C1.9 – Estimation

Look at the digit one place to the right of where you are rounding:

No. Leading zeros do not count as significant figures.

1. We keep 1 digit after the decimal point: 4
2. Check the next digit: 7  →  $7 \geq 5$, so round up
3. $4$ becomes $5$

1. We keep 2 digits after the decimal point: 4 and 7
2. Check the next digit: 3  →  $3 < 5$, so round down
3. The 7 stays as it is

1. The first significant figure is 8 (the thousands digit)
2. Check the next digit: 3  →  $3 < 5$, so round down
3. Replace the remaining digits with zeros

1. Leading zeros do not count. The first significant figure is 5
2. Check the next digit: 2  →  $2 < 5$, so round down
3. The 5 stays; the 2 is removed

1. Keep 2 decimal digits: 4 and 8
2. Check the next digit: 5  →  $5 \geq 5$, so round up
3. $8$ becomes $9$

1. Keep 2 decimal digits: 9 and 9
2. Check the next digit: 6  →  $6 \geq 5$, so round up
3. Add 1 to the 2nd decimal digit: $9 + 1 = 10$ → write $0$, carry $1$
4. Add the carry to the 1st decimal digit: $9 + 1 = 10$ → write $0$, carry $1$
5. Add the carry to the integer part: $9 + 1 = 10$

1. The first two significant figures are 4 and 7
2. Check the next digit: 6  →  $6 \geq 5$, so round up
3. $7$ becomes $8$. Replace the remaining digits with zeros

1. Leading zeros do not count. The first significant figure is 3, the second is 8
2. Check the next digit: 4  →  $4 < 5$, so round down
3. The 8 stays; all digits after are removed

1. Round each number to 1 s.f.:
   • $41.3 \approx 40$   (first s.f. = 4, next digit = 1 < 5, round down)
   • $9.79 \approx 10$   (first s.f. = 9, next digit = 7 ≥ 5, round up)
   • $0.765 \approx 0.8$   (first s.f. = 7, next digit = 6 ≥ 5, round up)
2. Substitute: $$\frac{40}{10 \times 0.8} = \frac{40}{8} = 5$$

1. Round each number to 1 s.f.:
   • $52.4 \approx 50$   (first s.f. = 5, next digit = 2 < 5, round down)
   • $3.86 \approx 4$   (first s.f. = 3, next digit = 8 ≥ 5, round up)
   • $19.3 \approx 20$   (first s.f. = 1, next digit = 9 ≥ 5, round up)
2. Substitute: $$\frac{50 \times 4}{20} = \frac{200}{20} = 10$$

1. Round each number to 1 s.f.:
   • $28.9 \approx 30$   (first s.f. = 2, next digit = 8 ≥ 5, round up)
   • $6.85 \approx 7$   (first s.f. = 6, next digit = 8 ≥ 5, round up)
   • $0.472 \approx 0.5$   (first s.f. = 4, next digit = 7 ≥ 5, round up)
2. Substitute: $$\frac{30 + 7}{0.5} = \frac{37}{0.5}$$
3. Dividing by $0.5$ is the same as multiplying by $2$: $$37 \times 2 = 74$$

1. Round each value to 1 s.f.:
   • $28 \approx 30$   (first s.f. = 2, next digit = 8 ≥ 5, round up)
   • $\$8.95 \approx \$9$   (first s.f. = 8, next digit = 9 ≥ 5, round up)
2. Estimated income $= 30 \times \$9 = \$270$

1. Formula: $\text{Area} = \text{length} \times \text{width} = 23.8 \times 11.4$
2. Remove decimals: $23.8 \times 11.4 = (238 \times 114) \div 100$
   There are 2 decimal places in total (1 in each number), so divide the whole-number answer by 100 at the end.
3. Long multiplication — $238 \times 114$:

1. Re-insert the decimal point: $27132 \div 100 = 271.32\text{ m}^2$
2. Reasonable accuracy: both measurements are given to 1 d.p. (3 significant figures), so give the answer to 3 significant figures: $271.32 \approx 271$

1. Formula: $\text{Speed} = \dfrac{\text{Distance}}{\text{Time}} = \dfrac{186}{2.4}$
2. Remove the decimal from the divisor by multiplying both by 10: $$\frac{186}{2.4} = \frac{1860}{24}$$
3. Long division — $1860 \div 24$:

1. Check each step:
   • $24 \times 7 = 168$;   $186 – 168 = 18$;   bring down $0$ → $180$
   • $24 \times 7 = 168$;   $180 – 168 = 12$;   add decimal, bring down $0$ → $120$
   • $24 \times 5 = 120$;   $120 – 120 = 0$ → exact
   Speed $= 77.5\text{ km/h}$
2. Reasonable accuracy: $186$ has 3 s.f., $2.4$ has 2 s.f. → give answer to 2 s.f.
   $77.5$ to 2 s.f.: first s.f. $= 7$, second s.f. $= 7$, check next digit $= 5 \geq 5$, round up → $78$

1. Leading zeros do not count. First s.f. $= 4$
2. Next digit $= 5 \geq 5$, round up: $4 \rightarrow 5$

1. Keep 1 digit after the decimal point: 4
2. Check the next digit: 5  →  $5 \geq 5$, round up
3. $4 \rightarrow 5$, giving $7.5$

1. Round each value to 1 s.f.:
   • $6.8 \approx 7$   (first s.f. $= 6$, next digit $= 8 \geq 5$, round up)
   • $247 \approx 200$   (first s.f. $= 2$, next digit $= 4 < 5$, round down)
2. Estimated fuel $= \dfrac{200}{100} \times 7 = 2 \times 7 = 14$ litres

1. Write the formula: $\text{Area} = \dfrac{1}{2} \times 8.3 \times 5.7$
2. Calculate $8.3 \times 5.7$ using long multiplication (treat as $83 \times 57$, then place decimal):

1. Both $8.3$ and $5.7$ have 1 decimal place: total 2 decimal places.
   $4731 \rightarrow 47.31$
   So $8.3 \times 5.7 = 47.31$
2. Check multiplication rows:
   • $83 \times 7$: $(80 \times 7) + (3 \times 7) = 560 + 21 = 581$   ✓
   • $83 \times 50$: $(80 \times 50) + (3 \times 50) = 4000 + 150 = 4150$   ✓
   • $581 + 4150 = 4731$   ✓
3. Area $= \dfrac{1}{2} \times 47.31$
   $47 \div 2 = 23$ remainder $1$;   $1.31 \div 2 = 0.655$
   Area $= 23.655\text{ cm}^2$

1. Keep 1 digit after the decimal point: 7
2. Check the next digit: 4  →  $4 < 5$, round down
3. The 7 stays as it is