Decimals in Context: Solve Problems and Find Mistakes | Grade 6 Math | LevelKite

One liter of paint covers $6.5$ square meters, and you have $0.8$ liter left. How much wall can you paint? A jug holds $4.5$ liters of juice, and each cup holds $0.25$ liter. How many cups can you fill? Before you calculate, think: would you multiply or divide, and about how big should each answer be?

You already know how to multiply two decimals and how to divide by a decimal. In this lesson you put those skills to work in two ways. First, you solve word problems, where you decide which operation the situation needs. Second, you find mistakes in decimal work, including your own, by testing whether an answer makes sense.

Each operation has a meaning, and the meaning tells you when to use it.

Multiply when you know the size of one group and the number of groups, and you want the total. The numbers you multiply are factors, and the result is the product. One group can be the amount for one liter, one meter or one item, and the number of groups can be a decimal. Paint that covers $6.5$ square meters per liter, used for $0.8$ liter, covers $6.5 \times 0.8$ square meters.
Divide when you know the total and the size of one group, and you want the number of groups. A $4.5$ -liter jug poured into $0.25$ -liter cups fills $4.5 \div 0.25$ cups.
Divide also when you know the total and the number of groups, and you want the size of one group. If $4$ people share $4.5$ liters equally, each person gets $4.5 \div 4$ liters.

In a division, the dividend is the amount you divide, the divisor is the number you divide by, and the quotient is the result. Every divisor in this lesson is a number other than zero, because division by zero is not defined.

Key concept

Three tests for a decimal answer

The situation chooses the operation. Then three tests tell you whether to trust the answer.

Meaning test. Does the operation fit the situation? Does the answer fit it too, such as a whole number for things that cannot be split?
Size test. Is the answer about as big as you predicted?
Inverse test. Does the inverse operation, the one that undoes yours, lead back to a number you started with? Expect an exact match for an exact answer and a close match for a rounded one.

Use the tests on your own work and on anyone else's. A failed test tells you to look for a mistake.

The size test starts before you calculate. You predict how big the answer should be. A prediction made with easy, rounded numbers is called an estimate.

Start with a positive quantity. Multiplying it by $0.8$ takes only $8$ tenths of it, so the product is less than the quantity. Multiplying by $1.2$ takes all of it and $2$ tenths more, so the product is greater. Multiplying by $1$ leaves it unchanged.

Dividing a positive quantity by a positive divisor works the other way around. Dividing by $0.25$ asks how many groups of one quarter fit. Four fit in every whole, so the quotient is greater than the quantity. Dividing by a divisor greater than $1$ gives a quotient less than the quantity, and dividing by $1$ leaves it unchanged.

Zero is the exception. Zero times any factor is zero, and zero divided by any divisor that is not zero is zero. A starting quantity of $0$ stays $0$ .

Size predictions when the starting quantity is positive

What you do to a positive quantity	The result is	Example starting from 2.5
Multiply by a factor greater than $1$	greater than the starting quantity	$2.5 \times 1.4 = 3.5$
Multiply by $1$	equal to the starting quantity	$2.5 \times 1 = 2.5$
Multiply by a factor between $0$ and $1$	less than the starting quantity	$2.5 \times 0.6 = 1.5$
Divide by a divisor greater than $1$	less than the starting quantity	$2.5 \div 1.25 = 2$
Divide by $1$	equal to the starting quantity	$2.5 \div 1 = 2.5$
Divide by a divisor between $0$ and $1$	greater than the starting quantity	$2.5 \div 0.5 = 5$

Worked example

Multiply in context: paint for a wall

One liter of paint covers $6.5$ square meters of wall. You have $0.8$ liter of paint left. How many square meters of wall can you cover?

Choose the operation from the meaning. You know the area that one liter covers and the number of liters, and you want the total area. That is the size of one group times the number of groups, so you multiply.
$6.5 \times 0.8$
Predict the size. The quantity 6.5 is positive and the factor 0.8 is between 0 and 1, so the product is less than 6.5. Half a liter would cover half of 6.5, which is 3.25, and 0.8 liter is more than half a liter. Expect an answer between 3.25 and 6.5.
$3.25 < 6.5 \times 0.8 < 6.5$
Multiply the digits as whole numbers. You can do this because 6.5 is 65 tenths and 0.8 is 8 tenths, so 65 times 8 counts the pieces of the product.
$65 \times 8 = 520$
Place the decimal point. Tenths times tenths make hundredths, so the 520 pieces are hundredths. Write 520 hundredths with two decimal places.
$6.5 \times 0.8 = 5.20 = 5.2$
Run the size test and the inverse test. The product 5.2 is between 3.25 and 6.5, as predicted. Now divide the product by the factor 0.8, which is not zero. Multiplying both numbers by 10 gives 52 divided by 8, which is 6.5, the other factor. The multiplication is correct.
$5.2 \div 0.8 = 52 \div 8 = 6.5$

Answer

You can cover $5.2$ square meters of wall.

The inverse test works because multiplication and division undo each other. You used it on a product in the paint problem. For a division, the divisor is never zero, and the test depends on the kind of quotient you have.

Exact quotient. An exact quotient has not been rounded. Quotient times divisor equals the dividend. For $4.5 \div 0.25 = 18$ , the test is $18 \times 0.25 = 4.5$ .
Rounded quotient. A rounded quotient is not the exact one, so quotient times divisor is only close to the dividend. For $7 \div 0.9 \approx 7.78$ , the test gives $7.78 \times 0.9 = 7.002$ , which is close to $7$ but not equal to it.

Before you test a quotient, ask what it counts.

Things you can split, such as juice. A decimal quotient is the complete answer, with nothing left over. When $4$ people share $4.5$ liters, each person gets $4.5 \div 4 = 1.125$ liters.
Things you cannot split, such as notebooks. The whole-number quotient, the whole-number part of the quotient, gives the count, and the amount left over is the remainder.

Do not invent a remainder. If the exact quotient is already a whole number, nothing is left over: $18$ cups use all $4.5$ liters. When there is a remainder, the inverse test adds it back:

\text{divisor} \times \text{whole-number quotient} + \text{remainder} = \text{dividend}

Worked example

Divide in context: whole notebooks and a leftover

You have $20$ dollars to spend on notebooks. Each notebook costs $3.25$ dollars. How many notebooks can you buy, and how much money is left over?

Choose the operation. You know the total, 20 dollars, and the size of one group, 3.25 dollars for each notebook. You want the number of groups, so you divide. The divisor 3.25 is not zero.
$20 \div 3.25$
Predict the size. The quantity 20 is positive and the divisor 3.25 is greater than 1, so the quotient is less than 20. For an estimate, round 3.25 to 3. At that price, 6 notebooks cost 18 dollars and 7 cost 21 dollars, so expect about 6 notebooks.
$3 \times 6 = 18 \text{ and } 3 \times 7 = 21$
Make the divisor a whole number. Multiply the dividend and the divisor by 100. You now count both amounts in cents, so the number of notebooks stays the same.
$20 \div 3.25 = 2000 \div 325$
Find the whole-number quotient. Six groups of 325 make 1950. Seven groups make 2275, which is more than 2000. So 6 groups fit, with 50 cents left.
$2000 - 6 \times 325 = 2000 - 1950 = 50$
Run the meaning test. A notebook cannot be split, so the answer is the whole number 6, not a decimal. It matches the estimate. The leftover 50 cents is the remainder. In dollars, 50 cents is 0.50.
$50 \text{ cents} = 0.50 \text{ dollars}$
Run the inverse test. There is a remainder, so use divisor times whole-number quotient plus remainder. The result equals the dividend, 20, so the answer is correct.
$3.25 \times 6 + 0.50 = 19.50 + 0.50 = 20$

Answer

You can buy $6$ notebooks and have $0.50$ dollars left over.

Common mistakes and the test that catches each one

Mistake	Wrong work	Test that catches it	Correct work
Wrong operation	Cups filled from the jug: $4.5 \times 0.25 = 1.125$	Meaning: the question asks how many groups fit, which is a division.	$4.5 \div 0.25 = 18$ cups
Decimal point misplaced in a product	$0.4 \times 0.3 = 1.2$	Size: positive $0.4$ times a factor between $0$ and $1$ must be less than $0.4$ .	$0.4 \times 0.3 = 0.12$
Decimal point moved in the divisor only	$3.6 \div 0.12$ written as $3.6 \div 12 = 0.3$	Inverse: $0.3 \times 0.12 = 0.036$ , not $3.6$ .	$3.6 \div 0.12 = 360 \div 12 = 30$
Decimal quotient given for whole items	$20 \div 3.25 \approx 6.15$ notebooks	Meaning: a notebook cannot be split.	$6$ notebooks, with $0.50$ dollars left over

Worked example

Find and fix the mistakes

Ribbon costs $2.4$ dollars per meter. Sam buys $7.2$ meters and cuts all of it into pieces $0.45$ meter long. Sam writes two lines. Cost: $2.4 \times 7.2 = 172.8$ dollars. Number of pieces: $7.2 \div 0.45 = 72 \div 45 = 1.6$ . Find and fix the mistake in each line.

Run the meaning test. The cost is the price of one meter times the number of meters, so multiplying is right. The number of pieces is the total length divided by the length of one piece, so dividing is right. The mistakes must be in the calculating.
Run the size test on the cost. Round each factor for an estimate: 2 times 7 is 14. Sam's 172.8 is more than ten times the estimate, so it cannot be right.
$2 \times 7 = 14$
Fix the cost. Sam's digits are right: 24 times 72 is 1680 plus 48, which is 1728. But 2.4 is 24 tenths and 7.2 is 72 tenths, and tenths times tenths make hundredths. The product needs two decimal places, and Sam wrote only one.
$2.4 \times 7.2 = 17.28$
Run the size test on the number of pieces. The length 7.2 is positive and the divisor 0.45 is between 0 and 1, so the quotient must be greater than 7.2. Sam's 1.6 is less. The inverse test fails too, because 1.6 times 0.45 does not give back 7.2.
$1.6 \times 0.45 = 0.72 \neq 7.2$
Fix the division. Sam multiplied the divisor by 100 to get 45, but multiplied the dividend by only 10 to get 72. The quotient stays the same when both are multiplied by the same number, as long as it is not zero. So multiply both by 100. Then 45 times 16 is 450 plus 270, which is 720.
$7.2 \div 0.45 = 720 \div 45 = 16$
Run the inverse test on both corrected answers. The cost 17.28 divided by the factor 7.2, which is not zero, gives back the other factor, 2.4. The quotient 16 is exact, so 16 times the divisor gives back the dividend, 7.2. No ribbon is left over, so there is no remainder.
$17.28 \div 7.2 = 172.8 \div 72 = 2.4 \text{ and } 16 \times 0.45 = 7.2$

Answer

The ribbon costs $17.28$ dollars, and Sam gets exactly $16$ pieces with no ribbon left over.

Takeaway. In a decimal word problem, the meaning of the situation chooses the operation. Multiply to find a total from equal groups. Divide, by a divisor that is not zero, to find how many groups there are or how big each one is. Then test the answer three ways. The meaning test asks whether the operation and the answer fit the situation, including whether a leftover makes sense. The size test asks whether the answer matches your prediction. The inverse test asks whether the inverse operation leads back to the numbers you started with, exactly for an exact answer and nearly for a rounded one. A failed test tells you to look for a mistake, in your own work or in someone else's.