Decimal fractions (Grade 9) — Mathematics Grades 8–11

Calculating with decimals, rounding off, and converting between fractions, decimals and percentages.

Gr 9 Term 1 — Decimal fractionsTerm 1

Practise Decimal fractions in the app →

What gets asked

Calculate an expression with decimals, with or without brackets.
Round off a decimal answer to a stated number of decimal places.
Solve a money or measurement problem using decimals.
Convert a number between fraction, decimal and percentage form.

You must be able to

Line up decimal commas correctly when adding or subtracting.
Place the decimal comma correctly after multiplying two decimals.
Round a decimal answer to the number of places a question asks for.
Change a number between a common fraction, a decimal and a percentage.

Traps that cost marks

Placing the decimal comma using only one of the two numbers being multiplied. Count the decimal digits in both numbers together to place the comma in the answer.
Rounding using the wrong digit. Look only at the digit right after the place you are rounding to: $2, 346$ to 1 decimal place is $2, 3$ .
Rounding a value part-way through a calculation, not just the final answer. Keep full accuracy until the last step, then round only the final answer.

Worked example

Nomvula buys 3,5 kg of mince at R64,80 a kilogram. Calculate the total cost.

Multiply: $3, 5 \times 64, 80 = 226, 80$ .
That answer is already in rands and cents.
The total cost is R226,80.

Rounding and equivalent forms

One number, three outfits: a decimal, a fraction and a percentage. Plus how to round it safely.

decimal fraction: A number written with a comma, where the digits after the comma stand for parts of one whole.
decimal comma: The comma that separates the whole part from the part smaller than one. In $3, 45$ the whole part is $3$ .
decimal place: One digit position after the comma. $3, 45$ has two of them.
round off: To swap a number for a shorter one that is close to it.
percentage: A number of parts out of $100$ . $85%$ means $85$ parts out of every $100$ .

A decimal fraction is a number with a decimal comma in it. Each place after the comma is ten times smaller than the one before: tenths, then hundredths, then thousandths. In $0, 07$ the $7$ sits in the hundredths place, so the number is seven hundredths.

To round off, look at one digit only: the first digit past the place you are keeping. If it is $5$ or more, the kept digit goes up by one. If it is smaller, the kept digit stays. Then drop the rest, in one move.

Before you multiply, count the decimal places in both numbers and add those two counts. $0, 3$ has one and $0, 4$ has one, so the answer carries two: $0, 3 \times 0, 4 = 0, 12$ .

A decimal becomes a common fraction through its last place. $0, 64$ has two decimal places, so the denominator is $100$ . Then simplify the fraction you get.

Multiply a decimal by $100$ and you have a percentage. Divide a percentage by $100$ and you have a decimal back. The comma moves two places, and the size of the number tells you which way.

Rules to remember

$0, 75 = \frac{75}{100} = \frac{3}{4}$
$0, 3 \times 0, 4 = 0, 12$
$\frac{3}{4} = 0, 75$ , and that is $75%$

Examples

Round off

2, 346

to one decimal place, and then to two decimal places.

Worked answer

For one place, keep the $3$ and look at the next digit, $4$ .
$4$ is less than $5$ , so the $3$ stays: $2, 3$ .
For two places, keep the $4$ and look at the next digit, $6$ .
$6$ is $5$ or more, so the $4$ goes up: $2, 35$ .

Answer: $2, 3$ and $2, 35$

Each rounding started again from $2, 346$ , so the first answer never fed into the second.

Write

0, 07

and

0, 64

as common fractions in simplest form.

Worked answer

$0, 07$ has two decimal places, so the denominator is $100$ : $\frac{7}{100}$ .
$7$ and $100$ share no factor but $1$ , so that one is finished.
$0, 64 = \frac{64}{100}$ .
Divide top and bottom by $4$ : $\frac{64}{100} = \frac{16}{25}$ .

Answer: $\frac{7}{100}$ and $\frac{16}{25}$

The count of decimal places chose the denominator, and only then did simplifying start.

Ayanda scored

\frac{17}{20}

in a test. Write that score as a decimal and as a percentage.

Worked answer

Divide the top by the bottom: $17 \div 20 = 0, 85$ .
To reach a percentage, multiply by $100$ .
$0, 85 \times 100 = 85$ .
Check it another way: $\frac{17}{20} = \frac{85}{100}$ .

Answer: $0, 85$ , or $85%$

All three forms came out of one division, and the check confirmed the comma had moved correctly.

Traps

Rounding using the wrong digit. Look only at the digit right after the place you are rounding to: $2, 346$ to 1 decimal place is $2, 3$ .
Chopping the extra digits off instead of rounding. $2, 38$ to one decimal place is $2, 4$ , not $2, 3$ . The dropped digits still get a say.
Expecting the answer to have as many decimal places as one of the numbers. Add the counts from both numbers. $0, 3 \times 0, 4$ needs two, so the answer is $0, 12$ .
Writing $\frac{1}{4}$ as $1, 4$ . The line means divide: $1 \div 4 = 0, 25$ .

Learn and practise “Rounding and equivalent forms” in the app →

Calculate with decimals

Where the comma goes in each of the four operations, and why squaring can make a number smaller.

product: The answer when you multiply. The product of $6$ and $7$ is $42$ .
square: A number multiplied by itself. The square of $0, 3$ is $0, 09$ .
square root: The number that multiplies by itself to give you the one you started with. The square root of $0, 09$ is $0, 3$ .
cube: A number multiplied by itself and then by itself again: $0, 2 \times 0, 2 \times 0, 2 = 0, 008$ .
cube root: The number that gives you back the one you started with when it is cubed.

To add or subtract, line the decimal commas up, one under the other. Fill the shorter number out with zeros so that every column holds a digit. $4, 8$ may be written as $4, 80$ .

To multiply, ignore the commas at first and multiply as though both were whole numbers. Then count the decimal places in both numbers together, and give the product that many.

To divide, move the comma in the divisor until it is a whole number. Move the comma in the other number by the same number of places. Then divide the usual way.

Squaring a number smaller than $1$ makes it smaller: $0, 3^{2} = 0, 09$ . The square root goes back the other way, so $0, 09 = 0, 3$ . A cube behaves the same, and the cube root brings you back.

In a problem with several steps, hold on to every digit until the end. Round off the final answer only. Rounding part-way through can push a money answer out by rands.

Rules to remember

$1, 2 \times 0, 05 = 0, 06$
$0, 2^{3} = 0, 008$
$0, 09 = 0, 3$

Examples

Calculate:

4, 8 - 2 \times (1, 5 + 0, 25)

Worked answer

Brackets first: $1, 5 + 0, 25 = 1, 75$ .
Then multiply: $2 \times 1, 75 = 3, 5$ .
Then subtract: $4, 8 - 3, 5 = 1, 3$ .

Answer: $1, 3$

The brackets came first, exactly as they would with whole numbers. Only the commas are new.

Calculate:

0, 6^{2} + 0, 64

Worked answer

$0, 6^{2} = 0, 6 \times 0, 6 = 0, 36$ .
The square is smaller than the number, because $0, 6$ is under $1$ .
$0, 64 = 0, 8$ , because $0, 8 \times 0, 8 = 0, 64$ .
Add them: $0, 36 + 0, 8 = 1, 16$ .

Answer: $1, 16$

The square shrank the number and the square root grew it, which is what those two do below $1$ .

Nomvula buys

3, 5

kg of mince at R64,80 a kilogram and pays with R250. How much change does she get?

Worked answer

Work out the cost: $3, 5 \times 64, 80 = 226, 80$ , so R226,80.
Keep all the digits; nothing gets rounded off yet.
Now subtract: $250 - 226, 80 = 23, 20$ .

Answer: R23,20

Rounding the cost to R227 first would have given R23, and the 20 cents would be gone.

Traps

Placing the decimal comma using only one of the two numbers being multiplied. Count the decimal digits in both numbers together to place the comma in the answer.
Rounding a value part-way through a calculation, not just the final answer. Keep full accuracy until the last step, then round only the final answer.
Lining decimals up by their last digit when adding. Line the commas up instead. Writing $4, 8$ as $4, 80$ makes the columns match.
Giving the square root of $0, 4$ as $0, 2$ . $0, 2 \times 0, 2 = 0, 04$ , not $0, 4$ . So $0, 04 = 0, 2$ instead.

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

Grade 9 Decimal fractions

What gets asked

You must be able to

Traps that cost marks

Worked example

Rounding and equivalent forms

Rules to remember

Examples

Traps

Calculate with decimals

Rules to remember

Examples

Traps

About this material