Estimating and Solving Word Problems with Decimals | Grade 7 Math | LevelKite

Sometimes about how much is all you need. Other times you have an exact answer and want to know whether it makes sense. An estimate helps both times.

An estimate is a quick answer that is meant to be close to the exact answer. You find it by replacing the numbers in a problem with nearby numbers that are easier to work with.

You will estimate four kinds of results:

a sum, the result of adding
a difference, the result of subtracting
a product, the result of multiplying
a quotient, the result of dividing

The symbol $\approx$ means is approximately equal to. Writing $6.2 \approx 6$ says that $6.2$ is close to $6$ . Every number you are given in this lesson is positive, which means greater than $0$ .

To estimate with decimals, take two steps:

Round each number. Rounding swaps a number for the nearest value at a place you choose, such as the nearest whole number or the nearest tenth. Choose a place that makes the arithmetic easy to do in your head.
Use the same operation on the rounded numbers.

Try $18.72 + 6.39$ . The nearest whole number to $18.72$ is $19$ , and the nearest whole number to $6.39$ is $6$ . So the estimate is $19 + 6 = 25$ . The exact sum is $25.11$ , which is only $0.11$ more than the estimate. The rounded numbers sit close to the originals, so their sum sits close to the exact sum.

The table shows the same two steps for each of the four operations.

The same two steps for each operation

Result	Problem	Rounded problem	Estimate	Exact answer
Sum	$18.72 + 6.39$	$19 + 6$	$25$	$25.11$
Difference	$42.15 - 17.8$	$42 - 18$	$24$	$24.35$
Product	$7.8 \times 5.1$	$8 \times 5$	$40$	$39.78$
Quotient	$31.92 \div 7.6$	$32 \div 8$	$4$	$4.2$

A number's rounding change is how far the number moves when you round it.

Tip: in a sum or a difference, each rounding change can push the answer off by at most its own size. So the estimate differs from the exact answer by no more than the rounding changes added together.

Sum row: $18.72$ moves up $0.28$ and $6.39$ moves down $0.39$ . The largest possible gap is $0.28 + 0.39 = 0.67$ . The real gap is $25.11 - 25 = 0.11$ , because the two changes partly cancel.
Difference row: $42.15$ moves down $0.15$ and $17.8$ moves up $0.2$ . The largest possible gap is $0.15 + 0.2 = 0.35$ . The real gap is $24.35 - 24 = 0.35$ , because both changes make the difference smaller.

Quotients sometimes need one more idea. In a division, the dividend is the number you divide, and the divisor is the number you divide by. In the table, rounding gave $32 \div 8$ , which is easy because $8 \times 4 = 32$ . But rounding can also leave a pair that is hard to divide in your head.

Compatible numbers are numbers close to the originals that are easy to compute with mentally. For a quotient, keep the rounded divisor, as long as it is not $0$ , and change the dividend to a nearby multiple of it. A multiple of a number is that number times a whole number. Dividing a multiple by the number gives that whole number back.

Worked example

Estimate a quotient with compatible numbers

Estimate $43.61 \div 8.9$ .

Round each number to the nearest whole number. The tenths digits, 6 in 43.61 and 9 in 8.9, are both 5 or more, so both numbers round up.
$43.61 \approx 44, \quad 8.9 \approx 9$
Test the rounded pair. 44 divided by 9 does not give a whole number, because 44 is not a multiple of 9. The multiples of 9 on either side of 44 are 36 and 45.
$9 \times 4 = 36, \quad 9 \times 5 = 45$
Choose the closer multiple. 45 is only 1 away from 44, while 36 is 8 away, so use 45 as the dividend. Then divide the compatible numbers. The quotient is 5 because 9 times 5 is 45.
$43.61 \div 8.9 \approx 45 \div 9 = 5$
Long division gives the exact quotient, 4.9. You can confirm it without repeating the division: multiply 4.9 by the divisor and check that the dividend comes back. Because 4.9 is 5 minus 0.1, multiply 8.9 by 5 and then take away 8.9 times 0.1.
$8.9 \times 4.9 = 8.9 \times 5 - 8.9 \times 0.1 = 44.5 - 0.89 = 43.61$
Find the gap between the estimate and the exact quotient. The gap is small, so the compatible numbers gave a close estimate.
$5 - 4.9 = 0.1$

Answer

A good estimate is $43.61 \div 8.9 \approx 5$ . The exact quotient, $4.9$ , is only $0.1$ away.

Key concept

Estimate, compute, compare

An estimate replaces each decimal with a nearby number that is easy to work with, then uses the same operation. It tells you about how large the exact answer should be. It works best when each rounding change is small compared with its number.

In a word problem, choose each operation from what the situation means, compute the exact answer, and compare it with your estimate. If the two are far apart, recheck both.

A word problem gives you quantities and a situation, but it usually does not name the operation. You choose the operation by asking what is happening to the quantities.

One kind of quantity appears often. A unit rate is the amount of one quantity that goes with $1$ unit of another quantity, such as $2.48$ dollars for each pound.

A reliable plan has four parts:

Say what you need to find and name its unit, such as kilometers or dollars.
Choose each operation from the situation, then estimate the answer.
Compute the exact answer.
Compare the exact answer with the estimate, then state the result in a sentence with its unit.

For part 2, the table that follows matches four kinds of situations to the four operations.

Let the situation choose the operation

What happens in the situation	Operation	Example
You join amounts into one total	Add	Walk $2.75$ kilometers, then $1.6$ more: $2.75 + 1.6 = 4.35$ kilometers
You take an amount away, or find how far apart two amounts are	Subtract	Cut $1.85$ meters off a $5$ -meter rope: $5 - 1.85 = 3.15$ meters left
You combine equal groups, or use a unit rate for a number of units	Multiply	Buy $3.5$ pounds at $2.48$ dollars per pound: $3.5 \times 2.48 = 8.68$ dollars
You split a total into equal parts, to find the size of one part or the number of parts	Divide	Cut $7.2$ meters of ribbon into $0.45$ -meter pieces: $7.2 \div 0.45 = 16$ pieces

Worked example

Use division to find a unit rate

A hiker walks $20.16$ kilometers in $4.8$ hours at a steady pace. How many kilometers does the hiker walk each hour?

Name the unknown and its unit. Let s be the hiker's speed in kilometers per hour, which is a unit rate. At a steady pace, the speed times the number of hours equals the distance, so s times 4.8 is 20.16. Division undoes multiplication by a number that is not 0, so divide the distance by the hours.
$s \times 4.8 = 20.16 \quad \text{so} \quad s = 20.16 \div 4.8$
Estimate before you compute. Round 20.16 to 20 and round 4.8 to 5. These are compatible numbers, because 20 is a multiple of 5.
$20.16 \div 4.8 \approx 20 \div 5 = 4$
Set up the exact division. Multiply the dividend and the divisor by 10 so that the divisor becomes the whole number 48. Multiplying both by the same number that is not 0 does not change the quotient.
$20.16 \div 4.8 = (20.16 \times 10) \div (4.8 \times 10) = 201.6 \div 48$
Divide by asking how many 48s make 201.6. Four 48s make 192, and taking 192 from 201.6 leaves 9.6. Two tenths of 48 is 9.6, which uses up the rest. So the quotient is 4 plus 0.2.
$201.6 \div 48 = 4 + 0.2 = 4.2$
Compare the exact answer with the estimate. The gap is small, so the decimal point is in a sensible place. A slip such as 42 or 0.42 would land far from 4.
$4.2 - 4 = 0.2$

Answer

The hiker walks $4.2$ kilometers each hour.

Worked example

Solve a multi-step problem and check it

You buy $3$ notebooks that cost $2.85$ dollars each and one pack of pens that costs $4.29$ dollars. You pay with a $20$ -dollar bill. How much change do you get?

Name the unknown and its unit. Let c be your change in dollars. The notebooks are 3 equal groups of 2.85 dollars, so multiply. The pens join that cost, so add. You take the total cost away from the 20 dollars you pay, so subtract. The parentheses hold the total cost.
$c = 20 - (3 \times 2.85 + 4.29)$
Estimate. Round 2.85 to 3 and round 4.29 to 4. Leave the 3 notebooks and the 20 dollars alone, because they are already whole numbers.
$c \approx 20 - (3 \times 3 + 4) = 20 - (9 + 4) = 20 - 13 = 7$
Compute the exact cost of the notebooks. 2.85 is 285 hundredths, so 3 notebooks cost 3 times 285, or 855, hundredths of a dollar. That is 8.55 dollars.
$3 \times 2.85 = 8.55$
Add the pens to find the total cost. You may add in any order, so add the whole dollars, 8 and 4, to get 12. Then add the parts of a dollar, 0.55 and 0.29, to get 0.84.
$8.55 + 4.29 = 12 + 0.84 = 12.84$
Subtract the total cost from the 20 dollars you pay. The cost is less than 20, so there is change left, and the difference is the distance from 12.84 up to 20. Count up: 0.16 takes you from 12.84 to 13, and 7 more takes you from 13 to 20.
$20 - 12.84 = 0.16 + 7 = 7.16$
Compare the exact answer with the estimate. The gap is small, so the size of the answer is reasonable.
$7.16 - 7 = 0.16$

Answer

You get $7.16$ dollars in change.

Takeaway. To estimate, replace each decimal with a nearby number that is easy to work with, and keep the same operation. The estimate tells you about how large the answer should be. In a word problem, let the situation choose the operation, compute the exact answer, and compare it with your estimate. A close match is a good sign that the size of your answer is sensible. A large gap tells you to recheck both.