# Working with fractions

People don’t like fractions. I don’t know why. They’re difficult to begin with, I know, but a few simple rules will help you add, subtract, multiply and divide.

Adding and subtracting are usually the easiest sums, but not when it comes to fractions. If fractions have the same denominator (the number on the bottom), then you can simply add or subtract the second numerator from the first, eg 4/5 – 3/5 = 1/5. If not, it would be like adding apples and oranges. They’re just not the same, so you first have to convert them into ‘pieces of fruit’ – or a common unit. The easiest way of doing that is by multiplying the denominators together. That guarantees that the new denominator is a multiple of both the others. Once you’ve found the right denominator, you can multiply each numerator by the denominator from the other fraction (because whatever you do to the bottom of the fraction you have to do to the top), add or subtract them and then simplify and/or convert into a mixed number if necessary, eg 2/3 + 4/5 = (2 x 5 + 4 x 3) / (3 x 5) = (10 + 12) / 15 = 22/15 = 1 7/15.

1. Multiply the denominators together and write the answer down as the new denominator
2. Multiply the numerator of the first fraction by the denominator of the second and write the answer above the new denominator
3. Multiply the numerator of the second fraction by the denominator of the first and write the answer above the new denominator (after a plus or minus sign)
4. Add or subtract the numerators and write the answer over the new denominator
5. Simplify and/or turn into a mixed number if necessary

Sample questions

1. 1/5 + 2/3
2. 2/7 + 3/5
3. 4/5 – 2/3
4. 7/8 – 3/4
5. 5/8 – 2/3

### Multiplication

This is the easiest thing to do with fractions. You simply have to multiply the numerators together, multiply the denominators together and then put one over the other, simplifying and/or converting into a mixed number if necessary, eg 2/3 x 4/5 = (2 x 4) / (3 x 5) = 8/15.

1. Multiply the numerators together
2. Multiply the denominators together
3. Put the result of Step 1 over the result of Step 2 in a fraction
4. Simplify and/or turn into a mixed number if necessary

Sample questions

1. 1/5 x 2/3
2. 2/7 x 3/5
3. 4/5 x 2/3
4. 7/8 x 3/4
5. 5/8 x 2/3

### Division

Dividing by a fraction must have seemed like a nightmare to early mathematicians, because nobody ever does it! That’s right. Nobody divides by a fraction, because it’s so much easier to multiply. That’s because dividing by a fraction is the same as multiplying by the same fraction once it’s turned upside down, eg 2/3 ÷ 4/5 = 2/3 x 5/4 = (2 x 5) / (3 x 4) = 10/12 = 5/6. You can even cut out the middle step and simply multiply each numerator by the denominator from the other fraction, eg 2/3 ÷ 4/5 = (2 x 5) / (3 x 4) = 10/12 = 5/6.

1. Multiply the numerator of the first fraction by the denominator of the second
2. Multiply the numerator of the second fraction by the denominator of the first
3. Put the result of Step 1 over the result of Step 2 in a fraction
4. Simplify and/or turn into a mixed number if necessary

Sample questions

1. 1/5 ÷ 2/3
2. 2/7 ÷ 3/5
3. 4/5 ÷ 2/3
4. 7/8 ÷ 3/4
5. 5/8 ÷ 2/3

### Simplifying fractions

By the way, to simplify a fraction, try dividing the denominator by the numerator first, eg 9/18 = 1/2. If that works, you don’t have to do anything else. If not, try dividing by the first few prime numbers, ie 2, 3, 5, 7 and 11. You don’t need to try the other numbers, because they’re all multiples of the primes, so they won’t work if the others don’t, eg 4 won’t work if 2 doesn’t work. Ideally, the quickest way would be to divide the numerator and denominator by the highest common factor (or HCF), but you don’t know what that is at the beginning, so it would take time to work it out. This way is a good compromise.

1. If possible, divide the numerator and denominator by the numerator
2. If the numerator doesn’t go exactly, start dividing by the smallest prime number that will go into both numbers, starting with 2, 3, 5, 7 and 11
3. Repeat Step 2 until the only number that goes into the numerator and denominator is 1

Sample questions

1. Simplify 14/28
2. Simplify 8/24
3. Simplify 4/12
4. Simplify 27/36
5. Simplify 30/50

### Turning improper fractions into mixed numbers

To turn an improper fraction into a mixed number, simply divide the numerator by the denominator to find the whole number and then put the remainder over the original denominator and simplify if necessary, eg 9/6 = 1 3/6 = 1 1/2.

1. Divide the numerator by the denominator
2. Write down the answer to Step 1 as a whole number
3. Put any remainder into a new fraction as the numerator, using the original denominator
4. Simplify the fraction if necessary

Sample questions

1. What is 22/7 as a mixed number?
2. What is 16/5 as a mixed number?
3. What is 8/3 as a mixed number?
4. What is 18/8 as a mixed number?
5. What is 13/6 as a mixed number?

### Turning mixed numbers into improper fractions

To turn a mixed number into an improper fraction, multiply the whole number by the denominator of the fraction and add the existing numerator to get the new numerator while keeping the same denominator, eg 2 2/5 = (10 + 2)/5 = 12/5.

1. Multiply the whole number by the denominator of the fraction
2. Add the answer to the existing numerator to get the new numerator
3. Write the answer over the original numerator
4. Simplify if necessary

Sample questions

1. What is 2 2/7 as an improper fraction?
2. What is 3 2/3 as an improper fraction?
3. What is 4 1/4 as an improper fraction?
4. What is 5 1/5 as an improper fraction?
5. What is 3 2/9 as an improper fraction?

There you go. Easy peasy lemon squeezy!