Chapter 4 · Whole number multiplication and division
Multi-Step Word Problems and Finding Mistakes
Solve word problems that need both multiplying and dividing by finding the hidden question first. Then spot and fix mistakes in solved problems.
Some word problems take more than one step. A problem like this is called a multi-step problem. You need two or more calculations to solve it.
Here is a small one. You have bags with marbles in each bag. You share all the marbles equally among friends. How many marbles does each friend get?
You cannot share the marbles until you know how many there are. The problem never asks How many marbles are there in all? But you must answer that first. A question like this is called a hidden question. The problem does not ask it, but you need its answer before you can go on.
- Hidden question: , so there are marbles in all.
- Question asked: , so each friend gets marbles.
In this lesson you will solve bigger problems the same way. You will also find and fix mistakes in problems that someone else has already solved.
Key concept
Answer the hidden question first
A multi-step problem asks one question, but another question hides inside it. Find the hidden question and answer it first. Write a label next to every result. A label is a word or two that tells what the number counts, such as marbles in all. Then use your result to answer the question that was asked. You are done only when your last label matches that question.
Worked example
Example 1: Muffins into boxes
A baker fills trays. Each tray holds muffins. She packs all the muffins into boxes, with muffins in each box. How many boxes does she fill?
Find the question asked: how many boxes? To pack boxes, you first need to know how many muffins there are. The problem does not ask for that number, so it is the hidden question: how many muffins are there in all?
Answer the hidden question. There are 12 trays with 18 muffins on each tray. That is 12 equal groups of 18, and joining equal groups means multiply. Split 18 into 10 and 8. Then 12 times 10 is 120, 12 times 8 is 96, and 120 plus 96 is 216. Write a label so you remember that 216 counts muffins.
Now answer the question asked. The 216 muffins are split into equal groups of 24, one group for each box. Finding how many equal groups there are means divide. Think: how many 24s make 216? Since 24 times 9 is 216, there are 9 groups. Label the result: it counts boxes.
Check your work. The problem asked for boxes, and your last label is boxes, so you answered the right question. Now check the calculation by multiplying. Nine boxes with 24 muffins in each box use all 216 muffins.
Answer
The baker fills boxes.
In Example 1, the number was not the final answer. It was a number you needed on the way. A number like this is called an intermediate result, which means an in-between result.
When you find an intermediate result, write down what it stands for. In Example 1 you wrote muffins next to . That label tells you what to do with the number: it is the amount you pack into boxes.
The table shows each step of the muffin problem three ways: in words, as a number sentence with symbols, and as a result with its label. Find in both rows. The result of the first step is the starting number of the second step.
| Step | In words | Number sentence | What the result stands for |
|---|---|---|---|
Hidden question | trays with muffins on each tray | muffins in all (intermediate result) | |
Question asked | muffins packed with in each box | boxes (final answer) |
Worked example
Example 2: Flashlights for camp cabins
A camp has campers. Each cabin has beds for campers, and every camper needs a bed. The camp puts flashlights in every cabin it uses. How many flashlights does the camp need?
Find both questions. The question asked is: how many flashlights? Flashlights go into cabins, so you need the number of cabins first. The hidden question is: how many cabins does the camp use?
Answer the hidden question. The 250 campers go into groups of 12, one group for each cabin. Finding how many groups of 12 there are means divide, so this time the first step is a division. Since 12 times 20 is 240, there are 20 full groups. Then 250 minus 240 leaves 10 campers, too few for another full group. So the remainder, which is the amount left over, is 10.
Write what each number stands for. The 20 means 20 full cabins. The remainder 10 means 10 campers who still need beds. If the camp used only 20 cabins, those 10 campers would have nowhere to sleep. They need one more cabin.
Now answer the question asked. Each of the 21 cabins gets 3 flashlights. That is 21 equal groups of 3, and joining equal groups means multiply. Label the result: it counts flashlights.
Check your work. The 21 cabins have 252 beds, which is enough for 250 campers. Your last label is flashlights, and the problem asked for flashlights.
Answer
The camp needs flashlights, which is for each of its cabins.
You can use the same habits to check a problem that someone else has solved. Checking the calculations is not enough. The work that ended with boxes had a correct multiplication and still gave a wrong answer.
Here are three kinds of mistakes to look for. One of them is choosing the wrong operation. An operation is a way of working with numbers, such as multiplying or dividing. The table shows each kind of mistake in a problem you have already seen. It also gives a question that catches the mistake.
| Kind of mistake | What it looks like, and why it fails | Question that catches it |
|---|---|---|
Wrong operation | Sharing marbles among friends, but multiplying: . No friend can get more than . | Does this step join equal groups (multiply) or split into equal groups (divide)? |
Ignored remainder | Using cabins when . Then campers have no bed. | What does the leftover amount mean in this story? |
Answered a different question | Writing boxes when counts muffins. The work stopped at the hidden question. | Does the label on the answer match the question that was asked? |
Worked example
Example 3: Find and fix the mistake
A printer prints pages each minute. It runs for minutes. The pages are stapled into booklets of pages each. How many booklets can be made? Kai solved it this way: first , then , so the answer is booklets. Find Kai's mistake and fix it.
Check Kai's first step and give its result a label. The printer makes 24 pages in each of 15 minutes. That is 15 equal groups of 24, so multiplying is right. The calculation is right too: 24 times 10 is 240, 24 times 5 is 120, and together they make 360. So 360 stands for pages.
Check the operation in Kai's second step. The story splits the 360 pages into equal groups of 18, one group for each booklet. Splitting into equal groups means divide, but Kai multiplied. The answer does not make sense either. Each booklet uses up 18 pages, so there must be fewer booklets than pages, not more.
Fix the second step by dividing. Think: how many 18s make 360? Since 18 times 2 is 36, 18 times 20 is 360. So there are 20 groups, and the result counts booklets.
Ask the other two checking questions. The division comes out even, so the remainder is 0 and no pages are left over. The last label is booklets, and the problem asked for booklets. Multiply to check the calculation.
Answer
Kai chose the wrong operation in the second step. Kai should divide: . So booklets can be made.
The big idea: a multi-step problem hides a question inside it. Find the hidden question and answer it first. Label every result so you know what it stands for. Keep going until your last label matches the question that was asked.
Use the same habits to check any solved problem, whether it is yours or someone else's. Ask three things. Is each operation right? What does the remainder mean? Does the answer match the question?