was famous for making quick, surprisingly accurate estimates from almost no information. He liked to ask his students questions such as "How many piano tuners are there in Chicago?" and expected them to reason it out: the population of the city, how many homes have a piano, how often a piano is tuned, and how many pianos one tuner can tune in a year. Each number is rounded, but the errors tend to cancel out, and the answer is usually the right size. Questions like this are now called Fermi problems. |
|
1938 Fermi wins the Nobel Prize in Physics |
Key Terms
|
Place value The value of a digit, which depends on its position in the number. |
Decimal place (d.p.) A digit after the decimal point. |
|
Significant figure (s.f.) Any digit of a number starting from the first non-zero digit. |
Estimate An approximate answer found by rounding the numbers first. |
|
Approximately equal to The symbol ≈, used for an estimate. |
Overestimate An estimate bigger than the exact answer. |
|
Underestimate An estimate smaller than the exact answer. |
|
Your Task: Estimation Station
12 minutes
|
Estimate each calculation, then use a calculator to find the exact answer and say whether your estimate was over or under. (a) 6.8 × 41.2 (b) 897 ÷ 2.9 (c) (19.6 × 0.48)/0.21 (d) √(98.6 + 23.1) (e) 3.14 × 7.9² 1. Estimate first, writing the rounded calculation. 2. Then use a calculator. 3. Say whether the estimate was over or under, and why. |
A good answer shows: (a) 7 × 40 = 280 (exact 280.16). (b) 900 ÷ 3 = 300 (exact 309.3). (c) (20 × 0.5)/0.2 = 50 (exact 44.8). (d) √120 ≈ 11 (exact 11.03). (e) 3 × 8² = 192 (exact 195.97).
Can I...?
☐ Use a known fact to work out related calculations.
☐ Multiply and divide by 0.1 and 0.01.
☐ Round to a given number of decimal places.
☐ Round to a given number of significant figures.
☐ Estimate a calculation by rounding to 1 s.f.
☐ Divide by a decimal such as 0.5 or 0.2.
☐ Estimate a square root.
☐ Say whether an estimate is over or under.
Summary
✓ A known fact plus place value gives many related answers.
✓ Significant figures start at the first non-zero digit; keep the place value when rounding.
✓ Estimate by rounding each number to 1 significant figure.
✓ Estimate square roots using the square numbers either side.
✓ Over or under: look at which way each number was rounded.
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EXAM FOCUS Work out an estimate for (3.78 × 48.2)/0.52 (3 marks) Write the rounded numbers down before you calculate. An answer with no rounded calculation shown can score zero, even if it is right. |