2.01c, 3.03a - Fractions of quantities

2.01c, 3.03a - Fractions of quantities

Can taking a fraction of an amount give more than you started with? Use 2.01c to calculate fractional amounts and compare two quantities, then use 3.03a to keep fraction calculations exact without a calculator.

A fraction of an amount

What would 74\frac74 of £24 mean, and could it exceed £24? Start with a smaller request: three quarters of £24. Split the £24 into four equal shares of £6, then take three shares. The result is £18.

In a calculation, “of” tells you to multiply: 34\frac34 of 2424 is 34×24\frac34\times24. Dividing by the denominator finds one part; multiplying by the numerator counts how many of those parts you need. These two steps may be done in either order, but dividing first often keeps whole-number arithmetic smaller.

A fraction of a quantity

ab of Q=ab×Q=(Q÷b)×a\frac ab\text{ of }Q=\frac ab\times Q=(Q\div b)\times a

Here b≠0b\ne0. Keep the units of QQ in the answer. For an ordinary shared amount, use a positive denominator.

Not on the 2027 formula sheets: learn it.

A unit fraction, such as 14\frac14, has numerator 11. It names one equal part. Three quarters means three of those quarter-sized parts, rather than three successive divisions by four.

Worked example 1: three quarters of an amount

Question. Work out 34\frac34 of £24.

Plan. Find one quarter, then take three quarters.

24÷4=6one quarter, in pounds6×3=18three equal shares\begin{aligned} 24\div4&=6 && \text{one quarter, in pounds} \\ 6\times3&=18 && \text{three equal shares} \end{aligned}

Answer. £18.

Check. The unused quarter is £6; 18+6=2418+6=24, so the four quarters rebuild the full amount.

Change only the denominator to eight. The number of shares selected stays three, but each share is smaller.

Worked example 2: three eighths of the same amount

Question. Work out 38\frac38 of £24.

Plan. Divide into eight equal shares and take three.

24÷8=3one eighth, in pounds3×3=9three eighths\begin{aligned} 24\div8&=3 && \text{one eighth, in pounds} \\ 3\times3&=9 && \text{three eighths} \end{aligned}

Answer. £9.

Check. Eighths are half the size of quarters, so three eighths of £24 must be half of £18. Half of £18 is £9.

For a positive amount, a positive proper fraction gives a smaller amount. A fraction of zero is zero.

Watch out

For 38\frac38 of £24, dividing by 33 and multiplying by 88 gives £64, which reverses the fraction. The denominator chooses the size of one share; the numerator counts shares.

Use smaller shares with a new total.

More than one whole

A numerator larger than the denominator asks for more shares than one whole contains. Five quarters of £24 means five shares of £6, even though the original £24 contains only four such shares.

[DIAGRAM: asset_name: Seven quarters of twenty-four pounds; asset_slug: seven-quarters-amount; caption: Each quarter of £24 is £6, so seven quarters total £42.; description: Two equal-length bars each divided into four equal cells. The first four cells are shaded blue and labelled one whole £24. Three cells in the second bar are shaded blue and the remaining cell is neutral. Each selected cell is labelled £6, and a caption states seven selected quarters total £42.]
Each quarter of £24 is £6, so seven quarters total £42.

This is multiplication by a scale factor above one. You may imagine extra copies of the original whole; there is no rule that a fraction must mean a piece smaller than one.

Worked example 3: five quarters

Question. Work out 54\frac54 of £24.

Plan. Find the value of one quarter, then count five quarters.

24÷4=6one quarter6×5=30five quarters\begin{aligned} 24\div4&=6 && \text{one quarter} \\ 6\times5&=30 && \text{five quarters} \end{aligned}

Answer. £30.

Check. 54=114\frac54=1\frac14, so take one whole £24 and another quarter £6: 24+6=3024+6=30.

Increase only the numerator from five to seven. The quarter-sized share remains £6.

Worked example 4: seven quarters

Question. Work out 74\frac74 of £24.

Plan. Use seven of the same quarter-sized shares.

24÷4=6one quarter6×7=42seven quarters\begin{aligned} 24\div4&=6 && \text{one quarter} \\ 6\times7&=42 && \text{seven quarters} \end{aligned}

Answer. £42.

Check. 74=134\frac74=1\frac34. One whole is £24 and three quarters are £18, so 24+18=4224+18=42.

The opening question is answered: taking a fraction can increase an amount when that fraction exceeds one. The method has not changed; only the number of shares has.

Watch out

Do not automatically reject an answer larger than the original quantity. For a positive quantity, 74\frac74 must produce a larger result because 74>1\frac74>1. Multiplication by 11 leaves it unchanged.

Keep counting equal shares beyond the first whole.

One quantity as a fraction of another

“What fraction of 50 is 30?” reverses the comparison: instead of finding an amount, you know both amounts and want their relationship. Put the amount being described above the comparison amount: 3050=35\frac{30}{50}=\frac35.

One quantity as a fraction of another

fraction=quantity being describedcomparison quantity\text{fraction}=\frac{\text{quantity being described}}{\text{comparison quantity}}

Use the same units for both quantities, and the comparison quantity must be nonzero. The units then cancel, leaving a fraction with no unit.

Not on the 2027 formula sheets: learn it.

Order matters. Thirty as a fraction of fifty is 35\frac35; fifty as a fraction of thirty is 53\frac53. A comparison fraction can exceed one if the quantity being described is larger.

Worked example 5: quantities in the same units

Question. Express 30 cm30\,\mathrm{cm} as a fraction of 50 cm50\,\mathrm{cm}.

Plan. Put thirty over fifty, then simplify.

fraction=3050same units, requested order=35divide both by 10\begin{aligned} \text{fraction}&=\frac{30}{50} && \text{same units, requested order} \\ &=\frac35 && \text{divide both by 10} \end{aligned}

Answer. 35\frac35.

Check. 35\frac35 of 50 cm50\,\mathrm{cm} is 50÷5×3=30 cm50\div5\times3=30\,\mathrm{cm}, the described amount.

Change only the unit of the comparison quantity from centimetres to metres. The written number remains fifty, but the actual comparison length is much greater.

Worked example 6: quantities in different units

Question. Express 30 cm30\,\mathrm{cm} as a fraction of 50 m50\,\mathrm{m}.

Plan. Use 1 m=100 cm1\,\mathrm{m}=100\,\mathrm{cm}, then compare in centimetres.

50 m=5000 cmmultiply by 100fraction=305000same units before dividing=3500cancel the common factor 10\begin{aligned} 50\,\mathrm{m}&=5000\,\mathrm{cm} && \text{multiply by 100} \\ \text{fraction}&=\frac{30}{5000} && \text{same units before dividing} \\ &=\frac3{500} && \text{cancel the common factor 10} \end{aligned}

Answer. 3500\frac3{500}.

Check. 5000÷500×3=305000\div500\times3=30, so 3500\frac3{500} of the comparison length gives 30 cm30\,\mathrm{cm}.

Writing the units on the conversion line helps you avoid comparing a count of centimetres with a count of metres.

Watch out

3050\frac{30}{50} does not compare 30 cm30\,\mathrm{cm} with 50 m50\,\mathrm{m}: its numbers use different units. Convert first, then form and simplify the fraction.

Use centimetres for both quantities.

Keeping exact fractional answers

Not every quantity divides into a whole number of units. Sharing 5 m5\,\mathrm{m} equally among three people gives each 53 m\frac53\,\mathrm{m}, or 123 m1\frac23\,\mathrm{m}. This is an exact value: it has not been rounded.

Fractions keep division exact throughout a calculation. A finite decimal may also be exact when it represents the fraction precisely, but rounding an ongoing decimal changes the value. Three lengths each rounded to 1.67 m1.67\,\mathrm{m} total 5.01 m5.01\,\mathrm{m}; three exact lengths of 53 m\frac53\,\mathrm{m} total exactly 5 m5\,\mathrm{m}. Retaining the fraction prevents that rounding drift.

Worked example 7: an exact share

Question. Find exactly 13\frac13 of 5 m5\,\mathrm{m}.

Plan. Multiply by the fraction and keep the division as a fraction.

13×5=1×53write 5 as 51=53exact division=123optional mixed number\begin{aligned} \frac13\times5&=\frac{1\times5}{3} && \text{write 5 as }\frac51 \\ &=\frac53 && \text{exact division} \\ &=1\frac23 && \text{optional mixed number} \end{aligned}

Answer. 53 m\frac53\,\mathrm{m}, or 123 m1\frac23\,\mathrm{m}.

Check. 53×3=5\frac53\times3=5, so three exact shares reconstruct the original length.

Change only the numerator from one to two. The unit share is unchanged, and two of those shares are needed.

Worked example 8: two exact shares

Question. Find exactly 23\frac23 of 5 m5\,\mathrm{m}.

Plan. Find one third, retain it as a fraction, then double it.

5÷3=53one exact share53×2=103two shares=313optional mixed number\begin{aligned} 5\div3&=\frac53 && \text{one exact share} \\ \frac53\times2&=\frac{10}{3} && \text{two shares} \\ &=3\frac13 && \text{optional mixed number} \end{aligned}

Answer. 103 m\frac{10}{3}\,\mathrm{m}, or 313 m3\frac13\,\mathrm{m}.

Check. The remaining third is 53 m\frac53\,\mathrm{m}; 103+53=153=5\frac{10}{3}+\frac53=\frac{15}{3}=5, so the pieces sum to the whole.

Exact fractional arithmetic also applies to fractional starting quantities. For example, 23\frac23 of 54 kg\frac54\,\mathrm{kg} is 23×54=56 kg\frac23\times\frac54=\frac56\,\mathrm{kg}. Cancel factors and simplify before considering a decimal; if an exact answer is requested, an approximation does not replace it.

Watch out

Do not replace 53\frac53 by 1.671.67 in an exact calculation. They are close, but unequal. A simplified fraction or an equivalent mixed number preserves the full value.

Keep both the starting amount and the result fractional.

Quick Recap

  • Divide by the denominator and multiply by the numerator to find a fraction of an amount.
  • Fractions greater than one scale a positive amount upwards.
  • To express one quantity as a fraction of another, use consistent units and the requested order.
  • Keep fractions through exact calculations, simplifying without rounding.

Seven quarters of £24 is £42: seven £6 shares, or one whole plus another three quarters.