C1.12; E1.12 - Percentage amounts and comparisons

C1.12; E1.12 - Percentage amounts and comparisons

A price rises from 80to80 to 92, then falls back to 80.Thetwochangesareboth80. The two changes are both 12, but are their percentages equal? C1.12 and E1.12 use percentages to calculate amounts, compare quantities and describe change with a clearly identified starting amount.

A percentage of an amount

What does 18% of 250 mean? Imagine dividing 250 into 100 equal shares and taking 18 of them. Each share is 250÷100=2.5250\div100=2.5, so 18 shares give 45.

Percentage

A percentage means a number of parts per hundred. Thus p%=p/100p\%=p/100, and 100%100\% represents the whole reference quantity. Percentages can exceed 100%: 125%=1.25125\%=1.25, or one and a quarter of the whole.

To find a percentage of a quantity, multiply the quantity by the percentage written as a fraction or decimal. Familiar percentages also support mental methods: 50%=1/250\%=1/2, 25%=1/425\%=1/4 and 10%=1/1010\%=1/10.

Worked example 1: a percentage below one hundred

Question. Calculate 18% of 250.

Plan. Convert 18% into a decimal and multiply.

18%=18÷100=0.1818% of 250=0.18×250=45\begin{aligned} 18\%&=18\div100=0.18\\ 18\%\text{ of }250&=0.18\times250=45 \end{aligned}

Answer. 45.

Check. 20%20\% of 250 is 50 and 2%2\% is 5. Subtracting gives 50−5=4550-5=45.

The next percentage exceeds 100%, so the answer must exceed the reference quantity.

Worked example 2: more than the whole

Question. Calculate 125% of 64 kg64\,\mathrm{kg}.

Plan. Use the multiplier 125/100=1.25125/100=1.25.

125% of 64=1.25×64=80 kg\begin{aligned} 125\%\text{ of }64&=1.25\times64\\ &=80\,\mathrm{kg} \end{aligned}

Answer. 80 kg80\,\mathrm{kg}.

Check. The whole is 64 kg64\,\mathrm{kg}, and another quarter is 16 kg16\,\mathrm{kg}. Their sum is 80 kg80\,\mathrm{kg}.

The decimal multiplier reveals the size of the result: one below 1 takes part of a positive quantity, and one above 1 gives more than that quantity.

Watch out

18%18\% means 0.180.18, not 18. Also, “125% of” asks for the total represented by 125%; it does not ask you to add another 125% to the whole.

Estimate whether the next answer should be above or below the given mass before calculating.

One quantity as a percentage

When 42 of 60 seats are occupied, the reference whole is all 60 seats. The occupied fraction is 42/6042/60. Multiplying this fraction by 100 expresses the same comparison per hundred.

One quantity as a percentage of another

Percentage=quantity being comparedreference quantity×100%\text{Percentage}=\frac{\text{quantity being compared}}{\text{reference quantity}}\times100\%

The reference quantity must be non-zero, and both quantities must use compatible units.

Not in the supplied formula list: learn the relationship.

The wording “A as a percentage of B” puts B in the denominator. Reversing the order changes the comparison.

Worked example 3: identifying the reference whole

Question. A hall has 60 seats and 42 are occupied. Express the occupied seats as a percentage of all the seats.

Plan. Divide the occupied number by the total, then multiply by 100.

Occupied fraction=4260=0.7Occupied percentage=0.7×100%=70%\begin{aligned} \text{Occupied fraction}&=\frac{42}{60}=0.7\\ \text{Occupied percentage}&=0.7\times100\%=70\% \end{aligned}

Answer. 70%.

Check. 70%70\% of 60 is 0.7×60=420.7\times60=42.

The next question adds a unit conversion. Grams and kilograms measure the same kind of quantity, but the numbers cannot be divided fairly until the units agree.

Worked example 4: comparing different units

Question. Express 750 g750\,\mathrm{g} as a percentage of 1.2 kg1.2\,\mathrm{kg}.

Plan. Write both masses in grams and use 1.2 kg1.2\,\mathrm{kg} as the reference.

1.2 kg=1200 gPercentage=7501200×100%=62.5%\begin{aligned} 1.2\,\mathrm{kg}&=1200\,\mathrm{g}\\ \text{Percentage}&=\frac{750}{1200}\times100\%\\ &=62.5\% \end{aligned}

Answer. 62.5%.

Check. 0.625×1200=7500.625\times1200=750. Since 750 is between half and all of 1200, 62.5% is sensible.

The unit conversion does not alter either mass. It allows their quotient to describe a ratio of like quantities.

Watch out

750÷1.2×100750\div1.2\times100 mixes grams with kilograms. Convert first. A percentage is a comparison, so your answer has the percent sign, rather than grams or kilograms.

For lengths, 1 km=1000 m1\,\mathrm{km}=1000\,\mathrm{m}. Multiply a length in kilometres by 1000 to express it in metres, then choose the reference quantity from the question's wording.

Percentage change

A change compares the difference with the amount before the change. The starting amount supplies the reference whole, even when the final amount is given more prominence in the question.

Percentage increase or decrease

Percentage change in size=size of changeoriginal amount×100%\text{Percentage change in size}=\frac{\text{size of change}}{\text{original amount}}\times100\%

For an increase, the size of change is final minus original. For a decrease, it is original minus final. State which direction applies. This relationship needs a positive original amount in the contexts used here.

Not in the supplied formula list: learn the relationship.

For example, an increase of 12 means twelve extra units. An increase of 12% means twelve extra units for every hundred original units. They are different descriptions.

Worked example 5: finding a percentage increase

Question. A price increases from 80to80 to 92. Calculate the percentage increase.

Plan. Find the increase and compare it with the original $80.

Increase=92−80=12Percentage increase=1280×100%=15%\begin{aligned} \text{Increase}&=92-80=12\\ \text{Percentage increase}&=\frac{12}{80}\times100\%=15\% \end{aligned}

Answer. 15% increase.

Check. 15%15\% of 80is80 is 12, and 80+12=9280+12=92.

Now reverse the journey between the same two prices. The change is still $12, but the starting amount has changed.

Worked example 6: the same difference with another base

Question. A price decreases from 92to92 to 80. Calculate the percentage decrease, correct to 3 significant figures.

Plan. Divide the 12decreasebytheoriginal12 decrease by the original 92.

Decrease=92−80=12Percentage decrease=1292×100%=13.043478…%\begin{aligned} \text{Decrease}&=92-80=12\\ \text{Percentage decrease}&=\frac{12}{92}\times100\%\\ &=13.043478\ldots\% \end{aligned}

Answer. 13.0% decrease, to 3 significant figures.

Check. Using the unrounded percentage gives 92×(12/92)=1292\times(12/92)=12, exactly the required decrease. A 12changeisasmallerfractionof12 change is a smaller fraction of 92 than of $80.

These examples differ in the denominator because the direction of the journey changes which price is original. The difference alone cannot determine a percentage change.

Watch out

For 80risingto80 rising to 92, 12÷92×10012\div92\times100 uses the final price and answers a different question. Always label the original amount before dividing.

The opening changes are therefore 15% up and approximately 13.0% down. Equal cash differences need not be equal percentage changes.

Quick Recap

  • To find a percentage of an amount, multiply by its fraction of 100.
  • Percentages over 100% represent more than the reference whole.
  • “A as a percentage of B” uses B as the reference quantity.
  • Percentage change compares the difference with the original amount.

The 12increaseand12 increase and 12 decrease have different percentages because their starting prices differ.