Rounding and estimation

The thousands digit is 8 and the next digit is 4.

Since $4 < 5$, round down: $18\,475 \approx 18\,000$.

The hundreds digit is 1 and the next digit is 8, which is 5 or more, so round up.

$6184 \to 6200$

8061 lies between 8000 and 8100.

The tens digit is 6, which is 5 or more, so round up: 8100.

Look at the units digit: it is 6, which is 5 or more, so round up.

$1476 \to 1480$

Look at the first decimal place: 19.4949. It is 4, which is less than 5, so round down.

Answer: 19

327 lies between 320 and 330.

The units digit is 7, which is 5 or more, so round up: $330$.

Look at the first decimal digit: 6, which is 5 or more.

So round up: 4.666 becomes 5.

Look at the hundreds digit: 29 381. It is 3, which is less than 5, so round down.

$29\,381 \approx 29\,000$

Look at the second decimal place: it is 8.

8 is 5 or more, so round the 5 up: $4.58 \approx 4.6$

8375 lies between 8000 and 9000.

The hundreds digit is 3, which is less than 5, so round down to 8000.

478 lies between 400 and 500.

The tens digit is 7, which is 5 or more, so round up: 500.

7829 lies between 7000 and 8000.

The hundreds digit is 8, which is 5 or more, so round up to 8000.

Look at the hundreds digit of 6324: it is 3, which is less than 5.

So round down to 6000.

Look at the second decimal place: it is 9, which is 5 or more, so round the 5 up.

$1.59 \approx 1.6$

2538 lies between 2500 and 2600.

The tens digit is 3, which is less than 5, so round down to $2500$.

Look at the third decimal place: it is 5.

5 or more rounds up, so $4.43|54$ becomes $4.44$.

The thousands either side are 5000 and 6000.

5689 is past the halfway point of 5500, so it rounds up to 6000.

(a) $13.82 \div 4.06 = 3.40394\ldots$

$\sqrt{3.40394\ldots} = 1.844977205\ldots$

(b) The third decimal is 4, so round down: 1.84

The expression is $\dfrac{\sqrt{35.2 + 1.7^3}}{4.6^2 - 8.91}$ (the square root covers the whole numerator).

Numerator: $\sqrt{35.2 + 4.913} = \sqrt{40.113} = 6.33348\ldots$

Denominator: $21.16 - 8.91 = 12.25$

(a) $6.33348\ldots \div 12.25 = 0.5170189759\ldots$

(b) $0.52$ (2 s.f.)

(a) $700 - 452 = 248$

(b) Round to 1 s.f.: $300 \times £10 = £3000$ and $400 \times £20 = £8000$

Estimate $= £3000 + £8000 = £11\,000$

(c) Overestimate: all four numbers were rounded up, so the true total is smaller.

(a) Round each number: $92 \approx 90$ and $1.63 \approx 2$

$90 \times 2 = 180$

(b) $29.6$ is $2.96 \times 10$ and $32$ is $3.2 \times 10$, so the answer is $9.472 \times 100 = 947.2$

(a) The 4th figure is 6, so round the 5 up: 87 600

(b) Top: $(3.2 + 3.7) \times 4.9 = 6.9 \times 4.9 = 33.81$

Bottom: $5.3 - 2.8 = 2.5$

$33.81 \div 2.5 = 13.524$

(a) The first two significant figures are 2 and 5; the next digit 3 rounds down: 2500

(b) The first significant figure is 8 (hundredths); the next digit 7 rounds up: 0.09

Round to 1 significant figure: $790 \approx 800$, $289 \approx 300$, $49 \approx 50$.

$\frac{800 \times 300}{50} = \frac{240\,000}{50} = 4800$

(a) Top: $29^2 - 4.6 = 836.4$. Bottom: $\sqrt{35 - 6.859} = \sqrt{28.141} = 5.30481...$

$836.4 \div 5.30481... = 157.668255...$

(b) The 4th significant figure is 6 and the next digit is 6, so round up: $157.7$

(a) The 4th figure is 7, so round up: $7357 \to 7360$

(b) Top: $\sqrt{17 + 16} = \sqrt{33} = 5.7445\ldots$

Bottom: $7.3^2 = 53.29$

$5.7445\ldots \div 53.29 = 0.1077981356$

Round each number: $3.4 \approx 3$ and $5.3 \approx 5$.

$3 \times 5 = 15$, and even $4 \times 6 = 24$, so the answer must be less than 24. 180.2 is far too big.

(a) Round: $513 \approx 500$ km and $0.81 \approx 0.8$ euros.

$500 \div 10 = 50$ lots of 10 km, $50 \times 0.8 = 40$ euros.

(b) Underestimate: both values were rounded down.

(a) Round: $3000$ miles, $15$ mph

Hours $= 3000 \div 15 = 200$

Days $= 200 \div 8 = 25$

(b) A higher speed means fewer hours, so he would need fewer days.

(a) Round: distance $\approx 3000$ miles, speed $\approx 15$ mph.

Hours $\approx 3000 \div 15 = 200$

Days $\approx 200 \div 8 = 25$

(b) He rides faster, so each day covers more miles and the race takes fewer days.

(a) Round: $4.2 \to 4$ and $50.1 \to 50$.

$4 \times 50 = 200$p a week, and $200 \times 13 = 2600$p $= £26$.

(b) Both 4.2 and 50.1 were rounded down, so the estimate is an underestimate. (The exact cost is about £27.35.)

Round: $5.7 \approx 6$, $8.2 \approx 8$, $0.26 \approx 0.3$

$6 \times 8 = 48$

$48 \div 0.3 = 480 \div 3 = 160$

(a) Round 5406 to 5400. One square: $5400 \div 6 = 900\text{ cm}^2$

Side $= \sqrt{900} = 30$ cm

(b) Underestimate: 5406 was rounded down to 5400, so each square's area and side were made slightly smaller than they really are.

(a) $\sqrt{63.5 \times 101.7} \approx \sqrt{64 \times 100} = \sqrt{6400} = 80$

(b) $0.23 = 2.3 \div 10$, so $(0.23)^6 = 148 \div 10^6 = 0.000148$

(c) $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

(a) Round 213 to 200 mph.

1 hour $= 60 \times 60 = 3600$ seconds.

Time for 1 mile $= 3600 \div 200 = 18$ seconds.

(b) Overestimate: 200 is slower than the real speed, so the real time is shorter.

(a) Round $1.899$ to $2$: $2 \times 10^{27} \times 0.3 = 0.6 \times 10^{27} = 6 \times 10^{26}$ kg.

(b) $1.899$ was rounded up to $2$, so the estimate is an overestimate.