Rounded Percentages and Truncation

Why rounded percentages may add up to 99.9% or 101% instead of 100%, why rounded amounts may not add up to the rounded total, what to do about it, and how cutting a number off differs from rounding it.

Before you start

To express one quantity as a percentage of another, divide the first by the second and multiply by 100%. For example, 13 out of 40 is $13 \div 40 \times 100\% = 32.5\%$ .

To round a number, look at the digit after the last one you keep. If it is 5 or more, round up. If it is less than 5, the last digit you keep stays as it is. So $26.4705\ldots\%$ is 26.5% correct to 1 decimal place, and it is also 26.5% correct to 3 significant figures. But $6.4705\ldots\%$ is 6.5% correct to 1 decimal place and 6.47% correct to 3 significant figures, because significant figures are counted from the first non-zero digit.

A rounding error is the difference between a rounded value and the exact value. When 7.38 is rounded to 7.4, the rounding error is $7.4 - 7.38 = 0.02$ . A rounding error is never more than half a unit of the place rounded to: at most 0.05 when rounding to 1 decimal place, and at most 0.5 when rounding to the nearest whole number. The word "error" here does not mean a mistake: it is the amount by which the rounded value is off.

Why Rounded Parts May Not Add Up to the Whole

When a whole is divided into parts, the percentages of the parts add up to exactly 100%. After each percentage has been rounded, the total may come to a little less or a little more. Nothing has been miscalculated: the rounding errors of the parts have added up.

Example 1: A total of less than 100%

The 240 Secondary 1 students of a school each choose one co-curricular activity (CCA): 98 choose a sport, 74 choose a performing art and 68 choose a uniformed group. Express each number as a percentage of the 240 students, correct to 3 significant figures, and add the three percentages.

Solution

Each percentage is the number of students divided by 240, multiplied by 100%. For sport, $98 \div 240 \times 100\% = 40.8333\ldots\%$

CCA	Number of students	Percentage on the calculator	Correct to 3 significant figures
Sport	98	$40.8333\ldots\%$	40.8%
Performing art	74	$30.8333\ldots\%$	30.8%
Uniformed group	68	$28.3333\ldots\%$	28.3%
Total	240	100%	99.9%

Total of the rounded percentages: $40.8\% + 30.8\% + 28.3\% = 99.9\%$

Each percentage was rounded down, and each lost $0.0333\ldots\%$ . The three losses add up to 0.1%, which is exactly what is missing from the total.

The exact percentages still add up to 100%. Keeping one more figure brings the total closer: $40.83\% + 30.83\% + 28.33\% = 99.99\%$ , which is 100% correct to 3 significant figures.

Sport 40.8%, performing art 30.8%, uniformed group 28.3% (each correct to 3 significant figures). The rounded percentages add up to 99.9%, not 100%, because of rounding.

Example 2: A total of more than 100%

The 150 students at a school camp vote for the activity of the last night: 67 vote for a campfire, 49 for a night walk and 34 for a talent show. Express each number of votes as a percentage of the 150 votes, correct to 3 significant figures, and add the three percentages.

Solution

Each percentage is the number of votes divided by 150, multiplied by 100%. For the campfire, $67 \div 150 \times 100\% = 44.6666\ldots\%$

Activity	Number of votes	Percentage on the calculator	Correct to 3 significant figures
Campfire	67	$44.6666\ldots\%$	44.7%
Night walk	49	$32.6666\ldots\%$	32.7%
Talent show	34	$22.6666\ldots\%$	22.7%
Total	150	100%	100.1%

Total of the rounded percentages: $44.7\% + 32.7\% + 22.7\% = 100.1\%$

This time each percentage was rounded up, and each gained $0.0333\ldots\%$ . The three gains add up to 0.1%, which is the extra in the total.

It is not possible for more than 100% of the students to vote. Each rounded percentage is correct. Only their total is off.

Campfire 44.7%, night walk 32.7%, talent show 22.7% (each correct to 3 significant figures). The rounded percentages add up to 100.1%, not 100%, because of rounding.

How far from 100% can the total be?

Each rounded percentage is off by at most half a unit of the place it was rounded to. Rounding errors in opposite directions partly cancel. Rounding errors in the same direction pile up, as they did in Examples 1 and 2.
A percentage between 10% and 100% that is correct to 3 significant figures is correct to 1 decimal place, so it is off by at most 0.05%. Three such percentages can be off by at most 0.15% altogether, so their total is 99.9%, 100.0% or 100.1%.
Percentages correct to the nearest whole number can each be off by as much as 0.5%. With four of them the total can be as much as 2% away from 100%.
The more parts there are, and the fewer figures each is rounded to, the further from 100% the total can be.
Rounded percentages may not add up to 100%. It does not happen every time: often the rounding errors cancel and the total is exactly 100%.

Rounded Amounts and the Rounded Total

The same thing happens with any quantities that are rounded and then added: sums of money, lengths, masses, numbers of people. The rounded parts may not add up to the rounded total.

Example 3: Rounded amounts of money

A school holds a charity fair over three days. It collects $1248 on Friday, $2349 on Saturday and $1746 on Sunday. The school newsletter gives the amount for each day, and the total, correct to the nearest $100.
(a) What four amounts does the newsletter print?
(b) Add the three rounded amounts for the days. Is the answer the same as the rounded total?

Solution

(a) Round each day's amount to the nearest $100. In each one the tens digit is 4, so the hundreds digit stays as it is.

Day	Amount collected	Correct to the nearest $100
Friday	$1248	$1200
Saturday	$2349	$2300
Sunday	$1746	$1700
Total	$5343	$5300

The exact total is $1248 + 2349 + 1746 = 5343$ dollars, which is $5300 correct to the nearest $100.

(b) The rounded amounts for the three days add up to $1200 + 2300 + 1700 = 5200$ dollars. This is $100 less than the rounded total, $5300.

Each day's amount was rounded down: by $48, by $49 and by $46. Together the three days lost $48 + 49 + 46 = 143$ dollars, so the sum of the rounded amounts is $143 below the exact total. The total itself, rounded once, lost only $43.

Every amount in the newsletter is correctly rounded. The total to print is $5300, because it comes from the exact amounts. A total found by adding the rounded amounts carries all three rounding errors.

(a) The newsletter prints $1200, $2300 and $1700 for the three days, and $5300 for the total.
(b) The rounded amounts add up to $5200, which is not the rounded total, $5300. The difference comes from rounding.

What to Do When the Parts Do Not Add Up

A total of 99.9% or 101% looks like a mistake, and it is tempting to "correct" it. Do not. The rounded figures are right. What is needed is to say that they are rounded.

What to do when rounded figures do not add up

Leave each rounded percentage as it is. Each one is the best value for its own part. Do not change one of them to force the total to 100%.
Find a total from the exact values, not by adding rounded ones. The percentages of all the parts of a whole total exactly 100%. In Example 3 the total to print is $5300, from the exact amounts, and not $5200.
Work out every percentage from its own numbers. Do not find the last one by subtracting the others from 100%. In Example 1, $100\% - 40.8\% - 30.8\% = 28.4\%$ , but the third percentage is 28.3%.
Say that the figures are rounded. Tables and charts often carry a line such as "Percentages may not add up to 100% because of rounding". You may write the same.

Example 4: The labels of a pie chart

In a survey, 250 Secondary 1 students each name their favourite hawker dish: 92 name chicken rice, 69 fishball noodles, 52 nasi lemak and 37 roti prata. The school magazine shows the result as the pie chart below, with each label correct to the nearest whole number.
(a) Check that each label is correctly rounded.
(b) Add the four labels.
(c) Kumar says that the chart must be wrong, and that the label for chicken rice should be changed to 36% so that the labels add up to 100%. Is he right?

Solution

(a) Each percentage is the number of students divided by 250, multiplied by 100%. For chicken rice, $92 \div 250 \times 100\% = 36.8\%$ , which is 37% correct to the nearest whole number. The table shows all four.

Dish	Number of students	Exact percentage	Label on the chart
Chicken rice	92	36.8%	37%
Fishball noodles	69	27.6%	28%
Nasi lemak	52	20.8%	21%
Roti prata	37	14.8%	15%
Total	250	100%	101%

All four labels are correctly rounded.

(b) $37\% + 28\% + 21\% + 15\% = 101\%$

Each percentage was rounded up: by 0.2%, by 0.4%, by 0.2% and by 0.2%. The four gains add up to 1%, which is the extra in the total.

(c) Kumar is not right. Chicken rice was named by 36.8% of the students, and the nearest whole number to 36.8 is 37. A label of 36% would be a wrong label, put there only to make the total look tidy. The sectors of the chart are drawn from the exact numbers, so they still fill the whole circle.

The magazine should leave the four labels as they are and print a line under the chart: "Percentages do not add up to 100% because of rounding."

(a) The labels 37%, 28%, 21% and 15% are all correctly rounded.
(b) The labels add up to 101%.
(c) Kumar is not right. The labels should be left as they are, with a note that they do not add up to 100% because of rounding.

Truncation: Cutting Off Instead of Rounding

This last section goes a little beyond what Secondary 1 asks for, but you will meet it on calculators and computers. There is a second way to shorten a number: simply cut its digits off.

Definition

To truncate a number is to cut off its digits after a given place without rounding. The digits that are kept are not changed.

Truncating a positive number never makes it larger. The truncated value is less than the number, or equal to it when every digit cut off is 0.

Example 5: Truncating and rounding the same numbers

Truncate each number to 2 decimal places, and round each number to 2 decimal places.
(a) 4.728
(b) 3.14159
(c) 19.996

Solution

(a) Truncated: cut off everything after the second decimal place, which leaves 4.72. Rounded: the digit after the second decimal place is 8, so round up to 4.73.

(b) Truncated: 3.14. Rounded: the digit after the second decimal place is 1, so the 4 stays as it is, which gives 3.14. The two agree, because the rounding did not round up.

(c) Truncated: 19.99. Rounded: the digit after the second decimal place is 6, so round up. Adding 1 to the last digit of 19.99 gives 20.00.

Number	Truncated to 2 decimal places	Rounded to 2 decimal places
4.728	4.72	4.73
3.14159	3.14	3.14
19.996	19.99	20.00

(a) Truncated 4.72, rounded 4.73.
(b) Truncated 3.14, rounded 3.14.
(c) Truncated 19.99, rounded 20.00.

Truncating and rounding compared (positive numbers)

When the first digit cut off is 0, 1, 2, 3 or 4, truncating and rounding give the same value.
When the first digit cut off is 5 or more, the rounded value is one unit of the last place kept more than the truncated value: 4.73 against 4.72, and 20.00 against 19.99.
A truncated value can be off by almost a whole unit of the last place kept. A rounded value is never off by more than half a unit. Truncating 4.728 to 4.72 is off by 0.008, and rounding it to 4.73 is off by only 0.002. That is why we round when we want the closest value.

Where you meet truncation

Ages. A person who is 12 years and 11 months old says "I am 12". An age is cut off to whole years, not rounded.
Calculators. A simple calculator with an eight-digit display may show $2 \div 3$ as 0.6666666: it cuts the digits off. A scientific calculator with a ten-digit display gives the decimal as 0.6666666667: it rounds the last digit.
Spreadsheets and computer programs. A spreadsheet has a function called TRUNC that cuts a number off, and a program that keeps only the whole-number part of 7.9 gets 7.
In your own answers, round unless a question tells you to truncate.

Common Mistakes

Taking a total of 99.9% or 101% for a wrong calculation. Check each percentage first. If every one is correctly rounded, the total is off only because of rounding.
Changing a percentage to make the total 100%, or finding the last percentage by subtracting the others from 100%. Each percentage is worked out from its own numbers and rounded on its own.
Adding rounded parts to find a total. In Example 3 the rounded amounts add up to $5200, but the total correct to the nearest $100 is $5300. Add the exact values, and round once.
(On truncation, a little beyond Secondary 1) Cutting the display off instead of rounding it. The calculator shows $\sqrt{7}$ as 2.645751311, which is 2.65 correct to 3 significant figures, not 2.64.
(On truncation, a little beyond Secondary 1) Taking "truncate" for another word for "round". Truncating 19.996 to 2 decimal places gives 19.99, and rounding it gives 20.00.

Practice

Work each question on paper, with a calculator where you need one, before you open its solution. Questions 7 to 10 are on truncation, which goes a little beyond Secondary 1.

Question 1

In a class of 40 students, 13 walk to school, 11 come by bus, 9 come by MRT and 7 come by car.
(a) Express each number as a percentage of the class, correct to the nearest whole number.
(b) Add the four rounded percentages.
(c) Explain why the total is not 100%.

ShowHide solution

(a) Walk: $13 \div 40 \times 100\% = 32.5\%$ , which rounds up to 33%.

Bus: $11 \div 40 \times 100\% = 27.5\%$ , which rounds up to 28%.

MRT: $9 \div 40 \times 100\% = 22.5\%$ , which rounds up to 23%.

Car: $7 \div 40 \times 100\% = 17.5\%$ , which rounds up to 18%.

(b) $33\% + 28\% + 23\% + 18\% = 102\%$

(c) Each exact percentage ends in .5, so each was rounded up by 0.5%. The four gains add up to $4 \times 0.5\% = 2\%$ . The exact percentages add up to 100%: $32.5 + 27.5 + 22.5 + 17.5 = 100$ .

(a) Walk 33%, bus 28%, MRT 23%, car 18%.
(b) The rounded percentages add up to 102%.
(c) Each of the four percentages was rounded up by 0.5%, which adds 2% to the total.

Question 2

Two classes of 36 students each vote for the colour of their class T-shirt. In class 1A, 15 vote for blue, 12 for red and 9 for green. In class 1B, 20 vote for blue, 11 for red and 5 for green.
(a) For class 1A, express each number of votes as a percentage of the 36 votes, correct to 1 decimal place, and add the three percentages.
(b) Do the same for class 1B.
(c) Do rounded percentages always fail to add up to 100%?

ShowHide solution

(a) Class 1A. Blue: $15 \div 36 \times 100\% = 41.666\ldots\%$ , which is 41.7%. Red: $12 \div 36 \times 100\% = 33.333\ldots\%$ , which is 33.3%. Green: $9 \div 36 \times 100\% = 25\%$ exactly, which is written 25.0% correct to 1 decimal place.

Total: $41.7\% + 33.3\% + 25.0\% = 100.0\%$ . Blue was rounded up by $0.0333\ldots\%$ and red was rounded down by the same amount, so the two rounding errors cancel.

(b) Class 1B. Blue: $20 \div 36 \times 100\% = 55.555\ldots\%$ , which is 55.6%. Red: $11 \div 36 \times 100\% = 30.555\ldots\%$ , which is 30.6%. Green: $5 \div 36 \times 100\% = 13.888\ldots\%$ , which is 13.9%.

Total: $55.6\% + 30.6\% + 13.9\% = 100.1\%$ . All three were rounded up, so the rounding errors pile up.

(c) No. In class 1A the rounding errors cancel and the rounded percentages add up to exactly 100%. In class 1B the errors are all in the same direction and the total is 100.1%.

(a) Class 1A: blue 41.7%, red 33.3%, green 25.0%. They add up to 100.0%.
(b) Class 1B: blue 55.6%, red 30.6%, green 13.9%. They add up to 100.1%.
(c) No. Rounded percentages add up to 100% when their rounding errors cancel, as in class 1A.

Question 3

A signboard at the start of a park trail gives the lengths of the three parts of the trail as 1.4 km, 2.8 km and 3.3 km, and the length of the whole trail as 7.4 km. All four lengths are correct to 1 decimal place. The exact lengths of the three parts are 1.36 km, 2.76 km and 3.26 km.
(a) Add the three lengths on the signboard.
(b) Find the exact length of the whole trail, and round it to 1 decimal place.
(c) Is anything on the signboard wrong? Explain.

ShowHide solution

(a) $1.4 + 2.8 + 3.3 = 7.5$ km, which is not the 7.4 km on the signboard.

(b) Exact length $= 1.36 + 2.76 + 3.26 = 7.38$ km, which is 7.4 km correct to 1 decimal place.

(c) Nothing is wrong. Each part is correctly rounded: 1.36 to 1.4, 2.76 to 2.8 and 3.26 to 3.3. Each was rounded up by 0.04 km, so the three rounded parts add up to $3 \times 0.04 = 0.12$ km more than the exact length: $7.38 + 0.12 = 7.5$ . The whole trail, rounded once, is 7.4 km.

(a) The three lengths on the signboard add up to 7.5 km.
(b) The whole trail is 7.38 km long, which is 7.4 km correct to 1 decimal place.
(c) Nothing is wrong. The rounded parts do not add up to the rounded total because of rounding.

Question 4

A class library has 90 books: 53 in English, 23 in Chinese and 14 in Malay.
(a) Find the percentage of the books in each language, correct to 1 decimal place.
(b) Devi finds the percentages for English and Chinese, and then works out the percentage for Malay as $100\% - 58.9\% - 25.6\%$ . What does she get? Is it correct to 1 decimal place?

ShowHide solution

(a) English: $53 \div 90 \times 100\% = 58.888\ldots\%$ , which is 58.9% correct to 1 decimal place.

Chinese: $23 \div 90 \times 100\% = 25.555\ldots\%$ , which is 25.6%.

Malay: $14 \div 90 \times 100\% = 15.555\ldots\%$ , which is 15.6%.

(b) Devi gets $100\% - 58.9\% - 25.6\% = 15.5\%$ . This is not correct to 1 decimal place: the percentage for Malay is 15.6%.

She used two rounded values. Each had been rounded up, so she subtracted a little too much. The three correct percentages add up to $58.9\% + 25.6\% + 15.6\% = 100.1\%$ because of rounding, and that is how they should be left.

(a) English 58.9%, Chinese 25.6%, Malay 15.6% (each correct to 1 decimal place).
(b) Devi gets 15.5%, which is not correct. The percentage for Malay is 15.6%.

Question 5

A class has a budget of $600 for its camp: $200 for transport, $150 for food, $140 for T-shirts and $110 for prizes. The treasurer shows each amount as a percentage of the budget, correct to 1 decimal place.
(a) Find the four percentages.
(b) Add the four percentages.
(c) Three students say what should be done. Aini: "The total is not 100%, so one of the percentages has been worked out wrongly." Ben: "Change the largest percentage so that the total is 100%." Chitra: "Leave the percentages as they are, and write under the table that they may not add up to 100% because of rounding." Who is right?

ShowHide solution

(a) Transport: $200 \div 600 \times 100\% = 33.333\ldots\%$ , which is 33.3%.

Food: $150 \div 600 \times 100\% = 25\%$ exactly, which is written 25.0%.

T-shirts: $140 \div 600 \times 100\% = 23.333\ldots\%$ , which is 23.3%.

Prizes: $110 \div 600 \times 100\% = 18.333\ldots\%$ , which is 18.3%.

(b) $33.3\% + 25.0\% + 23.3\% + 18.3\% = 99.9\%$

(c) Aini is not right. Each of the four percentages is correctly rounded. Three of them were rounded down by $0.0333\ldots\%$ , and the three losses add up to 0.1%.

Ben is not right. The largest percentage would become 33.4%, which is a wrong value for transport: transport takes $33.333\ldots\%$ of the budget.

Chitra is right. The exact percentages add up to 100%, as the amounts show: $200 + 150 + 140 + 110 = 600$ .

(a) Transport 33.3%, food 25.0%, T-shirts 23.3%, prizes 18.3%.
(b) The four percentages add up to 99.9%.
(c) Chitra is right: the percentages are left as they are, with a note that they may not add up to 100% because of rounding.

Question 6

In a class vote, each of three choices gets exactly one third of the votes. A pie chart of the result has three equal sectors.
(a) Each sector is labelled with its percentage, correct to the nearest whole number. What is each label, and what do the three labels add up to?
(b) What do the three labels add up to if each is correct to 1 decimal place? What if each is correct to 2 decimal places?
(c) Can the labels be made to add up to 100% by keeping more decimal places? How can each sector be labelled exactly?

ShowHide solution

(a) One third as a percentage is $1 \div 3 \times 100\% = 33.333\ldots\%$ , which is 33% correct to the nearest whole number. The three labels add up to $3 \times 33\% = 99\%$ .

(b) Correct to 1 decimal place each label is 33.3%, and $3 \times 33.3\% = 99.9\%$ . Correct to 2 decimal places each label is 33.33%, and $3 \times 33.33\% = 99.99\%$ .

(c) No. The 3s go on for ever, so wherever the decimal is rounded, each label is a little less than one third, and the three labels add up to a little less than 100%. More decimal places bring the total closer to 100% but never reach it.

The exact label is a fraction: $33\frac{1}{3}\%$ , or simply "one third". Then $3 \times 33\frac{1}{3}\% = 100\%$ .

(a) Each label is 33%, and the three labels add up to 99%.
(b) They add up to 99.9% with 1 decimal place, and to 99.99% with 2 decimal places.
(c) No. However many decimal places are kept, the total is a little less than 100%. The exact label is $33\frac{1}{3}\%$ .

Question 7

Truncate each number to 1 decimal place, and round each number to 1 decimal place. For which numbers are the two answers different?
(a) 8.36
(b) 6.02
(c) 0.449
(d) 29.97
(e) Ravi is 13 years and 10 months old. Write his age in years, truncated to a whole number and rounded to the nearest whole number. Which of the two is the age he gives when someone asks how old he is?

ShowHide solution

(a) Truncated: 8.3. Rounded: the digit after the first decimal place is 6, so round up to 8.4.

(b) Truncated: 6.0. Rounded: the digit after the first decimal place is 2, so the 0 stays as it is: 6.0.

(c) Truncated: 0.4. Rounded: the digit after the first decimal place is 4, so the 4 before it stays as it is: 0.4. (Look only at that one digit. The 9 after it plays no part.)

(d) Truncated: 29.9. Rounded: the digit after the first decimal place is 7, so round up: $29.9 + 0.1 = 30.0$ .

The two answers are different in (a) and (d), where the first digit cut off is 5 or more.

(e) Truncated: 13 years. Rounded: 10 months is more than half of a year, so his age rounds up to 14 years. He says "I am 13": ages are truncated.

(a) Truncated 8.3, rounded 8.4.
(b) Truncated 6.0, rounded 6.0.
(c) Truncated 0.4, rounded 0.4.
(d) Truncated 29.9, rounded 30.0.
The two are different in (a) and (d).
(e) Truncated 13 years, rounded 14 years. He gives his age as 13.

Question 8

Two simple calculators each show eight digits. For $5 \div 9$ , calculator A shows 0.5555555 and calculator B shows 0.5555556.
(a) Which calculator truncates the answer, and which rounds it?
(b) What does each calculator show for $1 \div 6$ ?
(c) What does each calculator show for $1 \div 3$ ? Why are the two displays the same?
(d) Truncate $7 \div 9 = 0.7777\ldots$ to 3 decimal places, and round it to 3 decimal places. Which of the two values is closer to $7 \div 9$ ?

ShowHide solution

(a) $5 \div 9 = 0.55555555\ldots$ , with 5s that go on for ever. Calculator A has cut the digits off after the seventh decimal place, so it truncates. Calculator B has looked at the next digit, a 5, and rounded the last digit up to 6, so it rounds.

(b) $1 \div 6 = 0.16666666\ldots$ Calculator A cuts it off and shows 0.1666666. Calculator B rounds it: the next digit is 6, so the last digit goes up, and it shows 0.1666667.

(c) $1 \div 3 = 0.33333333\ldots$ Both calculators show 0.3333333. The first digit cut off is 3, which is less than 5, so rounding does not round up and gives the same value as truncating.

(d) Truncated: 0.777. Rounded: the digit after the third decimal place is 7, so round up to 0.778. The truncated value is $0.000777\ldots$ below $7 \div 9$ , and the rounded value is $0.000222\ldots$ above it, so the rounded value is closer.

(a) Calculator A truncates and calculator B rounds.
(b) Calculator A shows 0.1666666 and calculator B shows 0.1666667.
(c) Both show 0.3333333, because the first digit cut off is less than 5.
(d) Truncated 0.777, rounded 0.778. The rounded value is closer to $7 \div 9$ .

Question 9

Three classes collect old newspapers for recycling. Class 1A collects 89 kg, class 1B collects 59 kg and class 1C collects 72 kg. A computer program shows each mass as a percentage of the total: 40.4%, 26.8% and 32.7%.
(a) Add the three percentages that the program shows.
(b) Work out each percentage, correct to 3 significant figures. Which of the percentages shown is not correctly rounded, and what did the program do to it?
(c) Add the three correctly rounded percentages.

ShowHide solution

(a) $40.4\% + 26.8\% + 32.7\% = 99.9\%$

(b) The total mass is $89 + 59 + 72 = 220$ kg.

Class 1A: $89 \div 220 \times 100\% = 40.4545\ldots\%$ . The digit after the third significant figure is 5, so round up: 40.5%. The program shows 40.4%, which is not correctly rounded. The program truncated the value: it cut the digits off after the third significant figure.

Class 1B: $59 \div 220 \times 100\% = 26.8181\ldots\%$ , which is 26.8%.

Class 1C: $72 \div 220 \times 100\% = 32.7272\ldots\%$ , which is 32.7%.

For classes 1B and 1C the first digit cut off is less than 5, so truncating and rounding give the same value.

(a) The percentages shown add up to 99.9%.
(b) Class 1A 40.5%, class 1B 26.8%, class 1C 32.7%. The 40.4% shown for class 1A is not correctly rounded: the program truncated it.
(c) The correctly rounded percentages add up to 100.0%.

Question 10

In a survey, 80 students each name their favourite sport: 27 name badminton, 22 football, 17 swimming and 14 basketball. A spreadsheet works out each number as a percentage of the 80 students.
(a) Find the four exact percentages.
(b) The spreadsheet is set to truncate each percentage to a whole number. What does it show, and what do the four values add up to?
(c) Round each percentage to the nearest whole number instead, and add the four values.
(d) Explain why truncated percentages of the parts of a whole can never add up to more than 100%.

ShowHide solution

(a) Badminton: $27 \div 80 \times 100\% = 33.75\%$ . Football: $22 \div 80 \times 100\% = 27.5\%$ . Swimming: $17 \div 80 \times 100\% = 21.25\%$ . Basketball: $14 \div 80 \times 100\% = 17.5\%$ .

(b) Truncated: 33%, 27%, 21% and 17%. Total: $33\% + 27\% + 21\% + 17\% = 98\%$

(d) Truncating never makes a positive number larger, so each truncated percentage is less than the exact one, or equal to it. The exact percentages add up to 100%, so the truncated ones add up to 100% or less. Here they lost $0.75\% + 0.5\% + 0.25\% + 0.5\% = 2\%$ .

Rounding moves some values up and some down, so its errors partly cancel. Here three values went up and one went down, and the total is only 1% over.

(a) Badminton 33.75%, football 27.5%, swimming 21.25%, basketball 17.5%.
(b) Truncated: 33%, 27%, 21% and 17%, which add up to 98%.
(c) Rounded: 34%, 28%, 21% and 18%, which add up to 101%.
(d) Each truncated percentage is less than or equal to the exact one, and the exact ones add up to 100%, so the truncated ones cannot add up to more than 100%.

Key Points

The exact percentages of all the parts of a whole add up to exactly 100%. After rounding, the total may be a little less or a little more, because the rounding errors of the parts add up.
The same happens with rounded amounts of money, lengths and other quantities: the rounded parts may not add up to the rounded total.
Leave each rounded figure as it is. Do not change one to force the total, and say that the figures are rounded.
Find a total from the exact values, and work out each part from its own numbers, not by subtracting rounded parts from 100%.
(On truncation, a little beyond Secondary 1) To truncate is to cut digits off without rounding. A positive number never becomes larger when it is truncated, and the result is the same as rounding only when the first digit cut off is less than 5.
(On truncation, a little beyond Secondary 1) A truncated value can be off by almost a whole unit of the last place kept, and a rounded value by half a unit at most. Round in your own answers unless a question tells you to truncate.