Maths- Decimals, Fractions, Recurring and Percentage

Question 1

Place value and ordering decimals

Answer 1

We use a decimal point to separate units from parts of a whole (tenths, hundredths, thousandths etc)
A tenth is 1/10 of a unit
A hundredth is 1/100 of a unit
A thousandth is 1/1000 of a unit
In the number 34.27, the value of the figure 2 is a tenth, and the value of the figure 7 is a hundredth.

Question 2

Adding and subtracting decimals

Answer 2

When adding and subtracting decimals add or subtract as normal, but make sure that you keep the decimal points aligned.
For example 4.27 + 2.3 =
   4.27
\+ 2.30
\_\_\_\_\_\_
   6.57
And to work out 5 - 0.24 we can write it:
    5.00
-   0.24
\_\_\_\_\_\_
    4.76
Note that we wrote 5 as 5.00. It is not essential to do this, but it helps.

Question 3

Multiplying and dividing decimals

Answer 3

Multiplying and dividing by 10, 100, 1000
Numbers can be written under their place value headings. For example 10.9 would be:

Thousands Hundreds Tens Units Tenths Hundredths
1 0 9
Multiply by 10 and the number will move one place value to the left.
10.9 x 10 = 109
Multiply by 100 and the number will move two place values to the left.
10.9 x 100 = 1090
Notice how the amount of zeros is the same as the amount of place values the number moves.
Divide by 10 and the number will move one place value to the left.
10.9 ÷ 10 = 1.09
These examples show you what happens.

Question 4

Action
Rule
Example

Answer 4

Multiply by 10
Move the whole number one place value to the left
2.45 x 10 = 24.5

Multiply by 100
Move the whole number two place values to the left
2.45 x 100 = 245

Multiply by 1000
Move the whole number three place values to the left
2.45 x 1000 = 2450
Divide by 10
Move the whole number one place value to the right
46.7 ÷ 10 = 4.67

Divide by 100
Move the whole number two place values to the right
46.7 ÷ 100 = 0.467

Divide by 1000
Move the whole number three place values to the right
46.7 ÷ 1000 = 0.0467

Question 5

Multiplying decimals

Answer 5

Multiplying decimals is the same as multiplying two whole numbers. You just need to remember the following:
If there is one digit after the decimal point in the question, there will be one digit after the decimal point in the answer.
If there are two digits after the decimal point in the question, there will be two digits after the decimal point in the answer etc.
For example to calculate 3.42 × 2 we work out 342 × 2 and then work out where to put the decimal point.
342
× 2
____
684
There are two digits after the decimal point in the question (3.42 x 2), so there will be two digits after the decimal point in the answer. Therefore 3.42 × 2 = 6.84
You can check you have the decimal point in the right place by estimating. 3.42 x 2 should be a little bit more than 3 x 2. So 6.84 looks like a sensible answer.

Question 6

Dividing decimals

Answer 6

When dividing a decimal by a whole number, divide as usual but keep the decimal points aligned
For example:
5.75 ÷ 5 =

Note that the decimal points are in the same place.
If you are dividing a decimal by another decimal, you need to use equivalent fractions.
For example, 2.42 ÷ 0.2 means which is the same as (we have multiplied the numerator and denominator by 10).
3.715 ÷ 0.005 means which is the same as We have multiplied the numerator and denominator by 1000.
Remember to always multiply the numerator and denominator by the same number. And make sure that the denominator is a whole number.

Question 7

Fractions of a quantity

Answer 7

How do we find 3/5 of 20?
Method 1 is to find 1/5 of 20, then multiply by 3.
1/5 of 20 is 20 ÷ 5 = 4.
We need 3/5 of 20, so we multiply 4 by 3.
3/5 of 20 = 4 × 3 = 12.
Method 2 is to multiply 3/5 by 20 (more on multiplying fractions here and here).
3/5 × 20 = 3/5 × 20/1 = 60/5 = 12

Question 8

Cancelling fractions

Answer 8

Sometimes you can divide the top and bottom of a fraction by the same number. This is called cancelling down. It is also called simplifying the fraction. You often have to write a fraction in its simplest terms. This means that you have to cancel it down until it cannot be canceled down any more

Question 9

One number as a fraction of another

Answer 9

Imagine that there are 10 questions in a test and you get 7 of them correct. You would say that you got
7 as a fraction of 10 is
In the same way, 4 as a fraction of 12 is and 20 as a fraction of 48 is
Easy? Yes, but just be careful with the units.
For example, 20p as a fraction of £2 is not (£2 = 200p). And 30cm as a fraction of 5m is (5m = 500cm).

Question 10

Multiplying and dividing fractions

Answer 10

Remember that to multiply fractions, you need to multiply the numerators together, and multiply the denominators together. Remember that you cannot cancel numbers that are both on the top of a fraction. Have a look at the examples below.
Example
2/3 x 1/2
We multilpy 2/3 by 1/2
So we have 2/3 x 1/2 = 2/6
Example
4/5 x 5/6
Multiply then cancel:
OR cancel between the top and the bottom then multiply:

Question 11

Dividing a fraction by a whole number

Answer 11

Three people share 3/4’s of a pie. They each get 1/4.
This means 3/4 divided by 3 would make the answer 1/4.
Another way of writing 3 is 3/1
So 3/4 ÷ 3/1 = 1/4
You get the same answer when you turn the fraction you divide by upside down, and multiply instead. For example,3/4 x 1/3 = 1/4.
The same is true with a whole number.
15 ÷ 3 = 5 and 15 x 1/3 = 5

Question 12

Dividing by a fraction

Answer 12

How do you divide 12 by 1/3? This isn’t the same as 12 divided by 3.
12 ÷ 1/3 means how many thirds are there in 12 whole units.
As there are 3 thirds in each whole unit, then there are 36 thirds in 12 whole units. Think how many 1/3’s there are in one pie, then 12 pies.
A simple way to divide by a fraction is to turn the fraction upside down and multiply.
It works with all fractions. For example dividing by 2/3 is the same as multiplying by 3/2.
10 ÷ 2/3 = 10/1 x 3/2 = 30/2 = 15/1 = 15
Notice that when you divide by a fraction the answer is larger than the number you started with.

Question 13

Reciprocals

Answer 13

The reciprocal of a number is 1 divided by that number.
Then the reciprocal of a fraction is the fraction turned upside down.
So the reciprocal of 3/4 is 4/3
For instance, to work out the reciprocal of 4:
We can think of 4 as 4/1
So 4 turned upside down is 1/4

Question 14

Fractions and decimals: Higher

Answer 14

If the prime factors of the denominator of a fraction in its simplest form are only 2 and/or 5 its decimal will terminate.
How do we know whether a fraction will give a terminating decimal? The rule is to find the prime factors of the denominator.
If the prime factors are only 2 and/or 5 the decimal will terminate.
Example
3/20 = 3/(2 x 2 x 5) = 0.15
However if we have 1/14 = 1/(2 x 7) this will not terminate
3/28
28 = 2 x 2 x 7
so the decimal will not terminate. It will be a recurring decimal.
7/40
40 = 2 x 2 x 2 x 5
The prime factors of 40 consist of 2s and 5s, so the decimal will terminate.
6/125
125 = 5 x 5 x 5
The decimal will terminate.
71/120
120 = 2 x 2 x 2 x 3 x 5
There is a 3 in there, so the decimal will recur.

Question 15

Percentage increase and decrease

Answer 15

n some questions you are given the cost price and the selling prices and have to find the percentage increase or decrease. This means you need to find one amount as a percentage of another. You form a fraction from the two amounts and multiply this by 100. Try these questions.

Question 16

Simple interest

Answer 16

With simple interest the amount of money borrowed remains fixed.
For example £400 is borrowed for 3 years at an interest rate of 5% pa (pa means per annum, or each year).
Interest for one year = 5% of £400
= (5/100 ) × 400
= £20
Interest for 3 years = £20 × 3 = £60.
You can write this in a formula.
Interest = P × R × T
P (principal) is the amount borrowed.
R is the rate of interest per year.
T is the time in years.

Question 17

Profit and loss

Answer 17

People often buy something at one price, and sell it on for another - eg, when they are selling things at a car boot sale.
If the selling price is greater than the buying price, a profit is made.
If the selling price is less than the buying price, there is a loss.