FN2, FN8, HN2, HN8 - Adding and subtracting fractions

FN2, FN8, HN2, HN8 - Adding and subtracting fractions

Learn to add and subtract fractions by counting equal-sized parts of the same whole. You will calculate exactly with proper fractions, improper fractions and mixed numbers, then apply the same ideas to negative values. This develops the fraction arithmetic in FN2 and HN2 and the exact calculation in FN8 and HN8.

Common denominators

If you have 23\frac23 of a metre of ribbon and add another 14\frac14 of a metre, do you have 37\frac37 of a metre? Adding ribbon cannot leave you with less than the amount you started with. The problem is that thirds and quarters are different-sized units.

You can add two thirds and one third immediately: together they make three thirds. The denominator stays 33 because the pieces are still thirds. To combine thirds with quarters, first express both lengths using the same-sized pieces.

Common denominator

A common denominator is a denominator shared by equivalent forms of the fractions in a calculation. For thirds and quarters, twelfths work because both 33 and 44 divide 1212.

Each third contains four twelfths, and each quarter contains three twelfths. Therefore 23=812\frac23=\frac8{12} and 14=312\frac14=\frac3{12}. The whole metre has not changed; only its subdivisions have changed.

[DIAGRAM: asset_name: Adding thirds and quarters; asset_slug: thirds-quarters-twelfths; caption: Two thirds and one quarter become eight twelfths and three twelfths.; description: Three equal-length bars show two thirds shaded blue, one quarter shaded sage, and their sum as eight blue twelfths followed by three sage twelfths. The combined bar has eleven of its twelve equal cells shaded.]
Two thirds and one quarter become eight twelfths and three twelfths.

The lowest common multiple gives the lowest common denominator, which usually keeps the numbers small. Any common multiple works, provided you scale each numerator with its denominator. Once the units match, add or subtract the numerators and simplify the result.

Worked example 1: adding different units

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

Plan. Rewrite both fractions in twelfths, then count the twelfths.

23+14=2×43×4+1×34×3make equivalent fractions=812+312both now count twelfths=1112add the numbers of pieces\begin{aligned} \frac23+\frac14 &=\frac{2\times4}{3\times4}+\frac{1\times3}{4\times3} &&\text{make equivalent fractions}\\ &=\frac8{12}+\frac3{12} &&\text{both now count twelfths}\\ &=\frac{11}{12} &&\text{add the numbers of pieces} \end{aligned}

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

Check. Two thirds needs another third to reach 11. Adding only a quarter must leave the total below 11, but above 23\frac23. Eleven twelfths fits both limits.

Change addition to subtraction. The equivalent fractions stay the same; the action on the numerators changes.

Worked example 2: subtracting the same units

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

Plan. Use twelfths again, then remove three of the eight twelfths.

23−14=812−312use equal-sized pieces=512subtract the numerators\begin{aligned} \frac23-\frac14&=\frac8{12}-\frac3{12} &&\text{use equal-sized pieces}\\ &=\frac5{12} &&\text{subtract the numerators} \end{aligned}

Answer. 512\frac5{12}.

Check. Add the removed amount back: 512+14=512+312=812=23\frac5{12}+\frac14=\frac5{12}+\frac3{12}=\frac8{12}=\frac23.

The common denominator names the unit; it is not another quantity to add.

Watch out

23+14≠37\frac23+\frac14\ne\frac37. Adding the denominators changes the size of the pieces. Rewrite as 812+312\frac8{12}+\frac3{12}, then add only the numerators.

Choose a common denominator for a new subtraction.

Mixed-number calculations

A mixed number combines whole units and a fraction of another unit. Converting it to an improper fraction lets you use the same method throughout a calculation.

For example, 213=732\frac13=\frac73: the two wholes contain six thirds, with one third more. Keep the denominator when converting. To turn an improper answer back into a mixed number, divide the numerator by the denominator and use the remainder as the fractional numerator.

You already know

213=2+13=2×3+13=732\frac13=2+\frac13=\frac{2\times3+1}{3}=\frac73. Multiply the whole-number part by the denominator, then add the numerator. A fraction written in its simplest form has no common factor greater than 11 in numerator and denominator.

With addition, a fractional total can cross a whole-number boundary. An improper fraction handles that crossing automatically.

Worked example 3: a total greater than a whole

Question. Work out 213+1342\frac13+1\frac34. Give your answer as a mixed number in its simplest form.

Plan. Convert both mixed numbers to improper fractions, use twelfths, then convert the answer back.

213+134=73+74count all the thirds and quarters=2812+2112use denominator 12=4912combine the numerators=4112since 49=4×12+1\begin{aligned} 2\frac13+1\frac34&=\frac73+\frac74 &&\text{count all the thirds and quarters}\\ &=\frac{28}{12}+\frac{21}{12} &&\text{use denominator }12\\ &=\frac{49}{12} &&\text{combine the numerators}\\ &=4\frac1{12} &&\text{since }49=4\times12+1 \end{aligned}

Answer. 41124\frac1{12}.

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

Now subtract the same mixed numbers. Here the fractional part of the first number is smaller, so subtracting the whole and fractional parts separately would require regrouping.

Worked example 4: subtracting across a whole

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

Plan. Use improper fractions so that no separate borrowing step is needed.

213−134=73−74convert both mixed numbers=2812−2112use a common denominator=712subtract the numerators\begin{aligned} 2\frac13-1\frac34&=\frac73-\frac74 &&\text{convert both mixed numbers}\\ &=\frac{28}{12}-\frac{21}{12} &&\text{use a common denominator}\\ &=\frac7{12} &&\text{subtract the numerators} \end{aligned}

Answer. 712\frac7{12}.

Check. Regroup 2132\frac13 as 116121\frac{16}{12}. Subtracting 19121\frac9{12} leaves 712\frac7{12}, agreeing with the improper-fraction method.

An answer can be a proper fraction even when both starting numbers are mixed numbers. Always follow the requested answer form.

Watch out

Do not subtract the smaller fractional part from the larger just to avoid a negative result. In 213−1342\frac13-1\frac34, the fraction calculation is 13−34\frac13-\frac34, in that order. Use improper fractions or regroup one whole correctly.

Try another subtraction that crosses a whole-number boundary.

Signed fractions

Signs describe direction as well as size, so use the signed-number rules before combining fraction units.

Adding a negative number moves left on a number line. Subtracting a negative reverses that move, so it moves right. Thus +(−14)+(-\frac14) means subtract a quarter, while −(−14)-(-\frac14) means add a quarter. Brackets keep the operation separate from the number's sign.

A minus sign in front of a fraction applies to the whole fraction: −23=−23-\frac23=\frac{-2}{3}. Keep denominators positive when rewriting. Similarly, −112-1\frac12 means −(1+12)=−32-\left(1+\frac12\right)=-\frac32, not −1+12-1+\frac12.

Worked example 5: adding a negative fraction

Question. Work out −23+(−14)-\frac23+\left(-\frac14\right). Give your answer in its simplest form.

Plan. Treat adding a negative as subtraction, then combine the signed numerators over 1212.

−23+(−14)=−23−14adding a negative moves left=−8−312write both amounts in twelfths=−1112combine signed numerators\begin{aligned} -\frac23+\left(-\frac14\right)&=-\frac23-\frac14 &&\text{adding a negative moves left}\\ &=\frac{-8-3}{12} &&\text{write both amounts in twelfths}\\ &=-\frac{11}{12} &&\text{combine signed numerators} \end{aligned}

Answer. −1112-\frac{11}{12}.

Check. The result must lie to the left of −23-\frac23. Its distance from zero is 1112\frac{11}{12}, which is greater than 812\frac8{12}, as required.

Change only the operation before the negative quarter. Subtracting it now moves right instead.

Worked example 6: subtracting a negative fraction

Question. Work out −23−(−14)-\frac23-\left(-\frac14\right). Give your answer in its simplest form.

Plan. Replace subtraction of a negative by addition, then use twelfths.

−23−(−14)=−23+14reverse the negative move=−8+312use equal-sized units=−512the negative amount is larger\begin{aligned} -\frac23-\left(-\frac14\right)&=-\frac23+\frac14 &&\text{reverse the negative move}\\ &=\frac{-8+3}{12} &&\text{use equal-sized units}\\ &=-\frac5{12} &&\text{the negative amount is larger} \end{aligned}

Answer. −512-\frac5{12}.

Check. Reverse the subtraction: −512+(−14)=−812=−23-\frac5{12}+\left(-\frac14\right)=-\frac8{12}=-\frac23, the starting value.

Opposite fractions cancel exactly: −37+37=0-\frac37+\frac37=0. A zero numerator gives zero when the denominator is nonzero. For a nonzero answer, simplify its magnitude and retain its sign.

Watch out

−23−(−14)-\frac23-(-\frac14) does not become 23+14\frac23+\frac14. Only the subtraction of the second negative changes to addition; the first fraction stays negative.

The ribbon question is now settled: equal-sized twelfths give 1112\frac{11}{12} of a metre. Negative values use the same common units; their signs tell you which way to combine them.

Quick Recap

  • Add or subtract equal-sized fraction units by using a common denominator.
  • Convert mixed numbers to improper fractions, or regroup wholes carefully.
  • Apply signed-number rules without changing unrelated signs.
  • Simplify the exact answer and check its direction and size.