Multi-Step Fraction Word Problems and Finding Mistakes | Grade 5 Math | LevelKite

You already know how to solve a fraction word problem that takes one calculation. Some stories need more than one. A multi-step problem is a problem that needs two or more calculations. You do them one after another, and the answer from one step becomes a number in the next step.

Picture a jug. You pour in some juice, pour in some more, and then drink some. To find how much is left, you first add the two pours. Then you subtract what you drank. Each step has its own job. In this lesson, you will name the job of each step before you calculate.

Every fraction is a fraction of something. That something is the whole. When a story says $\frac{3}{4}$ mile, it means 3 of the 4 equal parts of 1 mile. The whole is 1 mile.

A mixed number is a whole number and a fraction written together, like $2\frac{1}{2}$ . In $2\frac{1}{2}$ gallons, you have 2 full gallons and $\frac{1}{2}$ of 1 gallon. The whole is 1 gallon.

The whole matters because the same number can stand for very different amounts. $\frac{3}{4}$ of a mile is a distance. $\frac{3}{4}$ of an hour is a time. In a story, you can add or subtract fractions only when they are parts of the same whole. Then you measure every part against the same thing, so the answer has a clear meaning.

The same number can refer to different wholes

What the story says	Number	The whole
three fourths of a mile	$\frac{3}{4}$	1 mile
three fourths of an hour	$\frac{3}{4}$	1 hour
two and one half gallons	$2\frac{1}{2}$	1 gallon
one half of a short book	$\frac{1}{2}$	the short book
one half of a long book	$\frac{1}{2}$	the long book

Key concept

Name the whole and name the step

Before you calculate, name two things.

Name the whole. What is each fraction a part of? In a story, add or subtract fractions only when they share the same whole.
Name the step. What does this step find? If the story joins amounts, add. If the story takes an amount away, or asks what is left or how much more, subtract.

Then calculate one step at a time. At the end, look back and ask whether the size of your answer fits the story.

Worked example

Two steps, one whole

Nia has $\frac{7}{8}$ yard of ribbon. She uses $\frac{1}{4}$ yard on a card and $\frac{3}{8}$ yard on a gift. How much ribbon does she have left?

Name the whole. All three numbers count parts of 1 yard, so they share the same whole. You may add and subtract them.
First, find the ribbon Nia used. She used the card piece and the gift piece, so join them: add. In words, the ribbon used is the card piece plus the gift piece. Each fourth of a yard splits into 2 eighths, so one fourth is two eighths. Now both pieces are in eighths.
$\frac{1}{4} + \frac{3}{8} = \frac{2}{8} + \frac{3}{8} = \frac{5}{8}$
Next, find the ribbon left. Using ribbon takes it away, so subtract. In words, the ribbon left is the ribbon at the start minus the ribbon used. Both numbers are already in eighths. Two eighths make one fourth, so write the answer as one fourth.
$\frac{7}{8} - \frac{5}{8} = \frac{2}{8} = \frac{1}{4}$
Look back. The ribbon used and the ribbon left should add up to the ribbon Nia started with. Five eighths plus two eighths is seven eighths, so the answer fits the story.
$\frac{5}{8} + \frac{2}{8} = \frac{7}{8}$

Answer

Nia has $\frac{1}{4}$ yard of ribbon left.

Worked example

Mixed numbers and a number that does not belong

A fish tank has $6\frac{1}{2}$ gallons of water. Dana takes out $2\frac{1}{4}$ gallons, scrubs the glass for $\frac{1}{2}$ hour, and then pours in $3\frac{1}{8}$ gallons of fresh water. How many gallons of water are in the tank now?

Name the wholes. Three of the numbers count gallons of water, so their whole is 1 gallon. The one half counts hours, so its whole is 1 hour. It tells how long Dana scrubbed, not how much water there is. Leave it out of the calculation.
First, find the water left after Dana takes some out. Taking out means subtract. One half is the same as two fourths, so write both numbers in fourths. Then subtract the whole numbers and subtract the fourths.
$6\frac{1}{2} - 2\frac{1}{4} = 6\frac{2}{4} - 2\frac{1}{4} = 4\frac{1}{4}$
Next, find the water after Dana pours more in. Pouring in joins two amounts, so add. One fourth is the same as two eighths, so write both numbers in eighths. Then add the whole numbers and add the eighths.
$4\frac{1}{4} + 3\frac{1}{8} = 4\frac{2}{8} + 3\frac{1}{8} = 7\frac{3}{8}$
Look back. Dana took out less than 3 gallons and poured in more than 3 gallons, so the tank should end with more water than it started with. Seven and three eighths gallons is more than six and one half gallons, so the answer fits the story.
$7\frac{3}{8} > 6\frac{1}{2}$

Answer

The tank now has $7\frac{3}{8}$ gallons of water. The $\frac{1}{2}$ hour is a time, so it is not part of the answer.

Now switch jobs. Someone else has solved a problem, and you are the checker. Your job is to find the mistake, say why it is a mistake, and fix it.

You do not need a new method. Walk through the same plan and ask one question at each part. The table shows the four checks.

Four checks for solved work

Check	Ask yourself	A mistake looks like
The wholes	Is every fraction a part of the same whole?	Miles added to hours, or half of a small thing added to half of a big thing
The steps	Does each step add or subtract the way the story does?	A plus sign where the story takes an amount away
The calculations	If you work out each step again, do you get the same result?	A result that changes when you work it out again
The size	Does the size of the answer fit the story?	More left over than there was at the start

Worked example

Find and fix the mistake

A bowl has $2\frac{3}{4}$ cups of flour in it. A baker pours in $1\frac{1}{2}$ cups more, then scoops out $1\frac{1}{8}$ cups for muffins. How much flour is in the bowl now? Jo solved it like this. First line: $2\frac{3}{4} + 1\frac{1}{2} = 4\frac{1}{4}$ . Second line: $4\frac{1}{4} + 1\frac{1}{8} = 5\frac{3}{8}$ . Jo's answer: $5\frac{3}{8}$ cups. Find Jo's mistake and fix it.

Check the wholes. All three numbers count cups of flour, so the whole for each one is 1 cup. Jo may add and subtract them. There is no mistake here.
Check Jo's first line. Pouring in more flour joins two amounts, so adding matches the story. Now work it out again. One half is two fourths. Three fourths plus two fourths is five fourths, which is the same as one and one fourth. Jo's first line is correct.
$2\frac{3}{4} + 1\frac{1}{2} = 2\frac{3}{4} + 1\frac{2}{4} = 3\frac{5}{4} = 4\frac{1}{4}$
Check Jo's second line. Scooping out flour takes it away, so the amount in the bowl must go down. Jo added, so the amount went up from four and one fourth cups to five and three eighths cups. This is the mistake. The story subtracts here.
$5\frac{3}{8} > 4\frac{1}{4}$
Fix the second line. Change the plus sign to a minus sign. One fourth is the same as two eighths, so write both numbers in eighths. Then subtract the whole numbers and subtract the eighths.
$4\frac{1}{4} - 1\frac{1}{8} = 4\frac{2}{8} - 1\frac{1}{8} = 3\frac{1}{8}$
Check the size. After scooping, the bowl must have less flour than the four and one fourth cups it had before. Three and one eighth cups is less, so the new answer fits the story.
$3\frac{1}{8} < 4\frac{1}{4}$

Answer

Jo added in the second line, but scooping out flour means subtracting. The bowl has $3\frac{1}{8}$ cups of flour in it, not $5\frac{3}{8}$ cups.

Here is the idea to keep. A multi-step fraction problem is a chain of small steps, and each step has a job. Before you calculate, say what each fraction is a part of and what each step finds. In a story, you can add or subtract fractions of the same whole. You cannot combine fractions of different wholes.

The same habit makes you a good checker. When an answer looks wrong, ask about the wholes, the steps, the calculations and the size. Then you can say what went wrong and put it right.