Adding fractions needs a common denominator, because you can only combine pieces of the same size. Multiplying fractions needs none: multiply the tops together and the bottoms together.
How do you add fractions?
Make the denominators equal, add the numerators and keep the denominator unchanged.
For 2/3 + 3/4 the lowest common denominator is 12. Convert: 2/3 = 8/12 and 3/4 = 9/12. Add the tops: 8 + 9 = 17, so the sum is 17/12, which is 1 5/12.
A sense check helps. 2/3 is a little over half and 3/4 is over half, so the sum should be more than 1. The answer 1 5/12 is.
How do you multiply fractions?
Multiply numerator by numerator and denominator by denominator, then simplify.
For 3/5 × 10/9: tops 3 × 10 = 30, bottoms 5 × 9 = 45, so 30/45. Divide both by 15 to get 2/3. A shortcut is to cancel first: 10 and 5 share the factor 5, and 3 and 9 share the factor 3, giving 1/1 × 2/3 = 2/3.
Worked example in context
An original situation: a recipe needs 2/3 of a cup of sugar for the cake and 3/4 of a cup for the icing. How much sugar in all?
That is 2/3 + 3/4 = 17/12 = 1 5/12 cups. Now suppose you make 3/5 of the recipe. The sugar needed is 3/5 × 17/12 = 51/60, which simplifies to 17/20 of a cup. Check: 51 and 60 both divide by 3.
What does the common mistake look like?
A student adds 2/3 + 3/4 as 5/7, adding both the tops and the bottoms. The answer 5/7 is about 0.71, which is smaller than 3/4 (0.75) even though 2/3 was added to it. A sum cannot be smaller than one of its parts.
The repair is to think of the denominator as the name of the piece size. Thirds and quarters are different sizes, so they cannot be combined until both are renamed as twelfths.
Steps to follow
- To add: find a common denominator, convert both fractions, add the tops only.
- To multiply: multiply tops together and bottoms together, cancelling common factors first if you can.
- Simplify, then check that the answer is sensible in size.
Self-check
Work out 1/2 + 2/5 and 4/9 × 3/8.
Answer
1/2 + 2/5: common denominator 10, so 5/10 + 4/10 = 9/10. Sense check: half plus a bit less than half is just under 1.
4/9 × 3/8: tops 4 × 3 = 12, bottoms 9 × 8 = 72, so 12/72 = 1/6. Cancelling first, 4 and 8 share 4, and 3 and 9 share 3, giving 1/3 × 1/2 = 1/6.
Where to go next
If finding the common denominator slows you down, practise equivalent fractions and denominators. The next skill is converting between fractions, decimals and percentages. See the fractions hub for the full order.