Fractions
Adding mixed numbers
Split the job in two: add the whole numbers, add the fractions, then put them back together. The whole-number total is also your estimate.
The three-step method
STEP 1 — Add the WHOLE-NUMBER parts. Use this as your estimate. STEP 2 — Add the FRACTIONAL parts. Simplify, and if the result is an improper fraction, write it as a mixed number. STEP 3 — Add the answers from steps 1 and 2 together.
When the denominators already match
1 3/7 + 5 2/7 Step 1: 1 + 5 = 6. So the answer is just over 6. Step 2: 3/7 + 2/7 = 5/7. That is a proper fraction already in its simplest form. Step 3: 6 + 5/7 = 6 5/7
When they do not match
3 3/4 + 1 2/3 Step 1: 3 + 1 = 4. Both fractions are fairly big, so estimate around 5. Step 2: 3/4 + 2/3 needs a common denominator of 12: 3/4 = 9/12 and 2/3 = 8/12, so 9/12 + 8/12 = 17/12 That is improper, so write it as 1 5/12. Step 3: 4 + 1 5/12 = 5 5/12 The estimate of 'around 5' confirms the answer is sensible.
Why estimate at all?
The whole-number total from step 1 tells you roughly what to expect. If your final answer is nowhere near it, you know something has gone wrong — usually a forgotten carry from an improper fraction.
Worked examples
1 3/7 + 5 2/7
- Step 1 — whole numbers: 1 + 5 = 6.
- Step 2 — fractions: 3/7 + 2/7 = 5/7.
- Step 3 — put them together.
Answer: 6 5/7
3 3/4 + 1 2/3
- Step 1: 3 + 1 = 4, so expect about 5.
- Step 2: 3/4 = 9/12 and 2/3 = 8/12, so 17/12 = 1 5/12.
- Step 3: 4 + 1 5/12.
Answer: 5 5/12
Why the estimate matters
- The whole-number total tells you roughly what to expect.
- If your answer is nowhere near it, something is wrong.
Answer: Usually a forgotten carry from an improper fraction
Write 17/12 as a mixed number
- 17 ÷ 12 = 1 remainder 5.
Answer: 1 5/12
2 1/5 + 1 2/5
- Whole numbers: 2 + 1 = 3.
- Fractions: 1/5 + 2/5 = 3/5.
Answer: 3 3/5
Finding a common denominator for 3/4 and 2/3
- The lowest common multiple of 4 and 3 is 12.
Answer: Twelfths
Where you see this in real life
- Adding 1 3/4 hours and 2 1/2 hours of work: 3 hours plus 1 1/4 hours gives 4 1/4 hours in total.
- Measuring 2 2/3 m and 1 3/4 m of fabric — the estimate of 'about 4 1/2 m' tells you straight away whether your answer is plausible.
Key words
- Mixed number
- a whole number and a fraction together
- Improper fraction
- a fraction whose top is bigger than its bottom
- Proper fraction
- a fraction whose top is smaller than its bottom
- Equivalent fractions
- fractions of equal value written differently
- Simplest form
- a fraction cancelled as far as it will go
- Common denominator
- the same bottom number for both fractions
- Carry
- moving a whole number across from an improper fraction