Missing Mixed Numbers and Word Problems | Grade 5 Math | LevelKite

A number sentence is a statement written with numbers and symbols, such as $2 + 3 = 5$ . Sometimes one number is missing, and a question mark holds its place:

$1\frac{1}{4} + {?} = 3\frac{3}{4}$

The numbers in this lesson are mixed numbers. A mixed number is a whole number and a fraction together, such as $1\frac{1}{4}$ .

You already know how to add and subtract mixed numbers. Now you will learn how to choose which one to do, first in number sentences and then in word problems.

Picture a jug and one measuring cup. One full cup is the whole, the thing each fraction is a part of. You use the same cup every time, so every whole is the same size.

Pour in $1\frac{1}{4}$ cups. Then pour in $2\frac{1}{2}$ cups more. One half of a cup is the same amount as two fourths of that cup, so the jug holds $1\frac{1}{4} + 2\frac{2}{4} = 3\frac{3}{4}$ cups.
Now pour the $2\frac{1}{2}$ cups back out. The jug holds $3\frac{3}{4} - 2\frac{2}{4} = 1\frac{1}{4}$ cups again.

Taking away $2\frac{1}{2}$ undid adding $2\frac{1}{2}$ . Adding and subtracting are operations, or actions you do with numbers. Two operations that undo each other are inverse operations. Subtracting a number undoes adding that same number, and adding it undoes subtracting it.

Key concept

Addition and subtraction undo each other

Two parts join to make a total. All three numbers must count the same whole, or wholes of equal size, such as cups from one measuring cup. When they do, the same three numbers fit an addition sentence and a subtraction sentence.

To find a missing total, add the two parts.
To find a missing part, subtract the part you know from the total.

\text{part} + \text{other part} = \text{total} \qquad \text{total} - \text{part} = \text{other part}

So before you add or subtract, find the parts and the total. Each sentence you complete in this lesson has one plus sign or one minus sign.

Addition sentence: the plus sign joins the two parts. The total is on the other side of the equals sign.
Subtraction sentence: the total is the number just before the minus sign. The number you take away and the number left are the two parts.

In a sentence with no story, every number counts the same whole, the number $1$ . So you may add and subtract them.

The jug's three numbers fit four sentences. Each row of the table hides one number behind a question mark.

The jug's three numbers in four missing-number sentences

Sentence	What is missing	What to do	Missing number
$1\frac{1}{4} + {?} = 3\frac{3}{4}$	A part (the second pour)	Subtract: $3\frac{3}{4} - 1\frac{1}{4}$	$2\frac{2}{4} = 2\frac{1}{2}$
$? + 2\frac{1}{2} = 3\frac{3}{4}$	A part (the first pour)	Subtract: $3\frac{3}{4} - 2\frac{1}{2}$	$1\frac{1}{4}$
$? - 2\frac{1}{2} = 1\frac{1}{4}$	The total (all the water in the jug)	Add: $1\frac{1}{4} + 2\frac{1}{2}$	$3\frac{3}{4}$
$3\frac{3}{4} - {?} = 1\frac{1}{4}$	A part (the cups you pour back out)	Subtract: $3\frac{3}{4} - 1\frac{1}{4}$	$2\frac{2}{4} = 2\frac{1}{2}$

Worked example

A missing part in an addition sentence

Find the missing number: $? + 1\frac{2}{3} = 4\frac{5}{6}$

Find the parts and the total. The plus sign joins two parts. One part is missing. The other part is 1 and 2 thirds. The total, 4 and 5 sixths, is on the other side of the equals sign.
$? + 1\frac{2}{3} = 4\frac{5}{6}$
Use the inverse operation. Subtraction undoes addition, so subtract the part you know from the total. What is left is the missing part.
$? = 4\frac{5}{6} - 1\frac{2}{3}$
Give both fractions the same denominator (bottom number), so the pieces are the same size. Use sixths, because 3 times 2 is 6. Multiply the top and the bottom of 2 thirds by 2. Cut each third into 2 equal pieces and you get sixths, so 2 thirds and 4 sixths are the same amount of the same whole.
$1\frac{2}{3} = 1\frac{2 \times 2}{3 \times 2} = 1\frac{4}{6}$
Now all the pieces are sixths, so you can subtract. Subtract the whole numbers: 4 minus 1 is 3. Subtract the fractions: 5 sixths minus 4 sixths is 1 sixth.
$4\frac{5}{6} - 1\frac{4}{6} = 3\frac{1}{6}$
Check by putting your answer back into the sentence. The two parts should join to make the total. 3 plus 1 is 4, and 1 sixth plus 4 sixths is 5 sixths, so they do.
$3\frac{1}{6} + 1\frac{4}{6} = 4\frac{5}{6}$

Answer

The missing number is $3\frac{1}{6}$ , because $3\frac{1}{6} + 1\frac{2}{3} = 4\frac{5}{6}$ .

Worked example

A missing part in a subtraction sentence

Find the missing number: $5\frac{1}{4} - {?} = 2\frac{2}{3}$

Find the parts and the total. The number just before the minus sign, 5 and 1 fourth, is the total. You take away the missing number, and 2 and 2 thirds is left, so those two are the parts. Put the parts back together and you get the total again. That is addition undoing the subtraction.
$? + 2\frac{2}{3} = 5\frac{1}{4}$
A part is missing, so subtract the part you know from the total.
$? = 5\frac{1}{4} - 2\frac{2}{3}$
Write both fractions as twelfths, because 4 times 3 is 12 and 3 times 4 is 12. Multiply the top and bottom of 1 fourth by 3 to get 3 twelfths. Multiply the top and bottom of 2 thirds by 4 to get 8 twelfths. Each amount stays the same.
$5\frac{1}{4} - 2\frac{2}{3} = 5\frac{3}{12} - 2\frac{8}{12}$
These are twelfths of the same whole, and 3 twelfths is fewer than 8 twelfths. So regroup: trade 1 of the 5 wholes for 12 twelfths. With the 3 twelfths you already have, that makes 4 wholes and 15 twelfths. Now subtract. 4 minus 2 is 2, and 15 twelfths minus 8 twelfths is 7 twelfths.
$5\frac{3}{12} - 2\frac{8}{12} = 4\frac{15}{12} - 2\frac{8}{12} = 2\frac{7}{12}$
Check by putting your answer back into the first sentence. Use the regrouped total again. 4 minus 2 is 2, and 15 twelfths minus 7 twelfths is 8 twelfths. Divide the top and bottom of 8 twelfths by 4 to get 2 thirds, the same amount. The sentence is true.
$4\frac{15}{12} - 2\frac{7}{12} = 2\frac{8}{12} = 2\frac{2}{3}$

Answer

The missing number is $2\frac{7}{12}$ , because $5\frac{1}{4} - 2\frac{7}{12} = 2\frac{2}{3}$ .

A word problem is a story with a number sentence inside it. To find the sentence, ask the same question as before: what are the parts, and what is the total? Write a question mark for the number the story does not give. Then the rule is the same. Add to find a missing total. Subtract to find a missing part.

Check one more thing before you add or subtract. The amounts must count the same whole, or wholes of equal size. Two lengths in feet are fine, because every foot is the same length. Half of a small pizza and half of a large pizza are not equal amounts, so you cannot add those halves as fractions of one whole.

An estimate is a number that is close to the exact answer and quick to find in your head. The exact answer is the one you get by working out every step. An estimate tells you about how big that answer should be.

To estimate, round each mixed number. That means you swap it for the nearest whole number. To find the nearest one, compare the fraction part with $\frac{1}{2}$ of the same whole.

Less than $\frac{1}{2}$ : keep the whole number. $5\frac{1}{4}$ rounds to $5$ , because $\frac{1}{4}$ is less than $\frac{2}{4}$ , and $\frac{2}{4}$ equals $\frac{1}{2}$ .
$\frac{1}{2}$ or more: go up to the next whole number. $2\frac{2}{3}$ rounds to $3$ , because $\frac{2}{3}$ is $\frac{4}{6}$ , which is more than $\frac{3}{6}$ , and $\frac{3}{6}$ equals $\frac{1}{2}$ .

Then add or subtract the rounded numbers, the same way the problem does. For $5\frac{1}{4} - 2\frac{2}{3}$ , the estimate is $5 - 3 = 2$ . You found the exact answer earlier: $2\frac{7}{12}$ . Is that close enough to $2$ ?

How close is close enough? Rounding to the nearest whole number moves each mixed number by at most $\frac{1}{2}$ . Two numbers move by at most $\frac{1}{2} + \frac{1}{2} = 1$ in all. So when you add or subtract two mixed numbers that count the same whole, the estimate is at most $1$ away from the exact answer. $2\frac{7}{12}$ is $\frac{7}{12}$ away from $2$ , so it passes the check. If your answer is more than $1$ away, look for a mistake. An answer within $1$ is reasonable, but the check cannot prove that every step is right.

An estimate can be $0$ . For $2\frac{3}{4} - 2\frac{1}{2}$ , both numbers round to $3$ , so the estimate is $3 - 3 = 0$ . The exact answer is $2\frac{3}{4} - 2\frac{2}{4} = \frac{1}{4}$ , which is within $1$ of $0$ .

Worked example

A word problem with a missing total

Lena has a roll of ribbon. She cuts off $2\frac{7}{8}$ feet to wrap a gift. Now $3\frac{1}{4}$ feet are left on the roll. How many feet of ribbon were on the roll at first?

Find the parts and the total. The ribbon at first is the total, and the story does not give that number. The piece Lena cuts off and the piece left are the two parts. All the lengths are in feet, so they count wholes of the same size. Write the story as a sentence: start with the total, take away 2 and 7 eighths, and 3 and 1 fourth is left.
$? - 2\frac{7}{8} = 3\frac{1}{4}$
The total is missing, so add the two parts. Addition is the inverse operation here. It undoes the taking away.
$? = 3\frac{1}{4} + 2\frac{7}{8}$
Estimate first. Round each part to the nearest whole number. 1 fourth of a foot is less than half a foot (2 fourths), so 3 and 1 fourth rounds to 3. 7 eighths of a foot is more than half a foot (4 eighths), so 2 and 7 eighths rounds up to 3. The roll should be about 6 feet long.
$3 + 3 = 6$
Now find the exact answer. Multiply the top and bottom of 1 fourth by 2 to write it as 2 eighths. Add the whole numbers: 3 plus 2 is 5. Add the fractions: 2 eighths plus 7 eighths is 9 eighths. 8 eighths make 1 whole, so 9 eighths is 1 and 1 eighth. Regroup to get 6 and 1 eighth.
$3\frac{2}{8} + 2\frac{7}{8} = 5\frac{9}{8} = 6\frac{1}{8}$
Check the answer against the estimate. Both are lengths in feet, so you can compare them. 6 and 1 eighth feet is only 1 eighth of a foot more than the estimate of 6 feet. That is within 1 foot, so the answer is reasonable.
$6\frac{1}{8} - 6 = \frac{1}{8}$

Answer

The roll had $6\frac{1}{8}$ feet of ribbon at first, which is close to the estimate of $6$ feet.

Here is the idea to keep. Addition and subtraction undo each other. So when three numbers count the same whole, or wholes of equal size, two parts and their total fit both an addition sentence and a subtraction sentence. Ask what is missing. If it is the total, add the parts. If it is a part, subtract the part you know from the total. In a word problem, find the parts and the total in the story first. Then estimate with rounded numbers, solve, and check that your answer is close to your estimate.