Math Core

Recipes, rulers, maps and timers are full of fractions. In a word problem, the hardest part is often deciding what to do. Once you know whether to add or subtract, you can use the skills you already have. Then you check that your answer makes sense.

A plan for word problems

Read the problem and find the question.
Decide: are you putting amounts together (add) or finding what is left or how much more (subtract)?
Estimate with benchmarks like $0$ , $\dfrac{1}{2}$ and $1$ .
Solve with a common denominator.
Check: is your answer close to your estimate?

Add or subtract?

Add when amounts are combined: "in all," "total," "altogether," "more was added."

Subtract when you compare or take away: "how much is left," "how much more," "how much longer," "difference."

Estimating with benchmark fractions

Before you solve, round each fraction to the nearest benchmark: $0$ , $\dfrac{1}{2}$ or $1$ . For example, $\dfrac{5}{12}$ is close to $\dfrac{1}{2}$ (since $\dfrac{6}{12} = \dfrac{1}{2}$ ), and $\dfrac{7}{8}$ is close to $1$ .

7/8 is just one eighth away from 1.

An estimate helps you catch mistakes. If you estimate "about 1" and your answer is $\dfrac{2}{9}$ , something went wrong.

Worked example: Putting amounts together

Nia used $\dfrac{2}{3}$ of a cup of flour for pancakes and $\dfrac{3}{4}$ of a cup for muffins. How much flour did she use in all?

Decide: "in all" means add.

Estimate: each amount is a bit less than 1, so the total is a bit less than 2.

Solve:

\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12} = 1\frac{5}{12}

Check: $1\dfrac{5}{12}$ cups is between 1 and 2. That matches the estimate.

Worked example: How much more?

A green ribbon is $\dfrac{7}{8}$ of a yard long. A blue ribbon is $\dfrac{1}{2}$ of a yard long. How much longer is the green ribbon?

Decide: "how much longer" means subtract.

Solve:

\frac{7}{8} - \frac{1}{2} = \frac{7}{8} - \frac{4}{8} = \frac{3}{8}

The green ribbon is $\dfrac{3}{8}$ of a yard longer.

Check: about 1 minus $\dfrac{1}{2}$ is about $\dfrac{1}{2}$ , and $\dfrac{3}{8}$ is close to $\dfrac{1}{2}$ .

Worked example: Two steps

Leo had $5$ pounds of clay. He used $1\dfrac{3}{4}$ pounds for a bowl and $2\dfrac{1}{3}$ pounds for a vase. How much clay is left?

Step 1: find how much he used.

1\frac{3}{4} + 2\frac{1}{3} = 1\frac{9}{12} + 2\frac{4}{12} = 3\frac{13}{12} = 4\frac{1}{12}

Step 2: subtract from 5. Write 5 as $4\dfrac{12}{12}$ .

4\frac{12}{12} - 4\frac{1}{12} = \frac{11}{12}

Leo has $\dfrac{11}{12}$ of a pound left.

Check: he used about $2 + 2 = 4$ pounds, so about 1 pound should be left. $\dfrac{11}{12}$ is close to 1.

Common mistake

Watch out for extra information. "Maya ran $\dfrac{3}{4}$ mile on Monday and $\dfrac{1}{2}$ mile on Tuesday" does not always mean "add." If the question asks how much farther she ran on Monday, you subtract.

Practice

Practice 1

Sam drank $\dfrac{1}{2}$ of a bottle of water in the morning and $\dfrac{3}{8}$ of the bottle in the afternoon. How much of the bottle did he drink in all?

Type a number.

Practice 2

A pitcher is $\dfrac{2}{3}$ full of juice. Ana pours out $\dfrac{1}{2}$ of a pitcher. What fraction of the pitcher is still full?

Type a number.

Practice 3

Ben says $\dfrac{1}{2} + \dfrac{2}{5} = \dfrac{3}{7}$ . Which estimate shows his answer can't be right?

Practice 4

A worm crawled $\dfrac{3}{4}$ of a foot. A snail crawled $\dfrac{1}{3}$ of a foot. How much farther did the worm crawl, in feet?

Type a number.

Practice 5

A plant was $3\dfrac{1}{2}$ inches tall. It grew $1\dfrac{3}{4}$ inches. How tall is it now, in inches?

Type a number.

Practice 6

Ivy has a board $4\dfrac{1}{2}$ feet long. She cuts off a piece $2\dfrac{2}{3}$ feet long. How long is the board that is left, in feet?

Type a number.

Practice 7

Of the students in a class, $\dfrac{1}{2}$ walk to school and $\dfrac{1}{5}$ ride bikes. The rest ride the bus. What fraction of the class rides the bus?

Type a number.

Practice 8

A trail mix recipe uses $1\dfrac{1}{3}$ cups of nuts, $\dfrac{3}{4}$ cup of raisins and $\dfrac{1}{2}$ cup of seeds. How many cups of trail mix does it make?

Type a number.