Decimals: Missing Numbers, Word Problems and Mistakes | Grade 5 Math | LevelKite

You already know how to add and subtract decimals by lining up the decimal points. In this lesson you will use that skill in three ways: to find a missing number, to solve a word problem, and to catch a mistake.

One idea helps with all three. Think of a ribbon $3.75$ meters long that is cut into two pieces, $2.5$ meters and $1.25$ meters. The ribbon is the whole. The two pieces are its parts.

Put the pieces together: $2.5 + 1.25 = 3.75$ . This is an addition sentence. The two numbers you add are the parts, and the total is the whole.
Cut one piece off: $3.75 - 1.25 = 2.5$ . This is a subtraction sentence. The number you start with is the whole. The number you take away and the number that is left are the parts.

Key concept

Addition and subtraction undo each other

Addition and subtraction are inverse operations. This means each one undoes the other. If you add $1.25$ to a number and then subtract $1.25$ , you are back at the number you started with.

So two parts and their whole are linked in both directions:

If the whole is missing, add the two parts.
If a part is missing, subtract the part you know from the whole.

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

A missing-number sentence uses a box, $\square$ , to hold the place of a number you do not know yet. Before you add or subtract, ask one question: is the box a part or the whole?

The table shows the four places a box can sit. Every row uses the same three numbers: the parts $1.4$ and $2.5$ , and their whole, $3.9$ .

Four places for the box

Number sentence	The box is	What you do	Missing number
$\square + 1.4 = 3.9$	a part	Subtract: $3.9 - 1.4$	$2.5$
$1.4 + \square = 3.9$	a part	Subtract: $3.9 - 1.4$	$2.5$
$\square - 1.4 = 2.5$	the whole	Add: $2.5 + 1.4$	$3.9$
$3.9 - \square = 2.5$	a part	Subtract: $3.9 - 2.5$	$1.4$

In the first three rows, you do the opposite of the sign you see. The last row breaks that pattern. The sentence has a minus sign, and you still subtract, because you know the whole, $3.9$ , and one part, $2.5$ . So do not let the sign choose for you. Ask whether the box is the whole or a part.

When you have filled a box, put your number back into the sentence and work it out with care. If the two sides match, your number is right. If they do not, look for a slip.

Worked example

Find a missing part

Find the missing number in $4.6 + \square = 7.25$ .

Decide what the box is. This is an addition sentence, so the total, 7.25, is the whole. The number 4.6 is one part and the box is the other part.
A part is missing, so subtract the part you know from the whole. Subtraction is the inverse of addition, so subtracting 4.6 undoes the adding of 4.6 in the sentence.
$\square = 7.25 - 4.6$
Line up the decimal points and write 4.6 as 4.60, so both numbers show hundredths. The value is the same, because 6 tenths equals 60 hundredths. Now subtract each place. 5 hundredths minus 0 hundredths is 5 hundredths. Trade 1 one for 10 tenths: 12 tenths minus 6 tenths is 6 tenths. That leaves 6 ones, and 6 ones minus 4 ones is 2 ones.
$7.25 - 4.60 = 2.65$
Check by putting 2.65 back in the box. The two sides must match, and they do.
$4.60 + 2.65 = 7.25$

Answer

The missing number is $2.65$ , because $4.6 + 2.65 = 7.25$ .

A word problem hides a number sentence inside a story. Four moves help you find it and solve it.

First, find the sentence. Decide which amount in the story is the whole and which amounts are the parts. Write a number sentence with a box for the amount you need.
Next, estimate. An estimate is a quick answer that is close to the exact one. Swap each decimal for the whole number nearest to it, then add or subtract.
Then find the exact answer. Line up the decimal points and add or subtract.
Last, compare. Swapping changes each number by $0.5$ or less. So with two numbers, the estimate and the exact answer are never more than $1$ apart. If yours are farther apart than that, look for a mistake.

Worked example

Solve a word problem

A team's cooler held some water. The players drank $1.85$ liters, and then $4.4$ liters were left. How many liters of water were in the cooler at the start?

Find the sentence. The water at the start is the whole. It splits into two parts: the 1.85 liters the players drank and the 4.4 liters that were left. The story takes 1.85 away from the starting amount and leaves 4.4, so the box goes first.
$\square - 1.85 = 4.4$
The whole is missing, so add the two parts. Addition is the inverse of subtraction, so adding 1.85 undoes the taking away of 1.85.
$\square = 4.4 + 1.85$
Estimate. 4.4 is near 4 and 1.85 is near 2, so the answer should be near 6.
$4 + 2 = 6$
Find the exact answer. Line up the decimal points and write 4.4 as 4.40. Add each place. In the tenths place, 4 tenths plus 8 tenths is 12 tenths, so trade 10 tenths for 1 one and keep 2 tenths.
$4.40 + 1.85 = 6.25$
Compare. 6.25 is only 0.25 away from the estimate of 6, so the answer makes sense. Then put 6.25 back into the story sentence. Taking away 1.85 leaves 4.40, just as the story says.
$6.25 - 1.85 = 4.40$

Answer

The cooler held $6.25$ liters of water at the start.

Now switch jobs: you are the checker. Anyone can slip when adding or subtracting decimals. You have two tools for finding a slip.

The estimate check finds big mistakes. With two numbers, a big mistake is an answer more than $1$ away from your whole-number estimate.
The inverse check finds small mistakes too. Undo the work with the inverse operation, and take care as you do it. A right answer takes you back to the number you started with. A wrong answer does not.

Try your tools on this student's work for $3.2 + 1.45$ .

\begin{array}{r} 3.2 \\ +\;1.45 \\ \hline 1.77 \end{array}

The correct line-up, with decimal point under decimal point

Number	Ones	Tenths	Hundredths
$3.20$	$3$	$2$	$0$ (placeholder zero)
$1.45$	$1$	$4$	$5$
Total: $4.65$	$4$	$6$	$5$

Worked example

Find and fix a mistake

Leo worked out $7.3 - 4.16$ and got $3.26$ . He lined up the decimal points, but he left the hundredths place of $7.3$ empty. Is $3.26$ correct? If not, find the mistake and fix it.

Try the estimate check. 7.3 is near 7 and 4.16 is near 4, so the answer should be near 3. Leo's 3.26 is only 0.26 away from 3, so this check does not show a problem. A small mistake could still be hiding.
$7 - 4 = 3$
Try the inverse check. If 3.26 is right, adding back the 4.16 that Leo took away must give 7.3. It gives 7.42 instead, so 3.26 is wrong.
$3.26 + 4.16 = 7.42$
Find the mistake. In Leo's work, the 6 hundredths of 4.16 had nothing above it, so he copied the 6 straight down. That works in addition, since 0 + 6 = 6. But an empty place is worth 0, and 0 hundredths minus 6 hundredths is not 6 hundredths.
Fix it. Write the placeholder zero, so 7.3 becomes 7.30. Trade 1 tenth for 10 hundredths: 10 hundredths minus 6 hundredths is 4 hundredths. That leaves 2 tenths, and 2 tenths minus 1 tenth is 1 tenth. Last, 7 ones minus 4 ones is 3 ones.
$7.30 - 4.16 = 3.14$
Run the inverse check on the new answer. Adding 4.16 back gives 7.30, the number Leo started with, so 3.14 is right.
$3.14 + 4.16 = 7.30$

Answer

No, $3.26$ is not correct. Leo forgot the placeholder zero in $7.30$ . The correct answer is $7.3 - 4.16 = 3.14$ .

Keep this one idea: addition and subtraction undo each other. That is why two parts and their whole fit together, and why you can find any one of the three from the other two. Add the parts to find a missing whole. Subtract the part you know from the whole to find a missing part.

The same idea lets you check any answer, yours or someone else's. A whole-number estimate shows a big mistake fast, and the inverse operation finds a small one. Two habits stop the mistakes you met here before they start: line up the decimal points, and fill each empty place after the decimal point with a placeholder zero.