Decimal problems in words
A story hands you some decimals and a question. How do you tell whether to add, subtract, multiply or divide, and how do you know the answer makes sense?
Where you are 8 in Decimal arithmetic
When you reach for it
Wren keeps a fruit and vegetable stall at a Saturday farmers’ market. Everything on her table is weighed in kilograms and priced in dollars and cents, so nearly every number she meets has a decimal point. One customer asks what 2.6 kilograms of apples cost; another asks how much heavier one bag is than another.
The arithmetic is already in hand: adding and subtracting decimals, multiplying them, and dividing them, by long division when the divisor is whole and by scaling first when it is not. A story adds a part before the arithmetic and a part after it. Before: the story never says “multiply”, it says “what do 2.6 kilograms cost”, and you have to see the multiplication in it. After: the algorithm returns a number such as 5.357\ldots, and the story decides whether that means 5 costumes, 6 meters of fabric to buy, or 5.36 dollars.
The lesson takes two days. Day 1 is one-step stories: choosing the operation, estimating first and checking that the answer fits. Day 2 is stories with several steps, and quotients that do not come out even.
The procedure
Day 1: the action in the story picks the operation
Read for what happens, not for a key word. “Wren sold 2.5 kg more plums than pears” contains “more”, but if the plums are known and the pears are asked for, the answer comes from subtracting. What decides the operation is what the story does to the quantities, and there are only a few things it can do:
| What the story does | Operation | A story of that kind |
|---|---|---|
| puts amounts together, or one grows by another | add | two bags of tomatoes go into one box |
| takes an amount away, or compares two amounts | subtract | how much heavier is one bag than the other? |
| repeats one amount, or prices a quantity at a rate | multiply | 2.6 kg at 3.45 dollars per kilogram |
| splits an amount into equal parts, or asks how many parts of a given size fit | divide | 186.50 dollars raised equally by 5 runners |
Addition and subtraction undo each other, and so do multiplication and division, so a story can often be read two ways. “How much more does a runner who has raised 32.60 dollars need to reach 50 dollars?” is the subtraction 50 - 32.60, and also the missing number in 32.60 + {?} = 50. Both readings give the same answer.
Division answers two different questions. Splitting a total among a known number of people asks how much is in each part. Asking how many 1.4-meter pieces fit in a length of fabric asks how many parts there are when the size of each is known. Both undo a multiplication, so both are division.
Let the units check the choice. Units behave like the numbers they ride on. A price in dollars per kilogram times a mass in kilograms gives dollars:
Dollars divided by kilograms gives dollars per kilogram, a price; dollars divided by dollars per kilometer gives kilometers. Only amounts of the same kind in the same unit can be added or subtracted. If a planned calculation does not come out in the unit the question asks for, the operation is wrong.
Estimate before you calculate. Round each number to one that is easy to work with in your head (Estimating by rounding, Friendly numbers and deliberate over and under estimates) and do the chosen operation on the rounded numbers. The estimate says roughly what size of answer to expect, so a slipped decimal point stands out at once, and an estimate that makes no sense in the story points to the wrong operation before any exact work is spent.
Calculate exactly, then read the answer back into the story. Is it close to the estimate? Is it in the unit asked for? Is its size right against the other numbers? Two size facts help. Multiplying by a number less than 1 gives less than you started with: 0.6 kg of olives costs less than 1 kg does. Dividing by a number less than 1 gives more: 3 kg of beans fills 3 \div 0.5 = 6 bags of 0.5 kg. Then answer in a sentence with the unit.
Day 2: several steps, and quotients that do not come out even
Work back from the question. The question names the last quantity, and that quantity needs something the story did not give directly. “How much change from 50 dollars?” needs the amount spent; that needs the cost of each item; each cost is a price times an amount. Find the hidden quantities in that order, then come forward to the answer. Estimate the whole chain too, and keep each step’s unit in sight.
A quotient is not always the answer. Cutting 7.5 meters of fabric into 1.4-meter pieces gives 7.5 \div 1.4 = 5.357\ldots, and nobody can sew 5.357 costumes. What happens next depends on the question, never on one fixed rounding rule, just as for remainders in whole-number division.
- Round down when only complete things count. Five pieces of 1.4 meters can be cut, and a sixth would need more than is left.
- Round up when every last bit has to be covered. If the costumes need 16.8 meters and the shop sells only whole meters, 16 meters falls short, so 17 meters must be bought.
- Find the leftover when the question is about what remains. Five pieces use 5 \times 1.4 = 7 meters, and 7.5 - 7 = 0.5 meters are left over.
Money stops at the cent. A cent is a hundredth of a dollar and no coin is smaller, so money is written with two decimal places (Money written as decimals). A quotient in dollars that runs past the hundredths is rounded to the hundredths place (Rounding decimals), usually to the nearest cent. But when money has to cover a cost, the nearest cent can fall short, and the story may call for rounding up.
A long quotient can be stopped early. 40 \div 3 = 13.333\ldots never ends. To round to the nearest cent, the thousandths digit is the only one past the hundredths you need.
Worked examples
Examples 1 to 4 belong to Day 1, and Examples 5 to 7 to Day 2.
Same numbers, different questions
Wren has two bags of tomatoes. One holds 2.4 kg and the other 1.85 kg.
(a) What do the two bags weigh together?
(b) How much heavier is the first bag than the second?
Solution(a) The two amounts are put together into one total, so add. Estimate: about 2.5 + 2 = 4.5 kg. Lining up the places, with a zero written in so both numbers have hundredths:
The bags weigh 4.25 kg together, close to the estimate and more than either bag.
(b) Comparing two amounts asks for their difference, so subtract the smaller from the larger:
The first bag is 0.55 kg heavier. Check by adding back: 1.85 + 0.55 = 2.40.
The numbers were the same in both parts. Only the question changed, and it chose the operation.
Pricing by the kilogram
Kofi buys 2.6 kg of apples at Wren’s stall. Apples cost 3.45 dollars per kilogram. What does he pay?
SolutionEvery kilogram costs the same 3.45 dollars and Kofi buys 2.6 kilograms, so the price is repeated 2.6 times: multiply. The units agree, since dollars per kilogram times kilograms gives dollars.
Estimate: about 3 \times 3 = 9 dollars. Exactly, with the standard algorithm: 345 \times 26 = 8970, and the factors have 2 + 1 = 3 decimal places between them, so
Kofi pays 8.97 dollars. That agrees with the estimate, and it sits between the cost of 2 kg, 6.90 dollars, and of 3 kg, 10.35 dollars.
Two kinds of sharing
Runners in a school fun run collect sponsorship money for a charity.
(a) Mateo’s team of 5 runners raised 186.50 dollars, each runner raising the same amount. How much did each raise?
(b) Yuki’s sponsor pays 2.50 dollars for every kilometer she runs, and paid her 17.50 dollars. How far did she run?
Solution(a) A total is split into a known number of equal parts, and the question asks for the size of each part: divide. Estimate with a compatible number: 185 \div 5 = 37. Exactly,
Each runner raised 37.30 dollars. Check: 37.30 \times 5 = 186.50.
(b) Here the size of each part is known: each kilometer is worth 2.50 dollars. The question asks how many such parts make 17.50 dollars, division of the other kind. Dollars divided by dollars per kilometer leaves kilometers. Estimate: 10 dollars every 4 kilometers, so a little under 8 km. Scaling both numbers by 10 makes the divisor whole and leaves the quotient unchanged:
Yuki ran 7 km. Check: 7 \times 2.50 = 17.50.
Multiply or divide? Let the units decide
(a) A customer buys 0.35 kg of cheese priced at 18.40 dollars per kilogram. What does it cost?
(b) Kofi paid 7.14 dollars for 1.7 kg of plums. What is the price per kilogram?
Solution(a) A mass is priced at a rate, so multiply: dollars per kilogram times kilograms gives dollars. Judge the size first. Less than a kilogram must cost less than 18.40 dollars, and 0.35 is a little more than a third, while a third of 18 is 6. Expect a little over 6 dollars. 1840 \times 35 = 64400, and 2 + 2 = 4 decimal places give
The cheese costs 6.44 dollars, less than 18.40, as multiplying by a number less than 1 must give.
(b) The question asks for dollars per kilogram, and the units say how to get it: dollars divided by kilograms. Estimate: 7 \div 2 = 3.5, and 1.7 is less than 2, so a little more, around 4.
The plums cost 4.20 dollars per kilogram, written with two decimal places because it is money. Check: 4.20 \times 1.7 = 7.14. Multiplying instead would give dollars times kilograms, which is not a price at all.
Three steps to the change
Saoirse is sewing costumes for the school play. She buys 3 meters of red satin at 6.85 dollars per meter and 2.5 meters of lining at 3.40 dollars per meter, and pays with a 50-dollar note. How much change does she get?
SolutionWork back from the question. The change is what is left of 50 dollars once the amount spent is taken away. The amount spent is the satin’s cost plus the lining’s cost, and each cost is a price per meter times a number of meters.
Estimate the whole chain: satin about 3 \times 7 = 21 dollars, lining about 3 \times 3 = 9 dollars, so about 30 dollars spent and 20 dollars change. Now exactly:
Saoirse gets 20.95 dollars change, close to the estimate. Check by adding back: 29.05 + 20.95 = 50.
Fabric for the costumes
Saoirse’s pattern needs 1.4 meters of fabric for each costume.
(a) She has a piece 7.5 meters long. How many costumes can she cut from it, and how much fabric is left over?
(b) The play needs 12 costumes in all. If the shop sells fabric only in whole meters, how many meters should be bought?
Solution(a) The question is how many pieces of a given size fit into 7.5 meters: divide. Estimate: 7.5 \div 1.5 = 5. Exactly,
A costume needs a whole 1.4-meter piece, so only complete pieces count: round down to 5 costumes. They use 5 \times 1.4 = 7 meters, so 7.5 - 7 = 0.5 meters are left over, shorter than 1.4 meters, which confirms that a sixth costume does not fit.
The decimal part of the quotient, 0.357\ldots, is not the leftover length. It counts costumes: the leftover is about a third of one costume’s piece, and as a length that is \displaystyle \frac{5}{14} \times 1.4 = 0.5 meters.
(b) Twelve costumes need 12 \times 1.4 = 16.8 meters. Buying 16 meters would leave the last costume short, so here round up: buy 17 meters, with 17 - 16.8 = 0.2 meters to spare.
Cutting asks how many whole pieces the fabric holds, so it rounds down. Buying asks how much fabric covers every piece, so it rounds up.
Sharing a pledge that does not divide evenly
Linnea, Mateo and Kofi run the fun run as a team and promise the charity 40 dollars between them, split equally. How much should each pay?
SolutionA total is shared among a known number of people: divide. Estimate: about 13 dollars each.
The remainder at every step is 1, so the digit 3 repeats for ever. Money stops at the cent, and to the nearest cent the share is 13.33 dollars, since the thousandths digit is 3.
Now read that back into the story. Three payments of 13.33 dollars make
dollars, one cent short of the promise. If all three must pay the same and the 40 dollars must be covered, round up instead: each pays 13.34 dollars, which makes 40.02 dollars. If the amounts may differ, two pay 13.33 dollars and one pays 13.34 dollars, exactly 40 dollars. No share is both equal and exact; the story says which matters more.
Boundaries
The story must say enough. An operation can be chosen only when the story fixes what happens to the quantities and gives the numbers needed. “Wren sold 2.5 kg more plums than pears” gives neither amount, so it can be finished only once one of them is known.
The quantities must belong together. Only amounts of the same kind in the same unit can be added or subtracted. A mass in grams and one in kilograms must be brought to one unit first, and a mass can never be added to a price.
A rate must be steady. Multiplying 2.6 kg by 3.45 dollars per kilogram assumes every kilogram costs the same. If the price drops after the first 2 kilograms, one multiplication no longer gives the bill.
Estimates catch large errors, not small ones. An estimate exposes a misplaced decimal point or a wrong operation. It cannot tell 8.97 from 8.79; for that, check exactly by undoing the operation, as in Examples 1, 3 and 5.
Rounding a quotient is the story’s decision. Whole things to be made round down, amounts that must be covered round up, and money rounds to the cent. When the quotient is exact, as 17.50 \div 2.50 = 7, nothing is rounded at all.
Where this goes next
Pricing by the kilogram and paying by the kilometer are rates, and the price of one kilogram found by dividing in Example 4 is a unit price, which is how a large bag and a small bag are compared fairly. The habits of this lesson, choosing the operation from what happens, estimating first and letting the units and the story judge the answer, carry into every problem that arrives in words, whether its numbers are fractions, percentages or measurements.
Exercises
Exercises 1 to 11 go with Day 1 (Examples 1 to 4), Exercises 12 to 21 with Day 2 (Examples 5 to 7), and the reasoning group draws on both.
Day 1: choose the operation
For each story, say which operation it needs and why, then answer it.
- Wren sells 4.75 kg of potatoes in the morning and 6.8 kg in the afternoon. How many kilograms does she sell in all?
- A pumpkin on Wren’s stall weighs 5.62 kg and a squash weighs 3.9 kg. How much heavier is the pumpkin?
- Strawberries cost 4.80 dollars a box. What do 6 boxes cost?
- Wren cuts a 3.6 kg wheel of cheese into 8 equal wedges. How much does each wedge weigh?
- Wren’s honey jars each hold 0.35 kg. How many jars can she fill from 8.4 kg of honey?
- Mateo’s sponsor pays 1.75 dollars for each kilometer he runs, and Mateo runs 8.4 km. How much does the sponsor pay?
- Linnea has raised 63.45 dollars toward a target of 100 dollars. How much more does she need?
Day 1: estimate, then calculate
- Yuki buys 1.8 kg of pears at 2.90 dollars per kilogram. Estimate the cost, then find it exactly.
- Saoirse paid 11.34 dollars for 2.7 kg of cherries. Estimate the price per kilogram, then find it exactly.
- A price label says that 0.6 kg of olives at 12.50 dollars per kilogram costs 75 dollars. Use an estimate to show the label is wrong, then find the correct price.
- Kofi’s sponsor pledged 2.25 dollars for each lap of the track, and the pledge came to 27 dollars. Use the units to choose the operation, then find how many laps Kofi ran.
Day 2: more than one step
- A customer buys 3 jars of honey at 7.25 dollars each and a loaf of bread at 4.60 dollars, and pays with 30 dollars. How much change does the customer get?
- Mateo runs 5 km. One sponsor pays 1.20 dollars per kilometer, another pays 0.85 dollars per kilometer, and his aunt gives 10 dollars whatever the distance. How much does he raise?
- Saoirse makes 4 capes using 1.35 m of fabric each and 3 sashes using 0.6 m each, all from one fabric at 5.40 dollars per meter. What does the fabric cost?
- Three runners raised 48.50, 62.75 and 39.15 dollars, and their target was 200 dollars. How much is still needed?
Day 2: what the quotient means
- A roll holds 10 m of ribbon, and each costume needs a bow made from 0.8 m. How many bows can be cut, and how much ribbon is left?
- Each of 9 costumes needs 2.15 m of fabric, and the shop sells only whole meters. How many meters should be bought?
- Each of the 130 runners at the fun run is given 0.75 liters of water, poured from 2-liter bottles. How many bottles are needed?
- Wren sells a bag of 6 lemons for 2.50 dollars. What is the price of one lemon, to the nearest cent?
- Kofi, Linnea and Yuki share the 25-dollar fee for a market stall equally, each paying the same amount in whole cents. How much should each pay so that the fee is covered?
- Wren packs 14.5 kg of apples into bags of 1.5 kg. How many full bags does she make, and how much is left over?
Reasoning
- Write one story about Wren’s stall that uses 4.8 and 1.2 and needs division, and another that uses the same two numbers and needs multiplication. Answer both.
- In Exercise 16, 10 \div 0.8 = 12.5. Explain why the ribbon left over is 0.4 m and not 0.5 m.
- Linnea’s sponsor pays 0.80 dollars per kilometer, and she runs 6.5 km. Without multiplying, explain why the sponsor pays less than 6.50 dollars.
- Wren sold 2.5 kg more plums than pears. (a) If she sold 7.25 kg of plums, how many kilograms of pears did she sell? (b) If she sold 7.25 kg of pears, how many kilograms of plums did she sell? (c) Explain why the word “more” did not decide the operation.
Answers
- Add: 11.55 kg. The amounts are put together: 4.75 + 6.80 = 11.55.
- Subtract: 1.72 kg. Two masses are compared: 5.62 - 3.90 = 1.72.
- Multiply: 28.80 dollars. One price repeated six times: 4.80 \times 6 = 28.80.
- Divide: 0.45 kg. A total split into 8 equal parts: 3.6 \div 8 = 0.45.
- Divide: 24 jars. How many parts of 0.35 kg fit: 8.4 \div 0.35 = 840 \div 35 = 24.
- Multiply: 14.70 dollars. Dollars per kilometer times kilometers gives dollars: 1.75 \times 8.4 = 14.70.
- Subtract: 36.55 dollars. What is missing from the target: 100 - 63.45 = 36.55.
- About 6 dollars, from 2 \times 3; exactly 5.22 dollars. 18 \times 29 = 522, and two decimal places give 1.8 \times 2.9 = 5.22.
- About 4 dollars per kilogram, from 12 \div 3; exactly 4.20 dollars per kilogram. 11.34 \div 2.7 = 113.4 \div 27 = 4.2.
- 7.50 dollars. Less than 1 kg must cost less than 12.50 dollars, so 75 dollars is far too much. 12.50 \times 0.6 = 7.50.
- 12 laps. Dollars divided by dollars per lap gives laps: 27 \div 2.25 = 2700 \div 225 = 12.
- 3.65 dollars. Honey 3 \times 7.25 = 21.75; spent 21.75 + 4.60 = 26.35; change 30 - 26.35 = 3.65.
- 20.25 dollars. The sponsors pay 1.20 + 0.85 = 2.05 dollars per kilometer, so 2.05 \times 5 = 10.25; with the aunt’s 10 dollars, 20.25.
- 38.88 dollars. Capes 4 \times 1.35 = 5.4 m and sashes 3 \times 0.6 = 1.8 m make 7.2 m; 7.2 \times 5.40 = 38.88.
- 49.60 dollars. Raised 48.50 + 62.75 + 39.15 = 150.40; still needed 200 - 150.40 = 49.60.
- 12 bows, with 0.4 m left. 10 \div 0.8 = 12.5, and only whole bows count, so round down. 10 - 12 \times 0.8 = 10 - 9.6 = 0.4.
- 20 m. The costumes need 9 \times 2.15 = 19.35 m; 19 m falls short, so round up.
- 49 bottles. The water needed is 130 \times 0.75 = 97.5 liters, and 97.5 \div 2 = 48.75; 48 bottles fall short, so round up.
- 0.42 dollars. 2.50 \div 6 = 0.41666\ldots, and the thousandths digit is 6, so round up.
- 8.34 dollars each. 25 \div 3 = 8.333\ldots, but 3 \times 8.33 = 24.99 is one cent short, so round up: 3 \times 8.34 = 25.02.
- 9 full bags, with 1 kg left. 14.5 \div 1.5 = 9.666\ldots, rounded down for full bags; 14.5 - 9 \times 1.5 = 1.
- Stories vary. Division: Wren packs 4.8 kg of beans into bags of 1.2 kg, and 4.8 \div 1.2 = 4 bags. Multiplication: beans cost 4.80 dollars per kilogram and a customer buys 1.2 kg, so 4.80 \times 1.2 = 5.76 dollars. A division story asks how many parts or how much is in each; a multiplication story repeats an amount or prices a quantity at a rate.
- The 0.5 in 12.5 counts bows, not meters: half a bow’s worth of ribbon is left. Each bow takes 0.8 m, so half of one is 0.5 \times 0.8 = 0.4 m.
- At 1 dollar per kilometer, 6.5 km would earn 6.50 dollars. Each kilometer earns less than 1 dollar, so the total is less: multiplying 6.5 by a number less than 1 gives less than 6.5.
- (a) 4.75 kg. The pears are the smaller amount: 7.25 - 2.5 = 4.75. (b) 9.75 kg. The plums are the larger amount: 7.25 + 2.5 = 9.75. (c) The same sentence led to a subtraction in (a) and an addition in (b). What decided was which amount was known: going from the larger to the smaller subtracts, and going from the smaller to the larger adds.