Lesson 1.7 · Unit 1 · N-Q.3
Choosing an accuracy the situation actually supports
A calculator will give you ten digits for any calculation you ask it. Almost none of those digits mean anything, because the numbers you fed in did not have ten digits of accuracy either. Reporting all of them is not more precise; it is a claim about your measurements that you cannot support.
- An answer can be no more accurate than the data it came from. Multiplying a measurement good to two digits by one good to four does not produce a result good to six.
- Round at the end, never in the middle. Carry extra digits through the working and round only the final answer. Rounding early lets small errors accumulate.
- Let the context set the precision. Money is two decimal places. A count of people or objects is a whole number. A length measured with a ruler marked in millimeters is good to about a millimeter.
- Rounding direction can be decided by the situation, not by the digit. If 4.2 buses are needed, the answer is 5 buses, because four will not carry everyone.
- State the unit with every answer in context. "12.5" is not an answer; "12.5 gallons" is.
- An exact answer is better than a rounded one when it is available. Leave \( \frac{1}{3} \) or \( \sqrt{2} \) exact unless a decimal is asked for.
- Check that the size of the answer is sensible before reporting it. A negative time, a probability above 1, or a person's height of 300 cm is a signal to look again.
Where students lose marks: rounding partway through and then using the rounded value in later steps. If you round 0.666... to 0.67 and then multiply by 1,000, your answer is off by more than 3. Keep the full value in the calculator and round once, at the end.
The problem. A school is organizing a trip for 247 students. Each bus holds 44 students. Buses cost $385 each, and the cost is to be shared equally among the students. Find the number of buses needed and the cost per student, and justify the rounding at each stage.
Step one: find the exact number of buses required. \( 247 \div 44 = 5.6136\ldots \) buses.
Step two: decide how to round, using the context rather than the digit. Ordinary rounding of 5.61 would give 6, which happens to be right, but the reasoning matters: even 5.1 buses would require 6, because a fraction of a bus cannot be hired and five buses would leave students behind. The context forces rounding up regardless of the decimal.
Step three: check that 6 buses suffice. \( 6 \times 44 = 264 \) seats, which is more than 247, so everyone fits with 17 seats spare. And \( 5 \times 44 = 220 \), which is fewer than 247, so 5 is genuinely not enough. Six is the smallest workable number.
Step four: find the total cost. \( 6 \times \$385 = \$2{,}310 \).
Step five: divide among the students. \( 2310 \div 247 = 9.3522\ldots \) dollars per student.
Step six: round according to the context of money. Currency is expressed to the cent, so this is $9.35 per student. Reporting $9.3522267 would claim a precision that money does not have.
Step seven: check the rounding does not create a shortfall. At $9.35 each, \( 247 \times 9.35 = \$2{,}309.45 \), which is 55 cents short of the $2,310 needed. In a real situation the school would round the per-student charge up to $9.36, collecting \( 247 \times 9.36 = \$2{,}310.92 \), a small surplus. Which rounding is correct depends on whether the total must be covered, and saying so is part of the answer.
Step eight: state the result with units and the assumption. Six buses are needed, at a cost of about $9.35 per student, or $9.36 if the collection must cover the full hire cost. Notice that the two roundings in this problem went in different directions and for different reasons, neither of them the usual "five or more rounds up". The situation, not the digit, decided both.
Problems 1 to 5 are routine. Problems 6 to 10 are multi-step or ask for reasoning.
- Round 3.7482 to two decimal places.
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The third decimal is 8, so round the second up. 3.75
- Round 1,472 to the nearest hundred.
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1,500
- A price is calculated as $14.2857. State it as an amount of money.
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$14.29
- Give the units of an answer found by dividing dollars by hours.
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Dollars per hour
- A calculation gives 8.4 people. State a sensible answer.
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People are counted in whole numbers, and the right direction depends on the context; if these are people who must be seated or served, round up. 8 people, or 9 if everyone must be accommodated
- A rope 12.5 m long is cut into pieces 1.8 m long. How many whole pieces, and how much is
left over?
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\( 12.5 \div 1.8 = 6.944\ldots \), so 6 whole pieces can be cut; a seventh would require 12.6 m. Used length: \( 6 \times 1.8 = 10.8 \) m. Remaining: \( 12.5 - 10.8 = 1.7 \) m. Here the context forces rounding down, the opposite of the bus problem, because a partial piece is not a usable piece. 6 pieces, with 1.7 m left over
- A rectangle measures 4.2 cm by 7.9 cm. Find its area and state it to a sensible
precision.
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\( 4.2 \times 7.9 = 33.18 \) cm². Each measurement is given to two significant figures, so the product should not be reported to four. Rounding to two significant figures gives 33 cm². Reporting 33.18 would claim the measurements were accurate to a hundredth of a centimeter, which the given data does not support. About 33 cm²
- A student computes \( \frac{2}{3} \times 900 \) by first rounding \( \frac{2}{3} \) to
0.67. Find their answer, the correct answer, and the size of the error.
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Their answer: \( 0.67 \times 900 = 603 \). Correct: \( \frac{2}{3} \times 900 = \frac{1800}{3} = 600 \) exactly. The error is 3, which is small as a fraction but is a whole 3 units in a context where the answer might be a count. The rounding error of 0.00333 in the fraction was multiplied by 900, magnifying it 900 times. This is exactly why rounding belongs at the end. 603 against an exact 600; an error of 3
- A car travels 247 miles on 9.1 gallons. Compute the fuel economy and justify the
precision you report.
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\( 247 \div 9.1 = 27.142857\ldots \) miles per gallon. The fuel measurement is given to two significant figures, which is the weaker of the two inputs and therefore limits the result. Reporting 27.142857 would be absurd; even 27.14 claims more than the data supports. Rounding to two significant figures gives 27 miles per gallon, and quoting 27.1 would be defensible if the 9.1 is trusted to that tenth. About 27 miles per gallon, limited by the two-figure fuel measurement
- A theater has 340 seats. Tickets cost $18.50 and the show costs $4,000 to stage. Find the
number of tickets that must be sold to break even, and explain why the rounding direction
is not the usual one.
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Break even means revenue equals cost: \( 18.50t = 4000 \), so \( t = 4000 \div 18.50 = 216.216\ldots \) tickets. Ordinary rounding would give 216, but selling 216 tickets brings in \( 216 \times 18.50 = \$3{,}996 \), which is $4 short of covering the cost. The context requires meeting or exceeding the cost, so the answer must round up to 217 tickets, giving \( 217 \times 18.50 = \$4{,}014.50 \). The decimal .216 would normally round down; the situation overrides that, because a shortfall is not acceptable. It is also worth noting 217 is well under the 340 seats available, so break even is achievable. 217 tickets; the context requires covering the full cost, so it rounds up despite the decimal