Checking a Calculator Answer by Estimating

(a) $41.8 \times 0.0293 \approx 40 \times 0.03 = 1.2$. Only 1.22474 is near 1.2. The other two are ten times too large and ten times too small.

(b) $6.72 \div 0.028 \approx 7 \div 0.03 = 700 \div 3$, which is about 233. Only 240 is near it.

(a) Estimate: $14.6 \times 2.47 \approx 10 \times 2 = 20$. The display is positive and less than twice the estimate. Reasonable. It is well above the estimate because both numbers were rounded down. A closer estimate is $15 \times 2.5 = 37.5$.

(b) Estimate: $27.3 - 61.8 \approx 30 - 60 = -30$. The last step subtracts two numbers of similar size (60 is not as much as three times 30), so this estimate cannot judge. Find a limit from the numbers themselves: 61.8 is more than 27.3, so a larger number is taken from a smaller one, and the right answer must be below zero. The display is positive, on the wrong side of this limit. Not reasonable.

(c) Estimate: $902 \div 0.44 \approx 900 \div 0.4 = 9000 \div 4 = 2250$. The display is positive and close to the estimate. Reasonable. Dividing 902 by a number less than 1 gives more than 902, so a large answer is to be expected.

(d) Estimate: $0.52 \times 0.38 \approx 0.5 \times 0.4 = 0.2$. The display is about ten times the estimate. Not reasonable. A limit agrees: multiplying 0.52 by a number less than 1 must give less than 0.52.

(e) Estimate: $52.4 - 6.8 \times 7.3 \approx 50 - 7 \times 7 = 50 - 49 = 1$. The last step, $50 - 49$, subtracts two numbers of similar size, so this estimate cannot judge the display. Find limits for the part that is taken away. $6.8 \times 7.3$ is less than $7 \times 7.3 = 51.1$, so the right answer is more than $52.4 - 51.1 = 1.3$. And $6.8 \times 7.3$ is more than $6.8 \times 7 = 47.6$, so the right answer is less than $52.4 - 47.6 = 4.8$. The display lies between 1.3 and 4.8, so it has the right sign and nothing shows it to be wrong. But 4.8 is more than three times 1.3, so the limits are too far apart to judge its size. An estimate cannot settle it: key the calculation again with care.

(f) Estimate: $1.46 \times 14.3 \approx 1 \times 10 = 10$. The display is positive, but it is a little more than twice the estimate, which puts it in doubt. Each number lost nearly a third of its value when it was rounded down, so make a closer estimate: $1.46 \times 14.3 \approx 1.5 \times 14 = 21$. The display is very close to it. Reasonable.

The displays in (a), (c), (e) and (f) are in fact correct: in (e), $52.4 - 6.8 \times 7.3 = 52.4 - 49.64 = 2.76$, and in (f), $1.46 \times 14.3 = 20.878$. The correct answers to (b) and (d) are $-34.5$ and $0.1976$.

(a) Estimate: $39.5 + 13.2 - 10.1 \approx 40 + 10 - 10 = 50 - 10 = 40$. The last step, $50 - 10$, subtracts two numbers that are not of similar size (50 is more than three times 10), so the estimate can judge. The display is four times as large, which puts it in doubt. A limit settles it: $39.5 + 13.2$ is less than $40 + 20 = 60$, and 10.1 is then taken away, so the answer must be less than 60. Not reasonable.

With 13.2 keyed as 132, the estimate becomes $40 + 100 - 10 = 130$, which is near the display. Keying $39.5 + 132 - 10.1$ gives $161.4$, so the decimal point of 13.2 was left out. Correct answer: $39.5 + 13.2 - 10.1 = 42.6$.

(b) Estimate: $9.6 \div 0.32 \approx 10 \div 0.3 = 100 \div 3$, which is about 33. The display is about a tenth of that. It cannot be right, because dividing 9.6 by a number less than 1 must give more than 9.6. Not reasonable.

A misplaced decimal point is the first guess, but $9.6 \div 3.2 = 3$ exactly, not 3.072. Try a wrong key: $9.6 \times 0.32 = 3.072$. The multiplication key was pressed in place of the division key. Correct answer: $9.6 \div 0.32 = 960 \div 32 = 30$.

(c) Estimate: $18.5 \times 6.4 - 73.9 \approx 20 \times 6 - 70 = 120 - 70 = 50$. The last step, $120 - 70$, subtracts two numbers of similar size, so this estimate cannot judge, even though the display happens to be close to it. Find limits. $18.5 \times 6.4$ is more than $18 \times 6 = 108$ and less than $20 \times 6.4 = 128$, so the right answer lies between $108 - 73.9 = 34.1$ and $128 - 73.9 = 54.1$. The display lies between these limits, they have the same sign, and 54.1 is less than twice 34.1. Reasonable. It is in fact correct: $18.5 \times 6.4 - 73.9 = 118.4 - 73.9 = 44.5$.

(a) $79.5 \approx 80$, $33.21 \approx 30$ and $29.52 \approx 30$, so $79.5 - 33.21 \times 29.52 \approx 80 - 30 \times 30 = 80 - 900 = -820$.

(b) The last step, $80 - 900$, subtracts two numbers that are not of similar size (900 is more than three times 80), so the estimate can judge. Sign: both are negative. Size: Jason's answer is less than a quarter of the size of the estimate, which puts it in doubt.

A limit settles it: $33.21 \times 29.52$ is more than $30 \times 20 = 600$, so the right answer must be below $80 - 600 = -520$. Jason's answer is not. It is not reasonable.

(c) An answer near $-185$ would come from starting with about 800 instead of about 80, since $800 - 900 = -100$. So try 795 in place of 79.5.

Keying $795 - 33.21 \times 29.52$ gives $795 - 980.3592 = -185.3592$, exactly what Jason's calculator showed. He left out the decimal point of 79.5.

Keyed correctly: $79.5 - 33.21 \times 29.52 = 79.5 - 980.3592 = -900.8592$, which is close to the estimate.

(a) Estimate: $63.4 + 218.7 \approx 60 + 200 = 260$. The display is less than a third of the estimate, which puts it in doubt. A limit settles it: a positive number is added to 218.7, so the answer must be more than 218.7. Not reasonable.

Work backwards: $82.1 - 63.4 = 18.7$, which is 218.7 with the digit 2 dropped. Keying $63.4 + 18.7$ gives $82.1$, the display. Correct answer: $63.4 + 218.7 = 282.1$.

(b) Estimate: $7.5 \times 46.2 \approx 8 \times 50 = 400$. The display is about eight times the estimate, which puts it in doubt. A limit settles it: both numbers were rounded up, so 400 is an overestimate and the product must be less than 400. Not reasonable.

A decimal point left out would give $75 \times 46.2 = 3465$, which is not the display. Work backwards: $3346.5 \div 7.5 = 446.2$, which is 46.2 with the digit 4 keyed twice. Keying $7.5 \times 446.2$ gives $3346.5$, the display. (The extra 4 adds 400 to the number keyed, and so $7.5 \times 400 = 3000$ to the answer.) Correct answer: $7.5 \times 46.2 = 346.5$.

(a) Estimate: $95.2 - (12.6 + 40.9) \approx 100 - (10 + 40) = 100 - 50 = 50$. The last step, $100 - 50$, subtracts two numbers of similar size, so this estimate cannot judge. Find a limit: a positive number is taken away from 95.2, so the right answer must be less than 95.2. The display is more than that, on the wrong side of the limit. Not reasonable.

Without the brackets the estimate becomes $100 - 10 + 40 = 130$, which is near the display. Keying $95.2 - 12.6 + 40.9$ gives $82.6 + 40.9 = 123.5$, the display. The brackets were left out, so 40.9 was added where it should have been taken away. Correct answer: $95.2 - (12.6 + 40.9) = 95.2 - 53.5 = 41.7$.

(b) Estimate: $\dfrac{672}{4.8 \times 3.5} \approx \dfrac{700}{5 \times 4} = \dfrac{700}{20} = 35$. The display is 14 times the estimate, which is more than 10 times. Not reasonable.

Without brackets round the denominator the estimate becomes $700 \div 5 \times 4 = 140 \times 4 = 560$, which is near the display. Keying $672 \div 4.8 \times 3.5$ gives $140 \times 3.5 = 490$, the display. The brackets were left out, so the calculator multiplied by 3.5 where it should have divided by it. Correct answer: $672 \div (4.8 \times 3.5) = 672 \div 16.8 = 40$.

(a) Estimate: $8.4 \div 0.12 \approx 8 \div 0.1 = 80 \div 1 = 80$. Kumar's answer is less than a tenth of the estimate. It cannot be right, because dividing 8.4 by a number less than 1 must give more than 8.4. Not reasonable.

To divide by 0.12 on paper, both numbers are multiplied by 100: $8.4 \div 0.12 = 840 \div 12 = 70$. Multiplying 0.12 by 100 but 8.4 by only 10 gives $84 \div 12 = 7$, which is Kumar's answer. Correct answer: 70. Check with the inverse operation: $70 \times 0.12 = 8.4$.

(b) Estimate: $12.85 + 7.90 + 24.60 \approx 10 + 8 + 20 = 38$ dollars. Siti's total is about three times the estimate, which puts it in doubt. A limit settles it: the amounts are less than $13, $8 and $25, so the total is less than $13 + 8 + 25 = 46$ dollars. Not reasonable.

Siti's total is about $70 above the limit. That points to the 7 of $7.90: written one column too far to the left, it counts as 7 tens. Adding $12.85 + 79.0 + 24.60$ gives $116.45$, which is Siti's total. Correct total: $12.85 + 7.90 + 24.60 = 45.35$, so the amounts add up to $45.35.

(a) $29 \approx 30$ and $8.20 \approx 8$, so the total is about $30 \times 8 = 240$ dollars.

(b) 2378 is about ten times the estimate, so it is not reasonable. A limit agrees: fewer than 30 pupils each pay less than $10, so the total must be less than $30 \times 10 = 300$ dollars.

An answer ten times too large in a multiplication points to a decimal point. Keying $29 \times 82$ gives $2378$, so one slip that explains it is the decimal point of 8.2 left out. Another is as likely: keying $290 \times 8.2$ also gives $2378$, so a 0 may have been keyed after 29.

Correct total: $29 \times 8.2 = 237.8$, so the teacher collects $237.80. This is close to the estimate.

(a) $42.5 \approx 40$ and $2.86 \approx 3$, so the cost is about $40 \times 3 = 120$ dollars.

(b) The display is about an eighth of the estimate, which puts it in doubt. A limit settles it: each litre costs more than $2, so 42.5 litres cost more than $42.5 \times 2 = 85$ dollars. The display is not reasonable.

(c) A decimal point left out or misplaced would make the answer 10 or 100 times too large or too small, not about 8 times too small. So try a wrong operation key: $40 \div 3$ is about 13, which is near the display.

Keying $42.5 \div 2.86$ gives $14.86013986$, exactly what the calculator showed. He pressed the division key in place of the multiplication key.

Keyed correctly: $42.5 \times 2.86 = 121.55$, so the petrol costs $121.55. This is close to the estimate.

(a) $48.6 \approx 50$, $52.3 \approx 50$ and $55.1 \approx 60$, so the average is about $\dfrac{50 + 50 + 60}{3} = \dfrac{160}{3}$, which is about 53 kg.

(b) The display is more than twice the estimate, which puts it in doubt. A limit settles it: an average lies between the smallest value and the largest, so it cannot be more than 55.1 kg. The display is not reasonable.

(c) The numerator needs brackets on a calculator. Without them only 55.1 is divided by 3, and the estimate becomes $50 + 50 + 60 \div 3 = 50 + 50 + 20 = 120$, which is near the display.

Keying $48.6 + 52.3 + 55.1 \div 3$ gives $119.2666667$, exactly what the calculator showed. She left out the brackets round the numerator.

Keyed correctly: $(48.6 + 52.3 + 55.1) \div 3 = 156 \div 3 = 52$, so the average mass is 52 kg. This is close to the estimate.