Fraction Word Problems and Common Mistakes | Grade 6 Math | LevelKite

A fraction word problem gives you amounts as fractions or mixed numbers (a whole number with a fraction, such as $2\frac{1}{2}$ ) and asks a question about them. Before you compute, answer two questions.

What is the whole? The reference whole is the amount that counts as 1, such as 1 mile, 1 foot or 1 hour. Each fraction in the problem describes parts of a whole. You can add, subtract or compare fraction amounts only when they have the same reference whole.
What is happening? Decide what the story does with the amounts. It may join them, remove one from the other, or compare them. Read the full situation, not just one word. The word "more" can appear in an adding problem ("then she walks a quarter mile more") and in a subtracting problem ("how much more does she need?").

You already know how to add and subtract fractions and mixed numbers. Here you use those skills to solve word problems and to find mistakes in worked solutions.

Matching the situation to the operation, when all amounts have the same reference whole

What the story does	What you do	A question you might see
It joins two amounts into one total.	Add the amounts.	How far does she walk in all?
It removes part of an amount.	Subtract the part that is removed.	How much is left?
It compares two amounts.	Subtract the smaller amount from the larger.	How much longer is one than the other?
It gives a total and one of its two parts.	Subtract the known part from the total.	How much more does she need?

Key concept

Same whole, equal-sized parts, then count

A fraction counts parts. The denominator names the size of each part: it tells how many equal parts make 1 whole. The numerator tells how many of those parts you have.

You can add or subtract two fraction amounts by adding or subtracting their numerators only when both of these are true:

the amounts have the same reference whole, and
the amounts count equal-sized parts of that whole, which means they have the same denominator.

The denominator then stays the same, because the size of the parts does not change. For example, $\frac{3}{8}$ mile plus $\frac{2}{8}$ mile is $\frac{5}{8}$ mile: 3 eighths and 2 eighths of the same mile make 5 eighths of a mile.

Worked example

Joining two distances

Maya walks $\frac{3}{4}$ mile to the library and then $\frac{5}{8}$ mile to the park. How far does she walk in all?

Name the reference whole and the operation. Both distances have the same reference whole, 1 mile. The story joins the two distances into one total, so you add them.
$\frac{3}{4} + \frac{5}{8}$
Fourths and eighths are different-sized parts of the mile, so you cannot count them together yet. Each fourth splits into 2 eighths, so multiply the numerator and the denominator of three fourths by 2. This is the same as multiplying by 1, because 2 divided by 2 is 1, so the amount does not change.
$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$
Now both amounts count eighths of the same mile. Add the numerators to count the eighths. Keep the denominator 8, because the size of each part does not change.
$\frac{6}{8} + \frac{5}{8} = \frac{6 + 5}{8} = \frac{11}{8}$
Write the answer as a mixed number. Eight eighths make 1 whole mile, and 3 eighths are left over.
$\frac{11}{8} = \frac{8}{8} + \frac{3}{8} = 1\frac{3}{8}$
Check that the answer is reasonable. Both distances are measured in miles. Half a mile is 2 fourths or 4 eighths of a mile, so each distance is more than half a mile. The total must be more than 1 mile, and 1 and 3 eighths miles fits.
$\frac{3}{4} > \frac{2}{4} = \frac{1}{2} \qquad \frac{5}{8} > \frac{4}{8} = \frac{1}{2}$

Answer

Maya walks $1\frac{3}{8}$ miles in all.

Worked example

Finding what is left

A board is $5\frac{1}{4}$ feet long. You cut off a piece that is $2\frac{2}{3}$ feet long. How long is the piece that is left?

Name the reference whole and the operation. Both lengths have the same reference whole, 1 foot. The story removes a piece from the board, so you subtract that piece from the full length.
$5\frac{1}{4} - 2\frac{2}{3}$
Fourths and thirds are different-sized parts of a foot. Twelve is a multiple of both 4 and 3, so rename both fraction parts in twelfths. Each time, multiply the numerator and the denominator by the same number. That is the same as multiplying by 1, so each amount stays the same.
$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \qquad \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
You cannot take 8 twelfths away from only 3 twelfths. So regroup: trade 1 whole foot from the 5 feet for 12 twelfths. The length does not change, because 12 twelfths of a foot equal 1 whole foot.
$5\frac{3}{12} = 4 + \frac{12}{12} + \frac{3}{12} = 4\frac{15}{12}$
Now both lengths count twelfths of the same foot. Subtract the whole feet, and subtract the twelfths. Keep the denominator 12, because the parts are still twelfths.
$4\frac{15}{12} - 2\frac{8}{12} = (4 - 2) + \frac{15 - 8}{12} = 2\frac{7}{12}$
Check with the inverse operation, the operation that undoes another. Addition undoes subtraction, so the piece that is left plus the piece you cut off must equal the full board. Both pieces are counted in twelfths of a foot, so you can add them.
$2\frac{7}{12} + 2\frac{8}{12} = 4\frac{15}{12} = 5\frac{3}{12} = 5\frac{1}{4}$

Answer

The piece that is left is $2\frac{7}{12}$ feet long.

Now switch roles and become the checker. To find a mistake in a solution, read it one line at a time. Ask whether each line follows from the line before. These questions help.

Same whole? Do all the amounts have the same reference whole? When fractions have no units, the whole is the number 1.
Equal-sized parts? Do the fractions have the same denominator before the writer adds or subtracts numerators?
Renaming done right? Does each renaming multiply the numerator and the denominator by the same number?
Only the numerators? Once the parts are equal in size, does the writer add or subtract only the numerators and keep the denominator?
Regrouping done right? When the writer trades 1 whole for parts, is it the right number of parts? With twelfths, 1 whole is $\frac{12}{12}$ .
Sensible answer? For amounts greater than zero with the same whole, adding gives more than each amount, and subtracting gives less than the amount you start with. You can also add back what was subtracted. You must get the starting amount.

A common mistake: adding the numerators and adding the denominators. A student writes $\frac{1}{2} + \frac{1}{3} = \frac{1 + 1}{2 + 3} = \frac{2}{5}$ . Both fractions have the same whole, the number 1, so you can add them, but not this way. The answer $\frac{2}{5}$ is less than $\frac{1}{2}$ , because $\frac{2}{5} = \frac{4}{10}$ and $\frac{1}{2} = \frac{5}{10}$ . That cannot be right, because adding an amount greater than zero to $\frac{1}{2}$ must give more than $\frac{1}{2}$ . The method fails because halves and thirds are different-sized parts, so 1 half and 1 third are not 2 of any one kind of part. Adding the denominators also invents a new part size, fifths, that neither fraction uses. Correct work first renames both fractions as sixths of the same whole:

\frac{1}{2} + \frac{1}{3} = \frac{1 \times 3}{2 \times 3} + \frac{1 \times 2}{3 \times 2} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

More mistakes to look for. In rows 1 to 3, both amounts have the same whole, the number 1.

Mistaken work	What went wrong	Correct work
$\frac{2}{3} + \frac{1}{6} = \frac{3}{6}$	The student added the numerators before renaming the thirds as sixths.	$\frac{4}{6} + \frac{1}{6} = \frac{5}{6}$
$\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{1}{4} = \frac{2}{4}$	The student changed the denominator of $\frac{1}{2}$ to 4 but did not multiply the numerator by 2.	$\frac{3}{4} - \frac{2}{4} = \frac{1}{4}$
$1\frac{3}{4} + 2\frac{3}{4} = 3\frac{6}{4} = 3\frac{2}{4}$	The student lost 1 whole: $\frac{6}{4}$ equals $1\frac{2}{4}$ , not $\frac{2}{4}$ .	$3 + 1\frac{2}{4} = 4\frac{2}{4} = 4\frac{1}{2}$
$\frac{1}{2}$ of a small pizza plus $\frac{1}{2}$ of a large pizza makes 1 pizza.	The two halves come from different wholes, so they are not equal-sized parts.	Do not add the fractions. First measure both amounts with the same whole.

Worked example

Finding the mistake in a solution

Sam practiced piano for $2\frac{3}{4}$ hours on Saturday and $1\frac{5}{6}$ hours on Sunday. How much longer did he practice on Saturday? Sam wrote: Line 1: $2\frac{3}{4} - 1\frac{5}{6}$ . Line 2: $2\frac{9}{12} - 1\frac{10}{12}$ . Line 3: $1\frac{1}{12}$ hours. Find Sam's mistake and correct it.

Check line 1. Both times have the same reference whole, 1 hour. The question compares two amounts, so subtracting the smaller time from the larger time is the right operation. Line 1 is correct.
$2\frac{3}{4} - 1\frac{5}{6}$
Check line 2. Sam renamed both fraction parts as twelfths of an hour. Each time he multiplied the numerator and the denominator by the same number, so each amount is unchanged. Line 2 is correct.
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \qquad \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
Check line 3. Sam subtracted the whole hours to get 1. For the fraction parts he found 10 minus 9, but the problem needs 9 twelfths minus 10 twelfths. He switched the order instead of regrouping. Line 3 has the mistake.
$\text{Sam: } \frac{10}{12} - \frac{9}{12} = \frac{1}{12} \qquad \text{Needed: } \frac{9}{12} - \frac{10}{12}$
Correct the work. Regroup 1 whole hour from the 2 hours as 12 twelfths, so that there are enough twelfths to subtract from. The total time does not change, because 12 twelfths of an hour equal 1 hour.
$2\frac{9}{12} = 1 + \frac{12}{12} + \frac{9}{12} = 1\frac{21}{12}$
Both times now count twelfths of the same hour. Subtract the whole hours, and subtract the twelfths. Keep the denominator 12, because the parts are still twelfths.
$1\frac{21}{12} - 1\frac{10}{12} = (1 - 1) + \frac{21 - 10}{12} = \frac{11}{12}$
Check with the inverse operation. Addition undoes subtraction, so the answer plus Sunday's time must equal Saturday's time. The corrected answer passes. Sam's answer fails, because it gives 2 and 11 twelfths hours instead.
$\frac{11}{12} + 1\frac{10}{12} = 1\frac{21}{12} = 2\frac{9}{12} = 2\frac{3}{4}$

Answer

Sam subtracted the fraction parts in the wrong order instead of regrouping. He practiced $\frac{11}{12}$ of an hour longer on Saturday.

The takeaway. You can compare, add or subtract fraction amounts only when they have the same reference whole. To add or subtract them by counting parts, the parts must also be equal in size, so first rename the fractions with the same denominator. Then add or subtract the numerators and keep the denominator. In a word problem, name the whole first. Let the situation tell you whether the story joins, removes or compares amounts. When you check a solution, yours or someone else's, read it line by line. Test the answer with an estimate or with the inverse operation.