Two-Step Multiplication and Division Word Problems

What is a two-step multiplication and division word problem?

A two-step word problem is a mathematical puzzle presented in a real-world context that takes exactly two distinct mathematical operations to solve. In a two-step multiplication and division word problem, you will use a combination of multiplying and dividing.

The intermediate answer from the first step is required to complete the second step.

These problems might require you to multiply and then divide, divide and then multiply, multiply twice, or divide twice. Understanding the story is the key to choosing the right operations.

Find both operations

Word problems use specific language to indicate which operations are needed. You must look for clue words to determine the mathematical actions hidden in the story.

If a problem asks for a combined total based on equal groups, you have multiplication word problems. Look for phrases such as "times as many", "each", "total", or "in all".

If a problem requires sharing a total into equal groups or finding the number of equal groups, you have division word problems. Look for phrases such as "split equally", "divided by", "shared between", or "per group".

Choose a calculation order

Unlike a standard arithmetic equation where you follow strict order-of-operations rules, a word problem tells a narrative. The logical sequence of events in the story dictates your calculation order.

Identify what information is completely known at the beginning of the problem. If you know the size and number of initial groups, you must multiply first to find the total. If you start with a large known total that must be distributed before being expanded, you must divide first.

Use models and equation chains

Drawing a diagram helps organize the two operations visually. A bar model represents the quantities as rectangular blocks, making it easy to see whether you need to scale up through multiplication or partition down through division.

An equation chain records the logical steps algebraically. If a story states that you buy $5$ packs of $12$ pens and split all the pens equally among $4$ people, the chain linking the operations is $5\times 12=60$, followed by $60÷4=15$.

Step-by-step solving method

Follow a consistent method to ensure you complete both steps accurately and do not stop after finding the first number.

1. Read the problem carefully to understand the story.
2. Identify the given numbers and the final goal.
3. Determine the first operation and calculate the intermediate result.
4. Use the intermediate result in the second operation to find the final answer.
5. State the final answer with its correct units.

Interpret units and remainders

The unit of measurement will often change between the first and second steps. You might multiply packages to find a total number of items, and then divide those individual items into a completely different number of containers.

When the second step is division, you may encounter a remainder. Depending on the context of the story, you might need to drop the remainder, round up to the next whole number, or report it directly as a leftover quantity.

Worked examples

Example 1: Repacking items

Question: A store receives $6$ boxes of apples. Each box contains $24$ apples. The apples are then repacked into bags, with exactly $8$ apples in each bag. How many bags are needed?

Method:

1. Find the total number of apples by multiplying the number of boxes by the apples per box.

$6\times 24=144$ apples

1. Divide the total number of apples by the number of apples in each bag.

$144÷8=18$ bags

Answer: The store needs $18$ bags.

Check: Multiply the $18$ bags by the $8$ apples per bag to verify you have the $144$ apples from the first step ($18\times 8=144$).

Example 2: Sharing and scaling

Question: A baker makes $72$ cookies and splits them equally into $6$ display cases. A customer then buys all the cookies in $2$ of the display cases. How many cookies does the customer buy?

Method:

1. Find the number of cookies in each display case by dividing the total by the number of cases.

$72÷6=12$ cookies per case

1. Multiply the amount in one case by the number of cases bought.

$12\times 2=24$ cookies

Answer: The customer buys $24$ cookies.

Check: Divide the $24$ purchased cookies by $2$ to verify there are $12$ cookies per case ($24÷2=12$).

Example 3: Interpreting a remainder

Question: A farmer harvests $15$ rows of carrots, with $30$ carrots in each row. The carrots are then bundled into bunches of $8$. How many complete bunches can the farmer make, and how many carrots are left over?

Method:

1. Find the total number of harvested carrots by multiplying.

$15\times 30=450$ carrots

1. Divide the total by the bundle size to find the bunches and the remainder.

$450÷8=56$ remainder $2$

Answer: The farmer can make $56$ complete bunches with $2$ carrots left over.

Check: Multiply the $56$ bunches by $8$ and add the remainder of $2$ to confirm the starting total ($450$).

Checking both steps

Mistakes often happen when transitioning from the first step to the second, or by choosing the wrong operation entirely. To verify your work, read the question again to ensure your calculation order matches the story. Then, work backward from your final answer using inverse operations.

You can apply multiplication and division fact families to confirm small calculations. For larger numbers, reverse the division algorithm and checking process by multiplying the quotient by the divisor and adding any remainder. If working backward returns the numbers you started with, your steps are correct.

Frequently asked questions

Why do I get the wrong answer even if my math is correct?

If your arithmetic is correct but your answer is wrong, you likely chose the wrong calculation order. Ensure you calculate the intermediate total first before sharing or scaling it further.

Can a two-step problem have the same operation twice?

Yes. A problem might require you to multiply to find a total and then multiply again to find a cost, or divide a group and then divide it again.