2.01b - Adding and subtracting fractions

2.01b - Adding and subtracting fractions

How much is two thirds of a metre plus one quarter of a metre? Learn to count equal-sized fractional units, then use the same idea with subtraction, negative fractions and mixed numbers.

Common denominators

Suppose two ribbon pieces measure 23\frac23 metre and 14\frac14 metre. What is their combined length? Adding 22 and 11 does not yet help, because thirds and quarters are different-sized pieces.

With a common denominator, the pieces have the same size: 27+37=57\frac27+\frac37=\frac57. We add the counts, but keep the unit, sevenths. Subtraction works the same way: 57−27=37\frac57-\frac27=\frac37.

For different denominators, use equivalent fractions first. The lowest common multiple of the denominators gives the lowest common denominator. For thirds and quarters it is 1212, so 23=812\frac23=\frac8{12} and 14=312\frac14=\frac3{12}. Any common multiple works, but the lowest usually keeps the arithmetic smaller.

Worked example 1: adding unlike fractional units

Question. Work out 23+14\frac23+\frac14. Give the answer in its simplest form.

Plan. Convert both fractions to twelfths, then add the numerators.

23+14=812+312equivalent fractions=1112count the twelfths\begin{aligned} \frac23+\frac14&=\frac8{12}+\frac3{12} && \text{equivalent fractions} \\ &=\frac{11}{12} && \text{count the twelfths} \end{aligned}

Answer. 1112\frac{11}{12}; the ribbons total 1112\frac{11}{12} metre.

Check. 23\frac23 is one twelfth less than 34\frac34, so adding 14\frac14 must give one twelfth less than 11. That is 1112\frac{11}{12}.

Change only addition to subtraction. The fractional units still need to match.

Worked example 2: subtracting unlike fractional units

Question. Work out 23−14\frac23-\frac14. Give the answer in its simplest form.

Plan. Use the same denominator of twelve, then subtract the counts.

23−14=812−312both measured in twelfths=512subtract the numerators\begin{aligned} \frac23-\frac14&=\frac8{12}-\frac3{12} && \text{both measured in twelfths} \\ &=\frac5{12} && \text{subtract the numerators} \end{aligned}

Answer. 512\frac5{12}.

Check. Adding back 14=312\frac14=\frac3{12} gives 812=23\frac8{12}=\frac23, the starting amount.

We can answer the opening question exactly without rounding either fraction: their sum is 1112\frac{11}{12} metre.

Watch out

23+14≠37\frac23+\frac14\ne\frac37. Adding denominators changes the unit instead of counting equal pieces. Convert first, then add numerators while keeping the common denominator.

Find a common unit for this subtraction.

Signed fraction sums

A negative fraction is a negative number, so its sign follows the integer rules you already know. Rewriting −34-\frac34 as −912-\frac9{12} changes the size and number of parts, but keeps the negative sign.

Once all denominators match, calculate with the signed numerators. A minus in front of a fraction applies to its whole value. Subtracting a negative fraction adds its opposite: 14−(−12)=14+12=34\frac14-(-\frac12)=\frac14+\frac12=\frac34.

Worked example 3: adding across zero

Question. Work out −34+56-\frac34+\frac56.

Plan. Use twelfths, then add the signed numerators.

−34+56=−912+1012common denominator=−9+1012combine signed counts=112one positive twelfth remains\begin{aligned} -\frac34+\frac56&=-\frac9{12}+\frac{10}{12} && \text{common denominator} \\ &=\frac{-9+10}{12} && \text{combine signed counts} \\ &=\frac1{12} && \text{one positive twelfth remains} \end{aligned}

Answer. 112\frac1{12}.

Check. The positive amount 56\frac56 exceeds 34\frac34 by one twelfth, so a small positive answer makes sense.

Change only the operation to subtraction. You are now moving further in the negative direction.

Worked example 4: subtracting from a negative fraction

Question. Work out −34−56-\frac34-\frac56.

Plan. Convert to twelfths and subtract ten twelfths from negative nine twelfths.

−34−56=−912−1012equivalent fractions=−9−1012keep both signs=−1912nineteen twelfths below zero\begin{aligned} -\frac34-\frac56&=-\frac9{12}-\frac{10}{12} && \text{equivalent fractions} \\ &=\frac{-9-10}{12} && \text{keep both signs} \\ &=-\frac{19}{12} && \text{nineteen twelfths below zero} \end{aligned}

Answer. −1912-\frac{19}{12}, or −1712-1\frac7{12}.

Check. Adding back 56\frac56 gives −19+1012=−912=−34\frac{-19+10}{12}=-\frac9{12}=-\frac34.

For three or more terms, choose a denominator common to all of them and retain the original signs. For instance,

12−23+34=612−812+912=712\frac12-\frac23+\frac34 =\frac6{12}-\frac8{12}+\frac9{12} =\frac7{12}

You can combine the signed numerators as 6−8+96-8+9, working from left to right. Subtraction does not mean changing every later sign.

Watch out

−34−56-\frac34-\frac56 is not positive. Two negative contributions add to a negative total. A positive change would arise in −34−(−56)-\frac34-(-\frac56), because subtracting a negative adds its opposite.

Use common units and preserve the subtraction sign.

Calculating with mixed numbers

Mixed numbers combine whole units and fractional units. Converting them to improper fractions puts everything into one kind of unit, which is particularly useful when a subtraction would need borrowing.

For example, 213=732\frac13=\frac73, because two wholes contain six thirds and there is one more third. After conversion, use the same common-denominator method. You may leave an exact answer as a simplified improper fraction or convert it back to a mixed number, unless the question specifies one form.

Worked example 5: adding mixed numbers

Question. Work out 213+1342\frac13+1\frac34.

Plan. Convert both mixed numbers to improper fractions, then use twelfths.

213+134=73+74count thirds and quarters=2812+2112common denominator=4912add numerators=411248 twelfths make four wholes\begin{aligned} 2\frac13+1\frac34&=\frac73+\frac74 && \text{count thirds and quarters} \\ &=\frac{28}{12}+\frac{21}{12} && \text{common denominator} \\ &=\frac{49}{12} && \text{add numerators} \\ &=4\frac1{12} && \text{48 twelfths make four wholes} \end{aligned}

Answer. 41124\frac1{12}.

Check. Add wholes and fractions separately: 2+1=32+1=3, while 13+34=1312=1112\frac13+\frac34=\frac{13}{12}=1\frac1{12}. Together they give 41124\frac1{12}.

Change only addition to subtraction. Improper fractions avoid having to borrow from the whole number.

Worked example 6: subtracting mixed numbers

Question. Work out 213−1342\frac13-1\frac34.

Plan. Convert first, then subtract in the written order.

213−134=73−74improper fractions=2812−2112common denominator=712subtract the counts\begin{aligned} 2\frac13-1\frac34&=\frac73-\frac74 && \text{improper fractions} \\ &=\frac{28}{12}-\frac{21}{12} && \text{common denominator} \\ &=\frac7{12} && \text{subtract the counts} \end{aligned}

Answer. 712\frac7{12}.

Check. Add back 1341\frac34: 712+2112=2812=213\frac7{12}+\frac{21}{12}=\frac{28}{12}=2\frac13.

For a negative mixed number, convert the entire amount: −112=−32-1\frac12=-\frac32. Then, for example, −112+213=−96+146=56-1\frac12+2\frac13=-\frac96+\frac{14}{6}=\frac56. The sign rules and equal-unit reasoning work together; no new arithmetic rule is needed.

Watch out

Do not read 2132\frac13 as 2×132\times\frac13. It means 2+132+\frac13. Also, subtracting just the whole numbers and taking a positive difference between the fractional parts can reverse part of a subtraction.

The first amount is smaller in this final subtraction, so expect a negative answer.

Quick Recap

  • Add or subtract counts only after the fractional units match.
  • Keep the common denominator and simplify the result.
  • Negative fractions follow the same sign rules as integers.
  • Convert mixed numbers to improper fractions when that makes the calculation clearer.

The opening ribbons total 1112\frac{11}{12} metre: using twelfths lets us count both lengths in the same unit.