- Fraction - part of a whole
- Equivalent fractions - fractions which have the same value.
- Improper fraction - fraction in which numerator is greater than denominator.
- Mixed number - has a whole number and a fraction next to it.
- Converting mixed number to improper fraction -
Multiply the denominator and the whole number and add the numerator. denominator remains the same.
Example: 3 1/2
3 x 2 = 6 6 + 1 = 7 so 3 1/2 = 7/2
- Converting improper fraction to mixed number -
Divide the numerator by denominator, answer will be the whole number, remainder will be the numerator and denominator remains the same.
Example: 9/2
9 divided by 2 is 4 remainder 1 so 9/2 = 4 1/2
Scenario 1: If the denominators are the same, look at the numerators.
If the numerator is a bigger, the fraction is bigger.
Example: 1/6 < 2/6 < 3/6
Scenario 2: If the denominators are different but the numerators are the same, look at the denominator. As the denominator gets bigger, fraction gets smaller.
Example: 1/7 < 1/6 < 1/5
Scenario 3: If the denominators and numerators are different, make the denominators the same(by multiplying or dividing).
Example: find out which of the following fractions is bigger.
3/5 or 1/3
You can make them both fifteenths.
In 3/5, multiply the denominator by 3(always follow rule whatever done to the denominator should be done to the numerator). So 3/5 would become 9/15.
Similarly, 1/3 would become 5/15.
Now, you have two fractions with similar denominators which are easy to compare.
9/15 > 5/15 so 3/5 > 1/3
- Adding and subtracting fractions:
Before you add or subtract fractions, you always have make their denominators the same. Remember, you can do this only by multiplying or subtracting.
This is straight forward. Multiply the numerators and multiply the denominators.
3/5 x 1/2 = 3 x 1/5 x 2 = 3/10
3 x 2/3 is same as 3/1 x 2/3 = 3 x 2 /1 x 3 = 6/3 = 2
3/5 ÷ 2/3 follow the rule Keep, Change and Flip.
Keep 3/5, change ÷ into x and flip 2/3
So, it would be 3/5 x 3/2 which is equal to 9/10.
- Finding half of a fraction:
Example: find half of 3/5
3/5 ÷ 2 which would be 3/5 x 1/2 which is equal to 3/10
Basically, you multiply the denominator by 2.
So, half of 3/4 is equal to 3/8 and half of 2/3 is equal to 2/6 which is equal to 1/3
- Finding fraction of an amount or quantity:
Divide by the denominator and multiply by the numerator.
So, to find 3/4 of 60, divide 60 by 4 and multiply the answer you get by 3.
When you divide 60 by 4, you are actually finding 1/4 of 60.
There are three quarters in 3/4, so you have to multiply the answer by 3.
- Finding the original amount or quantity, when you know the fraction.
Example: If 3/4 of a number is 15, what is the number?
3/4 = 15.
So 1/4 = 5 because 15 ÷ 3 = 5
1 whole is 5 x 4 which is 20
So the number is 20
- Fractions, Percentages and Decimals are related.
1/2 = 50% = 0.5
1/4 = 25% = 0.25
3/4 = 75% = 0.75
1/10 = 0.1 = 0.1
1/5 = 20% = 0.2
2/5 = 40% = 0.4
3/5 = 60% = 0.6
4/5 = 80% = 0.8
1/100 = 1% = 0.01
1/8 = 12.5% = 0.125
1/3 = 33 1/3% = 0.33
From the above, you can understand how to find percentages of an amount of quantity.
To find 50%, divide the amount by 2 as 50% = 1/2
To find 25%, divide the amount by 4 as 25% = 1/4
To find 75%, divide the amount by 4 and multiply the answer by 3 as 75% = 3/4
To find 10%, divide the amount by 10 as 10% = 1/10
To find 20%, divide the amount by 5 as 20% = 1/5
and so on.
If you have to find 15% of something, find 10% and 5% and add the answers together.
Example: A scarf is sold at a discount of 20%. If the original price is £15, what is the discount and how much does it cost after discount?
Discount is 20%
20% of £15 = £15 ÷ 5 = £3 so discount = £ 3
Price after discount = £15 - £3 = £12
Remember, this type of question involves two steps.
If the question asks you to find sale price or discounted price, you have to find the sale or discount amount and take it away from the original price.
Key: Always read the question properly, make sure what has been asked and answer accordingly.
- Finding the whole amount when you know the percentage.
Example. A scarf was sold at a discount of 20%. After discount, it costs £24. What was the original price or the price before discount?
Discount is 20%. So, price after discount would be 80% because price after discount = original price - discount
price after discount = 100% - 20% = 80%
So, according to the question, 80% = £24
you can work out 10%, 10% = £24 ÷ £8 = £3
original price which is 100% = £3 x 10 = £30
Hint: 90 degrees represent 1/4
Example: 60 children were asked what their favourite flavour of ice-cream was. 30 said vanilla, 10 said chocolate and mint, 5 said chocolate and 15 said toffee ice-cream. If you show the results in a pie-chart, what angle would represent number of children who liked a) vanilla, b) chocolate and mint, c) chocolate and d) toffee ice-cream?
Work out the fraction first.
Fraction of children who liked vanilla = 30/60 = 1/2
There are 360 degrees in a circle, so would be represented by 180 degrees.
Fraction of children who liked chocolate and mint = 10/60 = 1/6
1/6 of 360 degrees = 360 ÷ 6 = 60 degrees
Fraction of children who liked chocolate = 5/60 = 1/12
1/12 of 360 degrees = 360 ÷ 12 = 30 degrees
- Multiplying decimals by 10, 100, 1000 etc.
When you multiply, the numbers move left. Number of places depend on the zeros in the multiplying number.
So 2.5 x 10 = 25
When you multiply a number by 10, the number becomes 10 times bigger and 25 is 10 times bigger than 2.5
2.5 x 100 = 250
250 is 100 times bigger than 2.5
- Dividing decimals by 10, 100, 1000 etc.
When you divide, the numbers move right. Number of places depend on the number of zeros the divisor has
25 ÷ 10 = 2.5
2.5 ÷ 10 = 0.25
When you divide a number by 10, it becomes 10 times smaller. So, 2.5 is 10 times smaller than 25.
2.5 x 3.1 = 7.75
because 25 x 31 =775 and 2.5 is 10 times smaller than 25 and 3.1 is 10 times smaller than 31, so the answer will be 100 times smaller than 775 which is 7.75
25 x 31 = 775
2.5 x 3.1 = 7.75
Also, 25 x 0.31 = 7.75 because 25 has not changed and 31 has become 100 times smaller, so the answer will be 100 times smaller than 775 which is 100 times smaller than 775
The idea is that the number with which you are dividing can not be a decimal. So, try to make it a whole number by multiplying by 10, 100, 1000 etc.
Example: 25 ÷ 0.5
multiply 0.5 by 10 to make it 5
You have to multiply 25 by 10 also because you are multiplying 0.5 by 10
25 x 10 ÷ 0.5 x 10 = 250 ÷ 5 = 50
The answer is 50
