Rounding to Suit the Situation

Deciding when an estimate is enough, rounding up or down by what the situation needs and not by the usual rule, and rounding the safe way to be sure of having enough or of reaching a target.

Before you start

The usual rule of rounding looks at the first digit that is dropped: 5 or more rounds up, less than 5 rounds down. To the nearest whole number, 4.2 is 4 and 7.69 is 8.

To estimate the result of a calculation, round each number to 1 significant figure (or to a convenient number) and work it out in your head: $38.6 \times 5.2 \approx 40 \times 5 = 200$ . The sign $\approx$ means "is approximately equal to".

An estimate that is more than the exact value is an overestimate, and one that is less is an underestimate. Here 200 is a slight underestimate of the exact value, 200.72.

When Is an Estimate Enough?

Many everyday questions do not need an exact answer. "Do I have enough money for lunch?" "About how long is the bus ride?" "Roughly how many people came?" A quick estimate answers each of them. Other questions do need the exact value: a cashier must work out exactly what you pay and exactly how much change to give you.

What an estimate is for

An estimate is a rough value that is close enough to the true value for what you need.

An estimate is enough when the number is used only to decide or to plan: is it enough, about how many, about how long.

Find the exact value when the number itself is what is wanted: a bill, change, a score, a count that goes into a record.

Example 1: Estimate or exact value?

For each situation, say whether an estimate is enough or the exact value is needed.
(a) A cashier at a supermarket works out the total of your shopping.
(b) Before she joins the queue, Mei wonders whether $10 will pay for a plate of chicken rice and a drink.
(c) A news report says how many people watched the fireworks at Marina Bay.
(d) A teacher records each pupil's marks for a test.
(e) Mr Lee decides what time to leave home so as to reach the airport by 6 p.m.

Solution

(a) The exact value. You pay the total, so it must be right to the cent.

(b) An estimate. Mei only needs to know that the total is not more than $10.

(d) The exact value. Each mark is counted, and it goes into a record.

(e) An estimate. The time a journey takes changes from day to day, so Mr Lee can only estimate it, and he should allow a little more to be safe.

An estimate is enough in (b), (c) and (e). The exact value is needed in (a) and (d).

Rounding Up or Down to Suit the Situation

A calculation often gives a decimal when the answer must be a whole number of things. You cannot hire 4.2 buses or buy 7.69 notebooks. The answer has to be rounded to a whole number, and it is the situation, not the usual rule of rounding, that says which way.

Rule: let the situation decide

Round up when everything must be covered: enough buses to seat every pupil, enough tins to paint the whole wall, enough boxes to pack every cake. Any part left over, however small, needs one more.
Round down when only complete ones count: how many you can buy with your money, how many whole pieces can be cut, how many boxes can be filled. A part of one, however large, is not one.

The digit after the decimal point does not decide which way to round. If the calculation gives a whole number, there is nothing to round.

Example 2: Rounding up

A school is taking 168 pupils and teachers to the Science Centre. Each bus has 40 seats. How many buses must the school hire?

Solution

Number of buses $= 168 \div 40 = 4.2$

4 buses have $4 \times 40 = 160$ seats, so 8 people would be left without a seat.

Everyone must have a seat, so round up. 5 buses have $5 \times 40 = 200$ seats, which is enough.

The usual rule of rounding would turn 4.2 into 4, and 8 people would be left behind.

The school must hire 5 buses.

Example 3: Rounding down

Rahul has $20. Notebooks cost $2.60 each. What is the greatest number of notebooks he can buy?

Solution

Number of notebooks $= 20 \div 2.60 = 7.69\ldots$

7 notebooks cost 7 × $2.60 = $18.20, which Rahul can pay.

8 notebooks cost 8 × $2.60 = $20.80, which is more than he has.

Only whole notebooks can be bought, so round down. Rahul has $20 − $18.20 = $1.80 left, which is not enough for another notebook.

The usual rule of rounding would turn 7.69 into 8, and Rahul would be 80 cents short.

Rahul can buy at most 7 notebooks.

Example 4: One calculation, two answers

A row of tiles is to be laid along the foot of a wall 2 m long. Each tile is 30 cm wide.
(a) How many whole tiles fit in the row?
(b) The gap at the end of the row is filled with a piece cut from one more tile. How many tiles are needed for the row altogether?

Solution

Work in one unit: $2\text{ m} = 200\text{ cm}$

Number of tiles $= 200 \div 30 = 6.66\ldots$

6 tiles take up $6 \times 30\text{ cm} = 180\text{ cm}$ , which leaves a gap of $200\text{ cm} - 180\text{ cm} = 20\text{ cm}$ . 7 tiles would take up $7 \times 30\text{ cm} = 210\text{ cm}$ , which is longer than the wall.

(a) Only whole tiles count, so round down.

(b) The gap of 20 cm must be covered as well, and that takes a piece of one more tile, so round up.

The usual rule of rounding turns 6.66… into 7. That is right for (b) and wrong for (a). The same calculation is rounded in two ways, because the two questions need different things.

(a) 6 whole tiles fit in the row.
(b) 7 tiles are needed altogether.

Rounding to suit the situation is a different decision from the usual rule of rounding. Sometimes the two give the same answer, but only by chance:

The calculation gives	Usual rule	What the situation needs
$168 \div 40 = 4.2$ buses	4	5 buses: everyone needs a seat
$20 \div 2.60 = 7.69\ldots$ notebooks	8	7 notebooks: only whole notebooks can be bought
34 cakes in boxes of 6: $34 \div 6 = 5.66\ldots$ boxes	6	6 boxes: every cake must be packed

The usual rule goes to the nearest whole number. The situation goes to the whole number that works, even when that one is further away:

So do not look at the digit after the decimal point. Ask what the question needs: must everything be covered, or do only complete ones count?

The situation decides in the same way when the answer is an amount of money and not a number of things. Money is rounded to the cent, and whether the last cent goes up or down depends on what the money has to do.

Example 5: Rounding money to suit the situation

Seven friends share the cost of a $50 present equally. How much must each of them pay, to the cent, so that the present is paid for in full?

Solution

Share of each friend $= 50 \div 7 = 7.142857\ldots$

By the usual rule of rounding this is $7.14 to the nearest cent. But $7 \times 7.14 = 49.98$ , which is 2 cents less than $50.

The whole cost must be covered, so round up to $7.15. Then $7 \times 7.15 = 50.05$ , which covers $50 with 5 cents to spare.

Each friend must pay $7.15.

Money that is given out goes the other way. If $50 of prize money is shared equally among 7 winners, each winner can be given only $7.14. The share is rounded down, whatever the next digit is, because 7 × $7.15 = $50.05 is more than there is to give. 7 × $7.14 = $49.98, and 2 cents are left over.

Overestimating to Be Safe

Rounding to the nearest value gives an estimate that may be a little too high or a little too low. Often that does not matter. It matters when you must be sure. A book that costs $13.20 is $10 to the nearest ten dollars, but a $10 note will not pay for it, however the price is rounded. To be sure, round every number the safe way, so that the estimate can only be wrong on the side that does no harm.

Rule: round the safe way

To be sure that you have enough (money, time, seats, paint), overestimate what you need: round every cost, or every amount needed, up. If you have at least as much as the overestimate, you certainly have enough.
To be sure that you have reached a target, underestimate what you have: round every amount down. If the underestimate reaches the target, the true total certainly does.

If the safe estimate does not settle the question, an estimate cannot answer it: work out the exact value.

Example 6: Is the money enough?

Mrs Lim has $30. She wants a tin of milk powder at $22.50, a box of cereal at $3.45 and a packet of buns at $2.80. Without working out the exact total, decide whether she has enough money.

Solution

She must be sure, so round every price up to the next dollar: $22.50 becomes $23, $3.45 becomes $4 and $2.80 becomes $3. (By the usual rule $3.45 would be rounded down to $3. Here it is rounded up.)

Overestimate of the total: $23 + $4 + $3 = $30

Every price was rounded up, so the true total is less than $30.

Check: the exact total is $22.50 + $3.45 + $2.80 = $28.75.

Yes, Mrs Lim has enough money.

Example 7: Has the target been reached?

Farid wants to cycle at least 50 km this week. So far he has cycled 12.6 km, 8.3 km, 15.9 km, 10.4 km and 6.7 km. Without working out the exact total, show that he has reached his target.

Solution

He must be sure, so round every distance down to a whole number of kilometres: 12, 8, 15, 10 and 6. (By the usual rule 12.6, 15.9 and 6.7 would be rounded up. Here every distance is rounded down.)

Underestimate of the total $= (12 + 8 + 15 + 10 + 6)\text{ km} = 51\text{ km}$

Every distance was rounded down, so the true total is more than 51 km, and 51 km is already more than the target of 50 km.

Check: the exact total is $(12.6 + 8.3 + 15.9 + 10.4 + 6.7)\text{ km} = 53.9\text{ km}$ .

The underestimate is 51 km, so Farid has cycled more than 50 km. He has reached his target.

Example 8: When the estimate cannot decide

Devi has $20. At a bookshop she picks a comic at $10.40, a notebook at $6.40 and a pen at $3.45. She rounds each price to the nearest dollar, gets $10 + $6 + $3 = $19, and says that she has enough money. Is she right?

Solution

Rounding to the nearest dollar has rounded all three prices down, so $19 is an underestimate. It cannot show that she has enough.

Round every price up instead: $11 + $7 + $4 = $22. This is more than $20, so the overestimate does not settle it either. All that the two estimates tell us is that the true total is between $19 and $22.

The estimates cannot decide, so work out the exact total: $10.40 + $6.40 + $3.45 = $20.25

$20.25 is more than $20.

No. The total is $20.25, so Devi is 25 cents short.

Common Mistakes

Using the usual rule of rounding when the situation decides. 4.2 buses means 5 buses, not 4. 7.69 notebooks means 7 notebooks, not 8.
Answering a different question. "How many boxes can be filled?" and "How many boxes are needed?" are rounded in opposite ways. Read which one is asked.
Rounding when there is nothing to round. $150 \div 6 = 25$ exactly, so 150 pupils at tables of 6 need 25 tables, not 26.
Rounding prices to the nearest dollar to see whether the money is enough. Some prices are then rounded down, so the estimate may be too low. Round every price up.
Rounding the wrong way for a target. To show that a target has been reached, round what you have down, not up.
Trusting an estimate that is too close to call. If the overestimate is more than the money you have, you may or may not have enough. Work out the exact total.

Practice

Work each question on paper before you open its solution.

Question 1

For each situation, say whether an estimate is enough or the exact value is needed.
(a) The change a hawker gives you
(b) The number of chairs to put out for a talk for parents
(c) The amount of rice to cook for a family dinner
(d) The number of votes for each pupil in a class election

ShowHide solution

(a) The exact value. Change is money owed to you, so it must be right to the cent.

(b) An estimate. Nobody knows exactly how many parents will come. It is safer to overestimate, so that nobody has to stand.

(d) The exact value. Votes are counted one by one, and one vote can decide who wins.

An estimate is enough in (b) and (c). The exact value is needed in (a) and (d).

Question 2

Each calculation has been done for you. Give the answer that the situation needs.
(a) 26 people go home by taxi, 4 people to a taxi: $26 \div 4 = 6.5$ . How many taxis are needed?
(b) 100 eggs are put into trays that hold 30 eggs each: $100 \div 30 = 3.33\ldots$ How many trays can be filled?
(c) Kumar has $50, and movie tickets cost $13 each: $50 \div 13 = 3.84\ldots$ How many tickets can he buy?
(d) 181 books go on shelves that hold 20 books each: $181 \div 20 = 9.05$ . How many shelves are needed?

ShowHide solution

(a) Everyone must get home, so round up. 6 taxis take only $6 \times 4 = 24$ people.

(b) Only full trays count, so round down. 3 trays use $3 \times 30 = 90$ eggs, and the 10 eggs left over do not fill a tray.

(c) Only whole tickets can be bought, so round down. 3 tickets cost 3 × $13 = $39. A fourth ticket would bring the cost to $52, which is more than Kumar has.

(d) Every book needs a place, so round up. 9 shelves hold $9 \times 20 = 180$ books, which leaves 1 book without a place.

(a) 7 taxis
(b) 3 trays
(c) 3 tickets
(d) 10 shelves

Question 3

A stall makes 230 curry puffs and packs them in boxes of 12.
(a) How many full boxes can the stall sell?
(b) How many boxes are needed to pack every curry puff?

ShowHide solution

$230 \div 12 = 19.16\ldots$

$19 \times 12 = 228$ , so 19 boxes are full and $230 - 228 = 2$ curry puffs are left over.

(a) Only full boxes count, so round down.

(b) The 2 curry puffs left over need a box as well, so round up.

The same calculation gives two different answers, because the two questions need different things.

(a) The stall can sell 19 full boxes.
(b) 20 boxes are needed to pack every curry puff.

Question 4

Four pupils each did a calculation correctly and then rounded the result. Say whether each pupil has given the answer that the situation needs. If not, give the right answer.
(a) Pens cost $4 each and Ali has $30. $30 \div 4 = 7.5$ , and Ali says, "5 rounds up, so I can buy 8 pens."
(b) 150 pupils are to sit at tables of 6. $150 \div 6 = 25$ , and Bala says, "Everyone must have a seat, so I round up: 26 tables."
(c) A lift carries at most 12 people at a time, and 50 people are waiting for it. $50 \div 12 = 4.16\ldots$ , and Chen says, "1 is less than 5, so the lift must make 4 trips."
(d) $100 of prize money is shared equally among 6 winners. $100 \div 6 = 16.66\ldots$ , and Darren says, "The next digit is 6, so I round up: each winner gets $16.67."

ShowHide solution

(a) Not right. Only whole pens can be bought, so round down. 7 pens cost 7 × $4 = $28, which Ali can pay. 8 pens cost 8 × $4 = $32, which is more than he has.

(b) Not right. $25 \times 6 = 150$ , so 25 tables seat every pupil exactly. The calculation gave a whole number, so there is nothing to round.

(c) Not right. 4 trips carry only $4 \times 12 = 48$ people, and 2 people would be left behind. Everyone must be carried, so round up.

(d) Not right. 6 × $16.67 = $100.02, which is 2 cents more than there is to give. Money that is given out is rounded down, to $16.66. Then 6 × $16.66 = $99.96, and 4 cents are left over.

(a) Ali can buy 7 pens.
(b) 25 tables are needed.
(c) The lift must make 5 trips.
(d) Each winner gets $16.66.

Question 5

The label on a tin of paint says that one tin covers $12\text{ m}^{2}$ of wall. Mr Ong wants to paint walls with a total area of $50\text{ m}^{2}$ .
(a) How many tins must he buy?
(b) A tin costs $38.90. Mr Ong takes the price as $40 to estimate what he will pay. Find his estimate. Is it too high or too low, and why is this the safe way to round?

ShowHide solution

(a) $50 \div 12 = 4.16\ldots$

4 tins cover only $4 \times 12\text{ m}^{2} = 48\text{ m}^{2}$ , which leaves $2\text{ m}^{2}$ of wall unpainted. The whole wall must be covered, so round up.

(b) Estimated cost: 5 × $40 = $200

The price was rounded up, so the estimate is too high. (The exact cost is 5 × $38.90 = $194.50.)

An overestimate is safe: if Mr Ong brings $200, he is sure to have enough.

(a) Mr Ong must buy 5 tins.
(b) His estimate is $200. It is an overestimate, so the money he brings is sure to be enough.

Question 6

Aisha must be at school by 7.30 a.m. She walks for 7 minutes to the bus stop, waits at most 8 minutes for a bus, rides for 18 minutes and walks for another 6 minutes to the school gate. She leaves home at 6.40 a.m. Without adding the exact times, show that she will not be late.

ShowHide solution

She must be sure, so overestimate the time she needs: round every time up to the next 10 minutes. The times of 7, 8 and 6 minutes each become 10 minutes, and 18 minutes becomes 20 minutes.

Overestimate of the time $= (10 + 10 + 20 + 10)\text{ minutes} = 50\text{ minutes}$

50 minutes after 6.40 a.m. is 7.30 a.m.

Every time was rounded up, so the journey takes less than 50 minutes.

Check: the exact time is at most $7 + 8 + 18 + 6 = 39$ minutes, so she arrives by 7.19 a.m.

The journey takes less than 50 minutes, so Aisha reaches school before 7.30 a.m. She will not be late.

Question 7

Mrs Goh has $50 with her. She chooses three T-shirts for her children, at $12.90, $15.80 and $21.20.
(a) Estimate the total by rounding each price to the nearest dollar.
(b) Can she be sure from the estimate in (a) that she has enough money? Find out whether she has.

ShowHide solution

(a) To the nearest dollar the prices are $13, $16 and $21, so the total is about $13 + $16 + $21 = $50.

(b) Two of the prices were rounded up and one was rounded down, so $50 may be too high or too low. She cannot be sure from it.

Round every price up: $13 + $16 + $22 = $51. This is more than $50, so the overestimate does not settle it either. It does not show that she is short: the true total is less than $51, and it may or may not be more than $50.

The safe estimate does not settle the question, so work out the exact total: $12.90 + $15.80 + $21.20 = $49.90

$49.90 is less than $50.

(a) The total is about $50.
(b) No, she cannot be sure from the estimate. The exact total is $49.90, so Mrs Goh has enough money, with 10 cents to spare.

Question 8

Four classes are collecting money for the school's charity drive. So far they have collected $78.50, $91.20, $66.80 and $84.30. Their target is $300 altogether. Without finding the exact total, show that they have reached the target.

ShowHide solution

To be sure that a target has been reached, underestimate what you have: round every amount down, here to the ten dollars below it.

Underestimate of the total: $70 + $90 + $60 + $80 = $300

Every amount was rounded down, so the true total is more than $300.

Check: the exact total is $78.50 + $91.20 + $66.80 + $84.30 = $320.80.

The underestimate is $300, so the classes have collected more than $300. They have reached their target.

Question 9

Wei Jie wants to run at least 20 km in a week. He runs on four days: 4.8 km, 6.3 km, 3.9 km and 5.6 km.
(a) Round every distance down to a whole number of kilometres and add. Does this underestimate show that he has reached his target?
(b) Find out whether he has reached it.

ShowHide solution

(a) Underestimate of the total $= (4 + 6 + 3 + 5)\text{ km} = 18\text{ km}$

18 km is less than 20 km, so the underestimate does not show that he has reached his target. It does not show that he has missed it either: the true total is more than 18 km, and it may or may not be as much as 20 km.

(b) The safe estimate does not settle the question, so work out the exact total: $(4.8 + 6.3 + 3.9 + 5.6)\text{ km} = 20.6\text{ km}$

20.6 km is more than 20 km.

(a) No. The underestimate is 18 km, which is less than the target.
(b) Yes. He has run 20.6 km, which is 0.6 km more than his target.

Question 10

128 pupils and 9 teachers are going on a learning journey to Gardens by the Bay. Each bus has 40 seats and costs $185 to hire.
(a) How many buses are needed?
(b) The school has set aside $800 for the buses. Use an overestimate to show that this is enough.
(c) The cost of the buses is shared equally among the 128 pupils. How much must each pupil pay, to the cent, so that the buses are paid for in full?

ShowHide solution

(a) Number of people $= 128 + 9 = 137$ , and $137 \div 40 = 3.425$

3 buses have only $3 \times 40 = 120$ seats. Everyone needs a seat, so round up.

(b) Round the cost of a bus up to $200. Overestimate of the cost: 4 × $200 = $800

The cost of a bus was rounded up, so the true cost is less than $800.

By the usual rule of rounding this is $5.78 to the nearest cent. But $128 \times 5.78 = 739.84$ , which is 16 cents less than $740.

The whole cost must be covered, so round up to $5.79. Then $128 \times 5.79 = 741.12$ , which covers $740.

(a) 4 buses are needed.
(b) The cost is less than 4 × $200 = $800, so $800 is enough.
(c) Each pupil must pay $5.79.

Key Points

An estimate is enough when the number is used only to decide or to plan. A bill, change, a score or a record needs the exact value.
When the answer must be a whole number of things, the situation decides: round up when everything must be covered, and round down when only complete ones count.
This is not the usual rule of rounding: 4.2 buses means 5 buses, and 7.69 notebooks means 7 notebooks.
One calculation can need two answers: 6.66… tiles means that 6 whole tiles fit and that 7 tiles are needed.
Money is rounded by the situation too: a share of a cost is rounded up to the cent, so that the whole cost is paid; a share of money given out is rounded down, so that there is enough to go round.
To be sure of having enough, overestimate what you need: round every cost up. To be sure of reaching a target, underestimate what you have: round every amount down.
If the safe estimate does not settle the question, work out the exact value.