Numbers – Fractions

Fractions

• . A quantity that tells us how many parts of a whole we have e.g   2/6 tells us we have 2 out of the 6 parts
• . Fractions consist of two numbers separated by a slash → top number being the numerator and bottom number being the denominator

Fractions to decimals:

1. Change the fraction to something equivalent out of 10, or a multiple of 10
2. Divide numerator by denominator

Example:

Q) Change 5/20 into a decimal

1. Multiply numerator and denominator by 5
2. Move decimal point to spaces to the left when dividing by 100

A) 0.25

Tips:

. If you can’t turn the denominator into a multiple of 10, use the short division method

 

Comparing Fractions:

1. Find a common denominator of both fractions 
2. Workout what you must multiply the denominator by to get the common denominator
3. Multiply your numerators with the answers from step 2
4. Compare only the numerators

Example: 

Q) Which fraction is larger, 4/7 or 3/5?

1. 35 is a common denominator 
2. 7 x 5 = 35 and 5 x 7 = 35 
3. 4 x 5 = 20 and 3 x 7 = 21
4. 21 > 20

A) 3/5 is larger

Multiplying Fractions:

1. Multiply the numerators together
2. Multiply the denominators together
3. Simplify your answer by cancelling down

Example:

Q) 5/8 x 6/7

1. Multiply the numerators 5 and 6
2. Multiply the denominators 8 and 7
3. 30/56 can be simplified by dividing by 2 = 15/28 → 15 and 28 share no common factors therefore this is the simplest form

A) 15/28

Dividing Fractions:

1. Keep the first fraction the same
2. Change the divide symbol into a multiply symbol
3. Flip the second fraction
4. Multiply the numerators by each other and the denominators by each other
5. Simplify your answer by cancelling down

Example:

Q) 2/5 ÷ 3/9

1. 2/5 remains the same
2. Multiply the fractions instead of dividing
3. 3/9 turns into 9/3
4. 2 x 9 = 18 and 5 x 3 = 15
5. 6 and 5 share no common factors so 6/5 is the simplest form

A) 6/5

Tips:

. After you change the sign and flip the second fraction, multiply the fractions as you normally would

 

Fraction of a quantity:

1. Divide the quantity by the denominator
2. Multiply answer from step 1 by numerator

Example:

Q) Find 3/5 of 65

1. 65 ÷ 5 = 13
2. 13 x 3 = 36

A) 36

Tips:

. ‘Of’ is another way of saying multiply

 

 Percentages – a quantity that tells us the proportion of a number out of 100

. e.g 9% = 9/100 = 0.09

Percentages to Fractions:

1. Put the percentage over 100 e.g 19% 19/100
2. Simplify the answer by cancelling down

Example:

Q) Change 24% into a fraction

1. 24% → 24/100
2. 24 and 100 share common factor 4, 6/25 is the simplest form

A) 6/25

Percentage of a quantity:

1. Put the percentage over 100 e.g 19% 19/100
2. Divide the quantity by the denominator
3. Multiply answer from step 1 by the numerator

Example:

Q) Find 45% of 50

1. 45% 45100
2. 50 ÷ 100 = 0.5
3. 0.5 x 45 = 22.5

A) 22.5

Percentage of a quantity:

1. Put the percentage over 100 e.g 19% 19/100
2. Divide the quantity by the denominator
3. Multiply answer from step 1 by the numerator

Tips:

. Another way to answer the question is find 10% → 100% ÷ 10 and 5% → 10% ÷ 2

. Multiply the 10% value by 4 → 10% x 4 = 40%

. Add the 40% value to the 5% value → 40% + 5% = 45%

 

Reverse Percentages:

1. Divide the percentage to a factor of 100
2. e.g 16% ÷ 4 = 4% → 4 is a factor of 100
3. Divide the answer by the same divisor
4. Find 100%

Example:

15% of an amount is 75.

Q) Workout the amount

1. 15% ÷ 3 = 5%, 5 is a factor of 100
2. 75 ÷ 3 = 25
3. 100% = 5% x 20 25 x 20 = 500

A) 500

Tips:

. A quicker method is to flip the fraction and multiply with the answer e.g 100/15 x 75

. 100 x 75 = 7500 → 7500 ÷ 15 = 500  

 

Mixed Numbers and Improper Fractions:

Mixed Numbers 

. A number written in the form of a whole number followed by a proper fraction

Improper Fractions 

. A fraction with a numerator bigger than the denominator

 

Mixed Number to Improper Fraction:

1. Multiply the whole number by the denominator
2. Add the answer from step 1 to the numerator
3. Put the answer from step 2 over the denominator
4. Simplify fraction by cancelling down

Example:

Q) Write 5 7/8 as a improper fraction

1. 5 x 8 = 40
2. 40 + 7 = 47 
3. 47/8
4. 47 and 8 share no common factors > 47/8 is the simplest form

A) 47/8

Tips:

. To write an improper fraction as a mixed number workout how many times the denominator goes into the numerator e.g 47 ÷ 8 = 5 remainder 7

. Put the remainder (7)  over the denominator (8) after the whole number (5) 5 7/8

 

Adding Mixed Numbers:

1. Add the fractions by making the denominators the same
2. Add the whole numbers
3. Simplify the fraction by cancelling down

Example:

Q) 5 1/2 + 3 3/10

1. 5/10+ 3/10= 8/10 –> 10 is a factor of 2 and 10
2. 5 + 3 = 8
3. 8/10= 4/5

A) 8 4/5

Tips:

. Add the fractions as you normally would make the denominators equal and add the numerators

 

Subtracting Mixed Numbers:

1. Make the denominators of the fractions the same and subtract them from each other
2. If the numerator from the first fraction is smaller than the second borrow 1 from the whole number
3. Subtract the whole numbers
4. Simplify the fraction 

Example:

Q) 6 1/5 – 4 3/7

1. 7/35 – 15/35 –> 7/35 is smaller than 15/35 so borrow 1 = 35/35
2. 7/35+ 35/35 = 42/35 –15/35 = 27/35
3. 5 – 4 = 1 because we borrowed 1 from 6 (6 – 1 = 5) subtract 4 from 5, not 6
4. 4. 27/35 is the simplest form

A) 1 27/35

Tips:

. The value for 1 = 35/35 because the fractions have the denominator 35

. Likewise if the fractions had a denominator of 15, the value of 1 would equal 15/15

. This is because any fraction with the same numerator and denominator equals 1

 

Multiplying Mixed Numbers:

1. Change the mixed numbers into improper fractions
2. Multiply the fractions
3. Simplify and change answer into mixed number

Example:

Q) 5 1/2 x 4 3/5

1. 5 1/2 = 11/2 and 4 3/5 = 23/5
2. 11/2 x 23/5 = 253/10
3. 253/10 is in its simplest form 253 ÷ 10 = 25 remainder 3

A) 25 3/10

Dividing Mixed Numbers:

1. Turn the mixed numbers into improper fractions
2. Keep the first fraction the same, change the division sign to multiply and flip the second fraction
3. Multiply the fractions
4. Simplify and turn answer into a mixed number

Example:

Q) 4 3/10 ÷  1 3/8

1. 4 3/10 = 43/10 and 1 3/8 = 11/8
2. 43/10 ÷ 11/8 → 43/10 x 8/11
3. 43/10 x 8/11= 344/110
4. 344/110 = 172/55 → 3 7/55

A) 3 7/55