Chapter 1 Ā· Whole number operations review
Plan and Check Multi-Step Word Problems
Plan two-step and multi-step word problems by finding hidden quantities first, solve them step by step, and judge whether the final answer is reasonable.
You cannot answer some word problems with one calculation. A number you need is missing from the problem, so you must work it out first.
Here is a small one. You buy packs of pencils with pencils in each pack. Then you give away pencils. How many pencils do you have left? The problem never says how many pencils you bought. Find that first: pencils. Only then can you subtract: pencils left.
That total, pencils, is a hidden quantity. A hidden quantity is an amount the problem does not state, but you need it before you can answer the question.
A two-step word problem takes two calculations, like this one. A multi-step word problem takes two or more calculations, so a two-step problem is the shortest kind. Longer problems have more hidden quantities.
Before you calculate, make a plan. Begin at the question and ask three things, in this order.
- What does the question ask for? Name the amount and its unit, such as books or kilometers.
- What do I need to know to find it? If the problem states it, use it. If it does not, you have found a hidden quantity.
- What comes first? Put each hidden quantity before the step that needs it.
So you plan backward from the question, and then you calculate forward from the facts. You already know how to choose the operation for one step. The new skill is putting the steps in order.
Key concept
Plan backward, solve forward, then check
A multi-step problem has one or more hidden quantities between the facts you are given and the answer you want. Plan by starting at the question and asking what you need to know. Solve by finding each hidden quantity in order, and label each result with its unit and what it measures. Check that your final number answers the question, is close to an estimate made with rounded numbers, and fits the story.
Worked example
Two steps: books on shelves
A library gets new books. It puts of them on a display table. It shelves the rest equally on shelves. How many books go on each shelf?
Start at the question. It asks for the number of books on each shelf. To find that, you need the number of books that go on the shelves. The problem does not state that number, so it is the hidden quantity. Find it first.
The library takes the display books out of the 252 new books, so subtract. That leaves 216 books to shelve.
Let b stand for the number of books on each shelf. Then 12 shelves hold 12 Ć b books, written 12b. The shelves hold the 216 books, so the two amounts are equal. An equation states that.
Divide both sides by 12. Dividing by 12 undoes multiplying by 12, and dividing two equal amounts by the same number (not zero) keeps them equal. For the arithmetic: 12 Ć 10 = 120, which leaves 216 ā 120 = 96, and 12 Ć 8 = 96. So b is 10 + 8 = 18.
Answer
The library puts books on each shelf.
You can also write both steps in one line. The parentheses hold the hidden quantity, so you work it out first. Replacing with its value, , changes how the line looks but not what it equals. That is why one line gives the same answer as two separate steps.
A problem with three or more calculations works the same way. It has more than one hidden quantity, and each result feeds a later step.
One habit matters more as problems get longer. Write every result with its unit and what it measures, such as 216 books to shelve. A bare number like is easy to use in the wrong place later.
Worked example
Three steps: a biking goal
Amara wants to bike kilometers this week. She bikes kilometers on each of school days. On Saturday she bikes kilometers. How many kilometers must she bike on Sunday to reach her goal?
Plan backward. The question asks for Sunday's distance in kilometers. For that you need the distance biked so far. For that you need the school-day total. So there are two hidden quantities, and the school-day total comes first.
Five equal distances of 4.9 kilometers means multiply. Split 4.9 into 4 and 0.9. Five of each part is still five of the whole, so the result does not change. The school-day total is 24.5 kilometers.
Add Saturday's distance to get the distance biked so far. Write 24.5 as 24.50, which has the same value, so that tenths line up with tenths and hundredths with hundredths. The distance so far is 33.75 kilometers.
Let k stand for the kilometers Amara must bike on Sunday. The distance so far plus Sunday's distance must equal her goal of 40 kilometers.
Subtract 33.75 from both sides. Subtracting 33.75 undoes adding 33.75, and taking the same amount from two equal distances leaves them equal. Count up to subtract: 33.75 + 0.25 = 34, and 34 + 6 = 40, so k is 0.25 + 6 = 6.25.
Answer
Amara must bike kilometers on Sunday to reach her goal of kilometers.
You have planned and solved. The last job is to judge your answer. An answer is reasonable when it could be true: it answers the question, its size is sensible, and it makes sense in the story.
To judge the size, use an estimate. You already know how to make one with rounded numbers. Run your plan again with easy numbers, then compare. In the biking problem, round to and to . Then , and . The exact answer, kilometers, is close to kilometers, so its size is reasonable.
| Check | Ask yourself | It can catch |
|---|---|---|
Right question | Does my number answer what was asked, with the right unit? | Stopping early and giving a hidden quantity as the answer |
Right size | Is my number close to an estimate made with rounded numbers? | A slipped decimal point or another arithmetic slip that changes the size |
Fits the story | Could this number really happen in this situation? | A wrong step that gives an impossible result, like shares that add up to more than the whole |
Worked example
Which answer is reasonable?
A school rents buses for a trip. Each bus costs dollars. The parent club pays dollars of the total. The students on the trip share the rest equally. Three classmates work out what each student pays: dollars, dollars and dollars. Which answer is reasonable?
Plan backward. One student's share needs the amount left to share. That amount needs the total cost of the buses. So the steps are multiply, subtract, then divide.
Estimate with that plan. Round 282 up to 300 and 96 up to 100. The total cost is about 900 dollars, about 750 dollars is left to share, and each student pays about 7.50 dollars.
Test 696 dollars. It matches the size of the amount that all 96 students share, about 750 dollars. It is not close to one share, about 7.50 dollars. It looks like a hidden quantity given as the final answer, so it fails the right-question check.
Test 72.50 dollars. It is about ten times the estimate. If about 100 students each paid about 70 dollars, they would pay about 7,000 dollars in all. The buses cost less than 900 dollars, so this does not fit the story. A slipped decimal point can cause an error like this.
Test 7.25 dollars. It is close to the estimate of 7.50 dollars, and it is an amount for one student. About 100 shares of 7.25 dollars make about 725 dollars, which fits the 750 dollars to share. It passes all three checks.
Reasonable is not the same as proven, so confirm with the exact steps: total cost, then the amount to share, then one share. For the division: 96 Ć 7 = 672, which leaves 696 ā 672 = 24, and 24 Ć· 96 = 0.25. So one share is 7 + 0.25 = 7.25 dollars.
Answer
Only dollars is reasonable, and the exact steps confirm it. Each student pays dollars.
Here is the idea to keep. A multi-step word problem is a chain: hidden quantities link the facts you are given to the answer you want. Plan backward from the question to find those links. Solve forward one step at a time, and label each result. Then check that your final number answers the question, is close to an estimate and fits the story. These checks cannot prove an answer right, but they tell you when to look again.