C1.6; E1.6 - Adding and subtracting fractions

C1.6; E1.6 - Adding and subtracting fractions

Adding fractions means combining counts of equal-sized parts. You will use common denominators to add and subtract, simplify the result, and handle improper fractions, mixed numbers and negative values. The methods are the same for Core and Extended.

Adding fractions with common denominators

If you pour half a litre and a third of a litre into a jug, why is the total not 25\frac25 of a litre? The tempting calculation adds the numerators and denominators, but a half and a third are different-sized units. We will return to the jug once we can count matching units.

Begin with parts that already match. Three sevenths and two sevenths make five sevenths, just as three apples and two apples make five apples. The count changes; the unit stays “sevenths”.

You already know

In 37\frac37, the denominator 77 tells you the size of each equal part of a whole. The numerator 33 counts those parts. Fractions with the same positive denominator count the same-sized parts.

For addition with a common denominator, add the numerators and keep the denominator. For subtraction, subtract the numerators and keep the denominator. Simplify the result, converting to a mixed number if requested.

Worked example 1: combining sevenths

Question. Work out 37+27\frac37+\frac27.

Plan. Count the total number of sevenths.

37+27=3+27the part size stays the same=57add the counts\begin{aligned} \frac37+\frac27&=\frac{3+2}{7} && \text{the part size stays the same}\\ &=\frac57 && \text{add the counts} \end{aligned}

Answer. 57\frac57.

Check. Remove the second amount: 57−27=37\frac57-\frac27=\frac37, recovering the first.

Change the first numerator from 33 to 66. The method stays the same even when the total crosses one whole.

Worked example 2: a total greater than one

Question. Work out 67+27\frac67+\frac27. Give your answer as a mixed number.

Plan. Add the sevenths, then collect seven of them into a whole.

67+27=6+27count equal parts=87eight sevenths=117seven sevenths make one\begin{aligned} \frac67+\frac27&=\frac{6+2}{7} && \text{count equal parts}\\ &=\frac87 && \text{eight sevenths}\\ &=1\frac17 && \text{seven sevenths make one} \end{aligned}

Answer. 1171\frac17.

Check. The first fraction is one seventh below one; adding two sevenths must take you one seventh above one.

The same counting works for subtraction: 67−27=47\frac67-\frac27=\frac47. If both fractions are identical, the result is zero: 37−37=0\frac37-\frac37=0. A zero numerator is allowed; a zero denominator is never allowed.

Watch out

37+27=514\frac37+\frac27=\frac5{14} is wrong: adding does not cut every seventh in half. Keep the denominator 77, giving 57\frac57. The wrong result 514\frac5{14} is even smaller than the first positive amount.

Different denominators and subtraction

Different denominators need a common unit before you add or subtract. Replacing fractions by equivalent fractions changes how you count the amount, without changing the amount itself.

For the opening jug, divide each half into three sixths and each third into two sixths. You can now count the same-sized parts:

12+13=36+26=56.\frac12+\frac13=\frac36+\frac26=\frac56.

[DIAGRAM: asset_name: Halves and thirds become sixths; asset_slug: halves_thirds_sixths; caption: One half is three sixths and one third is two sixths. Together they make five sixths.; description: Three equal-length bars aligned vertically and divided into six equal parts. The first has three pale blue parts shaded and is labelled one half equals three sixths. The second has two sage parts shaded and is labelled one third equals two sixths. The third has three blue parts followed by two sage parts and one unshaded part, labelled total five sixths.]
One half is three sixths and one third is two sixths. Together they make five sixths.

Choose a common denominator that is a multiple of both original denominators. Their lowest common multiple (LCM) usually keeps the arithmetic shortest. Dividing a part into smaller equal pieces multiplies its denominator and numerator by the same factor; do this separately for each fraction.

Worked example 3: adding quarters and sixths

Question. Work out 34+16\frac34+\frac16.

Plan. Use twelfths, because 1212 is the LCM of 44 and 66.

34+16=912+212multiply by three and two=9+212count twelfths=1112already simplest\begin{aligned} \frac34+\frac16&=\frac9{12}+\frac2{12} && \text{multiply by three and two}\\ &=\frac{9+2}{12} && \text{count twelfths}\\ &=\frac{11}{12} && \text{already simplest} \end{aligned}

Answer. 1112\frac{11}{12}.

Check. Three quarters leaves a gap of 14=312\frac14=\frac3{12} to one. Adding 212\frac2{12} leaves 112\frac1{12}, so the total is 1112\frac{11}{12}.

Change only the operation to subtraction. Keep the same equivalent fractions, but subtract their counts.

Worked example 4: subtracting the same fractions

Question. Work out 34−16\frac34-\frac16.

Plan. Express both fractions in twelfths, then subtract.

34−16=912−212make the units match=9−212subtract the counts=712already simplest\begin{aligned} \frac34-\frac16&=\frac9{12}-\frac2{12} && \text{make the units match}\\ &=\frac{9-2}{12} && \text{subtract the counts}\\ &=\frac7{12} && \text{already simplest} \end{aligned}

Answer. 712\frac7{12}.

Check. Add the removed amount back: 712+212=912=34\frac7{12}+\frac2{12}=\frac9{12}=\frac34.

Using a larger common multiple is also valid. For instance, twenty-fourths would give 1824−424=1424=712\frac{18}{24}-\frac4{24}=\frac{14}{24}=\frac7{12}. The extra simplification makes it less efficient, but it is still the same calculation.

Watch out

34−16=212\frac34-\frac16=\frac2{12} subtracts unlike numerators after multiplying only the denominators. Every equivalent fraction needs both numbers changed: 34=912\frac34=\frac9{12}, not 312\frac3{12}.

Negative fractions and mixed numbers

Fractions use the signed-number rules you already know. Adding a negative fraction moves left on the number line. Subtracting a negative reverses that move: 14−(−12)=14+12=34\frac14-(-\frac12)=\frac14+\frac12=\frac34.

For a negative mixed number, the minus sign applies to its entire value: −213=−(2+13)=−73-2\frac13=-(2+\frac13)=-\frac73. It is not −2+13-2+\frac13, which lies between −2-2 and −1-1. Converting mixed numbers to improper fractions first gives one count of equal parts and keeps this sign clear.

Worked example 5: subtracting mixed numbers

Question. Work out 213−1342\frac13-1\frac34. Give your answer in its simplest form.

Plan. Convert each mixed number to an improper fraction, then use twelfths.

213−134=73−74count thirds and quarters=2812−2112use a common denominator=712subtract and simplify\begin{aligned} 2\frac13-1\frac34&=\frac73-\frac74 && \text{count thirds and quarters}\\ &=\frac{28}{12}-\frac{21}{12} && \text{use a common denominator}\\ &=\frac7{12} && \text{subtract and simplify} \end{aligned}

Answer. 712\frac7{12}.

Check. Add back 1341\frac34: 712+2112=2812=73=213\frac7{12}+\frac{21}{12}=\frac{28}{12}=\frac73=2\frac13.

Now make only the first mixed number negative. The common denominator is unchanged, but both contributions to the final value are negative.

Worked example 6: a negative mixed number

Question. Work out −213−134-2\frac13-1\frac34. Give your answer as a mixed number in its simplest form.

Plan. Keep the minus sign on the entire first fraction, then combine the negative twelfths.

−213−134=−73−74convert entire mixed numbers=−2812−2112make equal-sized parts=−4912forty-nine twelfths below zero=−4112minus applies to the whole value\begin{aligned} -2\frac13-1\frac34&=-\frac73-\frac74 && \text{convert entire mixed numbers}\\ &=-\frac{28}{12}-\frac{21}{12} && \text{make equal-sized parts}\\ &=-\frac{49}{12} && \text{forty-nine twelfths below zero}\\ &=-4\frac1{12} && \text{minus applies to the whole value} \end{aligned}

Answer. −4112-4\frac1{12}.

Check. Adding back 74=2112\frac74=\frac{21}{12} gives −2812=−213-\frac{28}{12}=-2\frac13. Also, subtracting a positive amount must move the answer below −213-2\frac13.

You can also obtain a negative result by subtracting a larger positive fraction from a smaller one. For example, 25−45=−25\frac25-\frac45=-\frac25. The denominator remains positive; put the negative sign in front of the fraction to make the direction easy to read.

Watch out

−213=−53-2\frac13=-\frac53 is wrong because it treats the expression as −2+13-2+\frac13. Count the full magnitude first: two wholes and one third make seven thirds, so the negative mixed number is −73-\frac73.

Quick Recap

  • Add or subtract counts of equal-sized parts: keep the common denominator.
  • With different denominators, write equivalent fractions with a common denominator before combining the numerators.
  • Convert mixed numbers to improper fractions when that keeps the calculation clear. A minus before a mixed number applies to its whole value.
  • Simplify at the end, give the requested form, and check subtraction by adding the removed amount back.

The opening jug contains 56\frac56 of a litre, because a half is three sixths and a third is two sixths. Adding the denominators would change the units instead of combining the amounts.