About Decimal Fraction
A decimal fraction is simply a fraction whose denominator is a power of ten. Every calculation in the topic comes down to counting decimal places, and almost every wrong answer comes from counting them wrongly.
What you need to understand
- The number of decimal places in a product is the sum of the decimal places in the two numbers multiplied.
- A fraction gives a terminating decimal only if its denominator has no prime factors other than 2 and 5.
- Halves, quarters, fifths, eighths and twentieths all terminate. Thirds, sixths and sevenths do not.
- Trailing zeros after a decimal point do not change the value, so 0.50 and 0.5 are the same number.
- Multiplying by ten moves the decimal point one place to the right, and dividing by ten moves it one place to the left.
- Both numbers in a product being less than one guarantees that the answer is smaller than either of them.
Formulas to remember
How to work through these questions
- Count the decimal places in each number before multiplying, and add the counts to know where the point belongs.
- Multiply as though the points were not there, then insert the point that many places from the right.
- For a fraction to decimal question, divide the numerator by the denominator and stop when the remainder is zero.
- If the remainder never reaches zero the decimal recurs, and the question will expect a fraction as the answer.
- Check the size of the answer, because multiplying two numbers below one must give a smaller result than either.
Mistakes that cost marks
- Losing count of the places, so that 0.6 multiplied by 0.06 becomes 0.36 instead of 0.036.
- Moving the decimal point the wrong way when converting a fraction to a decimal.
- Writing 0.5 as 0.05 by shifting the point one place too far.
- Assuming every fraction gives a terminating decimal.
- Dropping the decimal point part way through a longer calculation and never recovering it.
Worked example
Find the value of 0.6 multiplied by 0.06.
- Ignore the decimal points and multiply: 6 x 6 = 36.
- Count the places: 0.6 has one decimal place and 0.06 has two, so the answer needs three.
- Count three places from the right in 36, which gives 0.036.
- Check: both numbers are below one, so the product must be smaller than either, and 0.036 is smaller than 0.06.
Answer: 0.036
Practice questions with answers
A few Decimal Fraction questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Convert 1/2 into a decimal.
Answer: Option B — with explanation
Divide the numerator by the denominator: 1 / 2.
2 goes into 1 giving 0.5 exactly, because the only prime factors of the denominator are 2 and 5.
Common mistakes
- moved the decimal point one place to the left: 0.05
- moved the decimal point two places to the right: 50
- moved the decimal point one place to the right: 5
Question 2
Find the value of $1.2 \div 3$.
Answer: Option D — with explanation
Write the dividend as a fraction: 1.2 = 12/10.
Dividing by 3 cancels the 3: the answer is 4/10 = 0.4.
Common mistakes
- moved the decimal point one place too far to the left: 0.04
- divided the divisor instead of the dividend: 0.3
- multiplied by ten instead of dividing: 4
Question 3
-
A
0.018
-
B
1.8
-
C
0.0018
-
D
0.18
Answer: Option A — with explanation
Ignore the decimal points first: 9 x 2 = 18.
The first number has 1 decimal place and the second has 2, so the answer must have 1 + 2 = 3 decimal places: 0.018.
Common mistakes
- counted only one decimal place: 1.8
- counted only two decimal places instead of three: 0.18
- counted four decimal places instead of three: 0.0018
Question 4
Answer: Option A — with explanation
Write the dividend as a fraction: 1.8 = 18/10.
Dividing by 2 cancels the 2: the answer is 9/10 = 0.9.
Common mistakes
- moved the decimal point one place too far to the left: 0.09
- multiplied by ten instead of dividing: 9
- divided the divisor instead of the dividend: 0.2
Question 5
Convert 2/4 into a decimal.
Answer: Option A — with explanation
Divide the numerator by the denominator: 2 / 4.
4 goes into 2 giving 0.5 exactly, because the only prime factors of the denominator are 2 and 5.
Common mistakes
- moved the decimal point two places to the right: 50
- moved the decimal point one place to the left: 0.05
- moved the decimal point one place to the right: 5
Frequently asked questions
Why is 0.6 multiplied by 0.06 equal to 0.036 and not 0.36?
Because the answer must have as many decimal places as the two numbers together. One plus two is three places, and 0.36 has only two.
Which fractions give terminating decimals?
Those whose denominators contain only 2s and 5s. Halves, quarters, fifths, eighths, tenths, sixteenths and twentieths all terminate.
Can a decimal carry on forever?
Yes, and then it is called a recurring decimal. One third is 0.3333 and the threes never stop, so the exact answer has to be given as a fraction.
How do I divide by a decimal?
Shift the decimal point in the divisor until it becomes a whole number, shift the point in the dividend the same number of places, and then divide as usual.
