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Fractions This section is designed to help you understand "fractions" and to lead you through a number of calculations involving fractions. Hopefully much of what follows is familiar to you.
A fraction is a part or subdivision of the whole and is written in the form x/y where x is the numerator and y is the denominator. In proper fractions the numerator is smaller than the denominator whereas in improper fractions the reverse is true. Hence, 1/2, 1/4 and 2/5 are all proper fractions whereas 5/3, 9/2 and 7/3 are all improper fractions. In proper fractions the value represented is always less than 1 but for improper fractions the value represented is always greater than 1.
Conversion of fractions We can convert fractions into other fractions by multiplication or division. For example, if we wanted to express 1/5 in terms of "tenths" we simply need to multiply the 5 by 2 to give us "10" as the denominator and also multiply the numerator by the same amount. Hence;
1/5 = 2/10 = 5/25 = 10/50 and so on.
We can complete the reverse procedure by division, hence;
25/75 = 5/15 = 1/3 (In this case we have divided both numerator and denominator by 5 each time).
These fractions are called equivalent fractions.
Check-point #1
Express the following fractions as indicated;
1/6 as x / 18
To convert 6 to 18 we must multiply by 3 so we multiply both numerator and denominator by 3,
1/6 = 3/18
You try:
1/3 as x/9
1/5 as x/30
1/9 as x/36
Answers
It is normal practice to express fraction as their "lowest forms", in other words with the smallest denominator whilst leaving the numerator as a whole number. So we would express 3/9 as 1/3, 36/48 as 3/4 and 12/ 24 as 1/2.
Addition/Subtraction of Fractions
To add or subtract fractions we must first convert them so they have a common denominator.
So to add 1/5 and 1/15 together we convert the 5 to 15;
(1/5 + 1/15) = (3/15 +1/15) = 4/15
Subtract 1/9 from 1/3
(1/3 – 1/9) = (3/9 – 1/9) = 2/9
Multiplying and Dividing Fractions
To multiply fractions we simply multiply the numerators and denominators as required.
Hence 1/5 x 2/6 = 2/30 or 1/15
(= 1 x 2 / 5 x 6)
But what happens if we have mixed numbers?
How can we multiply 1 ½ x 2 ¾ ?
For this calculation we must first convert the numbers into improper fractions;
1 ½ = 3/2
2 ¾ = 11/4
So 1 ½ x 2 ¾ = 3/2 x 11/4 = 33/8
We can now cancel down this improper fraction to give 4 1/8
Check-point #2
Complete the following calculations;
2/3 x 4/5
= (2x4)/(3x5) = 8/15
You try:
2/3 x 7/15
5/8 x 4/5
1 1/3 x 3 1/6
Answers
Division of fractions is little more involved. To divide fractions we invert (turn upside down) the fraction we wish to divide by and change the division into a multiplication.
An example is probably the best way to explain this.
So let’s do the division 1/5 ¸ 2/9
First invert 2/9 to give 9/2
Change the division sign to multiplication
1/5 x 9/2 = 9/10 and that’s our answer!
This might seem a little odd but if we take a more obvious example of how this might work. Suppose we have 3 ½ cakes and want to cut them up into quarters, the calculation would be;
3 ½ ¸ ¼
Following the rules above we invert the ¼ to give us 4/1 and change the division sign to a multiplication sign;
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