Two-Step Word Problems: Meaning, Methods, and Practice

What are two-step word problems?

Two-step word problems are mathematical puzzles that cannot be solved with a single calculation. They require breaking the problem into two distinct parts, where the answer to the first part is needed to solve the second part.

These situations are often called two operation word problems because they use two mathematical operations. Sometimes the operations are the same, such as adding twice. Other times, they use different operations, such as multiplying and then subtracting.

By treating them as two separate but connected math step problems, the final question becomes much easier to answer.

Find the first step

The most important task is locating the hidden question. The hidden question is the piece of information you need to calculate before you can answer the final question asked by the problem.

When you encounter math word problems, read the entire text carefully to determine what is known and what is missing. The first calculation will always involve the two pieces of given information that are directly related to each other.

For example, if you know a farmer has $50$ apples and sells $20$ of them, the first step is to calculate how many apples remain. You must find this hidden amount before you can share the remaining apples equally among $3$ baskets.

Represent each step

Drawing a picture or a diagram helps to clarify which mathematical operations are needed. Bar models are especially useful for representing choosing operations in word problems.

A bar model clearly displays the relationship between the parts and the whole. When the total is unknown, the model shows that you need to add or multiply. When a part is unknown, the model shows that you need to subtract or divide.

Use the answer from step one

The defining feature of a two-step procedure is that the intermediate answer must be carried forward into the next calculation.

Always record the result of the first step with its correct unit or meaning.

If you calculate that a baker has $40$ cookies left after a morning sale, write down "$40$ cookies remaining". Keeping track of exactly what this intermediate number represents prevents confusion when you use it in the second step to find out how many boxes are needed.

Choose the correct order

Executing the operations in the correct order is just as important as choosing the right operations.

When dealing with two step problems with four operations, carefully map out the sequence of events described in the text. Often, the events occur in chronological order, meaning you should perform the first described event as your first calculation. However, if the problem asks you to work backwards from a final amount, you must reverse the order of operations.

Worked examples

By breaking problems down into manageable pieces, even challenging combinations of operations become clear.

Here is a model of two-step addition and subtraction word problems:

Example 1: Finding the remaining quantity

Question: A library starts the week with $350$ books. On Monday, $85$ books are checked out. On Tuesday, $112$ books are checked out. How many books remain in the library?

Method:

1. Step 1: Find the total number of books checked out by adding the two amounts.

$85+112=197$

1. Step 2: Subtract the total number of checked-out books from the starting amount to find the remainder.

$350-197=153$

Answer: There are $153$ books remaining in the library.

Check: Add the remaining books back to the checked-out books to see if it equals the original total: $153+197=350$.

When problems involve grouping or sharing, we follow similar steps to solve two-step multiplication and division word problems.

Example 2: Calculating shared groups

Question: A teacher buys $4$ packs of pencils. Each pack contains $15$ pencils. The teacher shares all the pencils equally among $5$ tables. How many pencils does each table receive?

Method:

1. Step 1: Calculate the total number of pencils by multiplying the number of packs by the pencils per pack.

$4\times 15=60$

1. Step 2: Divide the total number of pencils by the number of tables to find the shared amount.

$60÷5=12$

Answer: Each table receives $12$ pencils.

Check: Multiply the pencils per table by the number of tables: $12\times 5=60$, which matches the total we found in the first step.

Common mistakes

When reasoning through two-step word problems, watch out for these frequent errors:

• Stopping after the first step: It is easy to find the answer to the hidden question and forget that another step is required to answer the final question. Always re-read the problem after your first calculation.
• Using the wrong operation: Mixing up sharing (division) with subtracting, or combining groups (multiplication) with simple addition, will lead to the wrong intermediate result.
• Ignoring units: If step one calculates dollars and step two requires pounds of flour, ensure your numbers align with what the text is asking. Keep track of what your intermediate number represents.

Frequently asked questions

What is the difference between two-step and multi-step word problems?

A two-step word problem requires exactly two mathematical operations to reach the final solution. Multi-step word problems can require three, four, or more connected calculations. The methods used to solve both types are identical; multi-step problems simply have more hidden questions to uncover along the way.

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

Yes. For example, if a store receives $20$ shirts, then receives $15$ more, and finally receives $10$ more, calculating the total requires adding twice: first $20+15=35$, and then $35+10=45$.