Decimal Word Problems and Finding Mistakes | Grade 6 Math | LevelKite

Decimals often appear when you measure: a run of $3.85$ kilometers, a rope $8.5$ meters long. A word problem gives you measured amounts and asks a question about them. You already know how to add and subtract decimals and how to complete a number sentence with a missing number. A word problem hides such a sentence inside a story. Your new job is to find it.

Every word problem in this lesson has a whole and its parts. The parts add up to the whole.

If you know all the parts and need the whole, you add. The result is called the sum.
If you know the whole and need a missing part, you subtract the parts you know from the whole. The result is called the difference.

Subtraction fits the second case because it undoes addition. Take the known parts away from the whole, and exactly the missing part is left.

Find the unknown, then choose the operation

Story	What is unknown	What you compute
You pour $1.5$ liters and then $0.75$ liters into an empty jug. How much is in the jug?	The whole: both pours together	Add: $1.5 + 0.75 = 2.25$ liters
A plant $12.4$ centimeters tall grows $2.75$ centimeters more. How tall is it now?	The whole: the old height and the growth together	Add: $12.4 + 2.75 = 15.15$ centimeters
You cut $2.6$ meters from a rope $8.5$ meters long. How much rope is left?	A part: the rope that is left	Subtract: $8.5 - 2.6 = 5.9$ meters
One bag weighs $4.2$ kilograms and another weighs $3.65$ kilograms. How much more does the first bag weigh?	A part: the extra weight of the heavier bag	Subtract: $4.2 - 3.65 = 0.55$ kilograms

This routine works for each word problem in this lesson:

Name the amounts and check that they share one unit.
Find the unknown. Is it the whole or a part? That tells you whether to add or subtract.
Estimate. Round each decimal to the nearest whole number and work it out in your head. This quick, rough answer is your estimate.
Compute. Combine only digits with the same place value. A digit's place value is what its position makes it worth, such as tenths or hundredths.
Check and answer. Compare the result with the estimate. Then use the inverse operation, the operation that undoes the one you used. Addition and subtraction undo each other. Finish with a sentence that names the unit.

Worked example

Parts known, whole unknown

Leo runs $3.85$ kilometers on Saturday and $2.3$ kilometers on Sunday. How far does he run in the two days?

Name the amounts. Both are distances in kilometers: 3.85 and 2.3. They are the two parts. The question asks for both days together, which is the whole. The parts are known and the whole is unknown, so add.
$3.85 + 2.3$
Estimate, so you know what size to expect. 3.85 rounds to 4 and 2.3 rounds to 2, so the total should be about 6 kilometers.
$4 + 2 = 6$
Give both numbers the same number of digits after the decimal point. Write 2.3 as 2.30. The value stays the same because 3 tenths equal 30 hundredths.
$2.3 = 2.30$
Add digits with the same place value, starting on the right. Hundredths: 5 + 0 = 5. Tenths: 8 + 3 = 11. Regroup, which means trading 10 of one place for 1 of the next larger place: 11 tenths are 1 one and 1 tenth. Ones: 3 + 2 + 1 = 6.
$3.85 + 2.30 = 6.15$
Check. 6.15 is only 0.15 away from the estimate of 6. The inverse operation agrees: subtraction undoes addition, so the total minus Sunday's distance gives back Saturday's distance.
$6.15 - 2.30 = 3.85$

Answer

Leo runs $6.15$ kilometers in the two days.

Worked example

Whole known, part unknown

A roll holds $12$ meters of ribbon. Dana cuts off $3.8$ meters for one gift and $2.15$ meters for another. How many meters of ribbon are left on the roll?

All three amounts are lengths in meters. The 12 meters are the whole. Its parts are the two cut pieces and the ribbon that is left. A part is unknown, so subtract. Both pieces come off the roll, so add them first and subtract their total once.
$12 - (3.8 + 2.15)$
Estimate. 3.8 rounds to 4 and 2.15 rounds to 2. About 6 meters are cut off, so about 6 meters should be left.
$12 - (4 + 2) = 6$
Add the two cut pieces. Write 3.8 as 3.80 so that both numbers show hundredths. Hundredths: 0 + 5 = 5. Tenths: 8 + 1 = 9. Ones: 3 + 2 = 5.
$3.80 + 2.15 = 5.95$
Get ready to subtract that total from the whole. Write 12 as 12.00, since a whole number has 0 tenths and 0 hundredths. Then regroup the other way: trade 1 one for 10 tenths, and 1 of those tenths for 10 hundredths. That leaves 11 ones, 9 tenths and 10 hundredths.
$12 = 12.00$
Subtract place by place. Hundredths: 10 minus 5 is 5. Tenths: 9 minus 9 is 0. Ones: 11 minus 5 is 6.
$12.00 - 5.95 = 6.05$
Check. 6.05 is close to the estimate of 6. Then use the inverse operation: adding the cut pieces back must rebuild the whole roll.
$6.05 + 5.95 = 12.00$

Answer

There are $6.05$ meters of ribbon left on the roll.

Key concept

Same place values, checked two ways

The whole and its parts tell you which operation to use. Place value tells you how to carry it out: combine only digits with the same place value, hundredths with hundredths, tenths with tenths, ones with ones. Lining up the decimal points puts matching place values in the same column.

A correct result passes two checks. Its size fits a whole-number estimate, and the inverse operation leads back to a number you already know.

Now turn these ideas on someone else's work. To identify a mistake is to find the step where the work first goes wrong and to say why.

Start with the two checks, and know what each one can tell you.

The estimate catches big mistakes. Rounding to the nearest whole number moves a number by at most $0.5$ . So when two decimals are rounded and then added or subtracted, the exact answer is at most $1$ away from the estimate. A result farther away than that must be wrong. A closer result can still be wrong.
The inverse operation tests the exact value. A difference plus the amount subtracted must equal the starting amount. A sum of two numbers minus one of them must equal the other. If you do this check correctly and it does not come out this way, the work has a mistake.

A failed check shows that something is wrong. To find where, go through the work one place value at a time, from right to left.

\text{Mistaken: }\begin{array}{r} 6.4 \\ +\;2.38 \\ \hline 3.02 \end{array} \qquad \text{Correct: }\begin{array}{r} 6.40 \\ +\;2.38 \\ \hline 8.78 \end{array}

Three more mistakes to look for

Mistake	What the work shows	What is true
A regrouped 1 is not added	$3.85 + 2.3$ : the tenths make $11$ , but the ones are added as $3 + 2 = 5$ . Result: $5.15$	$11$ tenths are $1$ one and $1$ tenth, so the ones are $3 + 2 + 1 = 6$ . The sum is $6.15$ .
The smaller digit is taken from the larger in every place	$7.2 - 3.56$ : uses $6 - 0 = 6$ hundredths and $5 - 2 = 3$ tenths. Result: $4.36$	The digits of $3.56$ are the ones being taken away. Regroup to subtract them: $7.20 - 3.56 = 3.64$ .
A place that was regrouped is not reduced	$8.75 - 3.48$ : after $15 - 8 = 7$ hundredths, the tenths are still taken as $7 - 4 = 3$ . Result: $5.37$	One tenth was regrouped, so $6$ tenths are left: $6 - 4 = 2$ . The difference is $5.27$ .

Worked example

Find and fix a mistake

Mia subtracts $9.3 - 4.67$ and gets $4.77$ . She lined up the decimal points. In the hundredths place she wrote $7$ . In the tenths place she regrouped and found $13 - 6 = 7$ . In the ones place she found $8 - 4 = 4$ . Is her result correct? If not, find the mistake and correct it.

Try the estimate. 9.3 rounds to 9 and 4.67 rounds to 5, so the difference should be about 4. Mia's 4.77 is less than 1 away from 4, so the estimate cannot decide. It only catches big mistakes.
$9 - 5 = 4$
Test the exact value with the inverse operation. Her result plus the amount subtracted must give back 9.3. It gives 9.44, so her result is wrong.
$4.77 + 4.67 = 9.44 \neq 9.3$
Find where the work first goes wrong, starting on the right. 9.3 shows no hundredths digit, which means 0 hundredths. Mia had to subtract 7 hundredths from 0 hundredths. She copied the 7 down instead. That is the mistake.
$9.3 = 9.30$
Correct the work. Regroup 1 tenth as 10 hundredths: 10 minus 7 is 3, and 2 tenths are left. Regroup 1 one as 10 tenths: 12 minus 6 is 6, and 8 ones are left. Ones: 8 minus 4 is 4. Her tenths digit changes too, because one mistake can change the places after it.
$9.30 - 4.67 = 4.63$
Check the corrected result with the inverse operation. This time the sum returns to the starting number.
$4.63 + 4.67 = 9.30$

Answer

Mia's result is not correct. Her mistake is in the hundredths place: she wrote down $7$ instead of subtracting $7$ hundredths from $0$ hundredths. The correct difference is $4.63$ .

Takeaway. In a decimal word problem, find the whole and its parts. If the whole is unknown, add the parts. If a part is unknown, subtract the known parts from the whole. Then compute by combining digits with the same place value, with the decimal points lined up.

To judge a result, yours or someone else's, test it. An estimate shows whether its size is sensible, and the inverse operation shows whether its value is right. When a check fails, go through the work from right to left, one place at a time, until you reach the first step that is not true.