Decimal fractions (Grade 8) — Mathematics Grades 8–11

Ordering, rounding and calculating with decimal fractions.

Gr 8 Term 3 — Decimal fractionsTerm 3

Practise Decimal fractions in the app →

What gets asked

Order or round a list of decimal numbers.
Add, subtract, multiply or divide decimal fractions.
Predict how many decimal places an answer will have.
Solve a money or measurement problem using decimals.

You must be able to

Line up the decimal comma correctly when adding or subtracting.
Work out the number of decimal places in a product before calculating.
Divide a decimal by another decimal by adjusting both numbers first.
Round a money answer to two decimal places.

Traps that cost marks

Judging $0, 03$ to be bigger than $0, 3$ because it has more digits after the comma. Line up the decimal commas and compare place value column by column. $0, 3$ is bigger.
Lining up decimal digits from the right, like whole numbers, instead of lining up the comma. Always line up the decimal commas in a column, then add or subtract as normal.
Giving too few decimal places in a product, e.g. $0, 4 \times 0, 2 = 0, 8$ . Count the decimal places in both numbers first. Two decimal places in, two out: $0, 4 \times 0, 2 = 0, 08$ .
Multiplying only the divisor by 10 or 100 to clear the comma, and leaving the other number as it is. Whatever you do to the divisor, do to the number being divided too.

Worked example

A prepaid electricity voucher costs R2,85 per unit. How much for 6 units?

Multiply $2, 85 \times 6$ .
Ignore the comma first: $285 \times 6 = 1710$ .
Put the comma back two places from the right: R17,10.

Order, round and convert

How to tell which decimal is bigger, how to round it off, and how to write it as a fraction.

decimal fraction: A number with a decimal comma in it. The digits after the comma are parts of one whole. $0, 7$ is seven tenths.
place value: What a digit is worth because of where it stands. After the comma come tenths, hundredths, thousandths.
round off: To swap a number for the nearest number at a chosen place. $3, 47$ rounds off to $3, 5$ .
common fraction: A number written as one whole number over another, like $\frac{25}{100}$ .
percentage: A number of parts out of $100$ . Twenty-five parts out of $100$ is $25%$ .

Every digit after the comma has its own place value. Tenths come first, then hundredths, then thousandths. To compare two decimal fractions, line up the commas and read from the left. The first column where they differ decides it. More digits does not make a number bigger.

To round off, first choose the place you are keeping. Then look at the one digit just after it, and nothing else. If it is $5$ or more, the kept digit goes up by one. If it is smaller, it stays. Do not just chop the rest off.

Counting in decimal steps works like counting whole numbers. Keep the same step the whole way. After $0, 8$ comes $0, 9$ . The tenths are then full, so the next one is $1, 0$ and not $0, 10$ .

The place value of the last digit gives the bottom number. In $0, 25$ the last digit is in the hundredths place, so this is $\frac{25}{100}$ . A percentage is out of $100$ too, so the same number is $25%$ . Move the comma two places right to get a percentage.

Rules to remember

$0, 5 = \frac{5}{10} = \frac{1}{2}$
Multiplying by $100$ moves the comma two places right

Examples

Put these in order, smallest first:

0, 4

;

0, 25

;

0, 309

Worked answer

Line the commas up so the columns match.
Read the tenths first: $4$ , then $2$ , then $3$ .
The tenths already decide it, so stop.
In order they are $0, 25$ ; $0, 309$ ; $0, 4$ .

Answer: $0, 25$ ; $0, 309$ ; $0, 4$

The number of digits told you nothing. Only the tenths mattered.

Round off

4, 249

to one decimal place

Worked answer

One decimal place means keeping the $2$ in the tenths place.
Look at the next digit only. It is $4$ , which is smaller than $5$ .
So the $2$ stays, and $4, 249$ rounds off to $4, 2$ .

Answer: $4, 2$

Only the first digit after the kept place gets a say.

Write

0, 35

as a common fraction and as a percentage

Worked answer

The last digit is in the hundredths place, so the bottom number is $100$ .
That gives $\frac{35}{100}$ .
Top and bottom both divide by $5$ , so it is also $\frac{7}{20}$ .
A percentage is out of $100$ , so $0, 35$ is $35%$ .

Answer: $\frac{35}{100}$ , or $\frac{7}{20}$ , and $35%$

The place value of the last digit gave the bottom number.

Traps

Judging $0, 03$ to be bigger than $0, 3$ because it has more digits after the comma. Line up the decimal commas and compare place value column by column. $0, 3$ is bigger.
Chopping the extra digits off instead of rounding, so $2, 78$ becomes $2, 7$ . The next digit is $8$ , which is $5$ or more, so $2, 78$ rounds off to $2, 8$ .
Counting $0, 8$ ; $0, 9$ ; $0, 10$ and carrying on. Ten tenths make one whole. After $0, 9$ comes $1, 0$ , and then $1, 1$ .
Moving the comma one place instead of two, so $0, 25$ becomes $2, 5%$ . Multiply by $100$ , which moves the comma two places. So $0, 25$ is $25%$ .

Learn and practise “Order, round and convert” in the app →

Calculate with decimals

Adding, subtracting, multiplying and dividing money and measurements that have a comma in them.

placeholder zero: A zero put at the end of a decimal so both numbers have the same digits after the comma. It changes nothing.
product: The answer you get when you multiply. The product of $0, 4$ and $0, 2$ is $0, 08$ .
divisor: The number you are dividing by. In $5, 4 \div 6$ the divisor is $6$ .
square: To square a number is to multiply it by itself. $0, 3$ squared is $0, 09$ .
square root: The number that multiplies by itself to give it. The square root of $0, 09$ is $0, 3$ .

To add or subtract, line up the commas, not the last digits. Fill any short number with placeholder zeros so every column has a digit. Then work column by column as usual, and bring the comma straight down into the answer.

To multiply, forget the commas and multiply as whole numbers. Then count the decimal places in the two numbers you started with. The product gets that many places. One place and one place makes two, so $0, 4 \times 0, 2 = 0, 08$ .

To divide by a decimal, make the divisor a whole number. Multiply it by $10$ or $100$ , and do exactly the same to the other number. Then divide as usual. Dividing by a number under $1$ makes the answer bigger: $54 \div 0, 6 = 90$ .

Squaring a decimal doubles its decimal places. One place squared gives two: $0, 3 \times 0, 3 = 0, 09$ . Cubing gives three: $0, 2 \times 0, 2 \times 0, 2 = 0, 008$ . A square root works back the other way.

Rules to remember

$0, 4 \times 0, 2 = 0, 08$
Putting a zero at the end of a decimal does not change it

Examples

You have R50. Bread costs R18,75 and two litres of milk cost R27,50. How much is left?

Worked answer

Add the two amounts: $18, 75 + 27, 50 = 46, 25$ .
Write R50 as $50, 00$ so both have two decimal places.
Line the commas up and subtract: $50, 00 - 46, 25 = 3, 75$ .

Answer: R $3, 75$

The placeholder zeros gave every column something to subtract from.

A prepaid electricity voucher costs R2,85 for one unit. What do

6

units cost?

Worked answer

Forget the comma and multiply: $285 \times 6 = 1710$ .
Only one of the numbers has decimal places, and it has two.
So the product has two places: $17, 10$ .

Answer: R $17, 10$

The whole-number sum was the easy half. Counting the places made it right.

A tap fills

0, 6

litre bottles. How many can be filled from

5, 4

litres?

Worked answer

The divisor is $0, 6$ . Multiply it by $10$ to get $6$ .
Do the same to the other number: $5, 4 \times 10 = 54$ .
Now divide: $54 \div 6 = 9$ .
The answer is bigger than $5, 4$ , and here that is correct.

Answer: $9$ bottles

Both numbers grew ten times, so the answer to the division stayed the same.

Traps

Lining up decimal digits from the right, like whole numbers, instead of lining up the comma. Always line up the decimal commas in a column, then add or subtract as normal.
Giving too few decimal places in a product, e.g. $0, 4 \times 0, 2 = 0, 8$ . Count the decimal places in both numbers first. Two decimal places in, two out: $0, 4 \times 0, 2 = 0, 08$ .
Multiplying only the divisor by 10 or 100 to clear the comma, and leaving the other number as it is. Whatever you do to the divisor, do to the number being divided too.
Halving to find a square root, so the square root of $0, 04$ is given as $0, 02$ . Look for the number that multiplies by itself. $0, 2 \times 0, 2 = 0, 04$ , so the answer is $0, 2$ .

Learn and practise “Calculate with decimals” in the app →

Estimate, check and solve

Knowing roughly what the answer should be, so that a wrong one cannot slip past you.

estimate: A quick rough answer, worked out with easy numbers. You use it to check the real answer.
decimal place: One of the places after the decimal comma. $2, 85$ has two decimal places.
unit: What a number is counted in, such as metres, litres or cents.
cent: One hundredth of a rand. There are $100$ cents in a rand, so money has two decimal places.

Predict before you calculate. When you multiply, add the decimal places of the two numbers. $1, 43$ has two places and $0, 2$ has one, so the product has three. Knowing that first means you cannot put the comma in the wrong spot afterwards.

Dividing does not follow that rule. $4, 8 \div 0, 6 = 8$ , a whole number with no decimal places at all. The divisor and the number being divided both change when you clear the comma, so the places do not simply add up.

To estimate, round off each number to something easy and work it out in your head. Then compare that with what the calculator says. If the two are far apart, you keyed something in wrongly. Round off only at the end, never in the middle.

In a word problem, get the units the same before you start. Centimetres and metres cannot be added as they stand. A money answer is rounded off to two decimal places, because $100$ cents make one rand.

Rules to remember

Decimal places in a product: add the places of the two numbers
$100$ cents make R1, so money is written with two decimal places

Examples

How many decimal places will

1, 43 \times 0, 2

have? Then work it out.

Worked answer

$1, 43$ has two decimal places and $0, 2$ has one.
Two plus one is three, so the product has three places.
Multiply as whole numbers: $143 \times 2 = 286$ .
Put the comma three places from the right, giving $0, 286$ .

Answer: three places; $0, 286$

The prediction told you where the comma goes before you did the sum.

Estimate, then work out, what

6

prepaid electricity units cost at R2,85 each.

Worked answer

Round off R2,85 to R3 for the estimate.
$3 \times 6 = 18$ , so expect about R18.
Now do it properly: $2, 85 \times 6 = 17, 10$ .
R17,10 sits close to R18, so it is sensible.

Answer: about R $18$ ; exactly R $17, 10$

A slip that gave R171,00 would have stood out at once.

A shelf plank is

2, 4

m long. You cut off

85

cm. How much is left, in metres?

Worked answer

The units differ, so change $85$ cm into metres: $0, 85$ m.
Write $2, 4$ as $2, 40$ so both have two decimal places.
Line the commas up and subtract: $2, 40 - 0, 85 = 1, 55$ .

Answer: $1, 55$ m

Taking $85$ away from $2, 4$ would have given a nonsense length.

Traps

Adding the decimal places when dividing, because that is the rule for multiplying. That rule is only for products. $4, 8 \div 0, 6 = 8$ , with no decimal places.
Taking whatever the calculator shows, with no estimate to hold it against. Work out a rough answer first. If the two are far apart, key it in again.
Adding centimetres to metres in one sum, without changing them first. Put both into the same unit first. $85$ cm is $0, 85$ m.
Leaving a money answer as R $12, 5$ or as R $12, 456$ . Money is rands and cents, so round off to two decimal places: R12,50.

Learn and practise “Estimate, check and solve” in the app →

Grade 8 Decimal fractions

What gets asked

You must be able to

Traps that cost marks

Worked example

Order, round and convert

Rules to remember

Examples

Traps

Calculate with decimals

Rules to remember

Examples

Traps

Estimate, check and solve

Rules to remember

Examples

Traps

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