1.6 Percentages

Learning objectives

1.6A Meaning of percentageUnderstand that a percentage is a number of parts per 100.1.6B Express a given number as a percentage of another numberexpress a given number as a percentage of another number 1.6C Express a percentage as a fraction and as a decimalexpress a percentage as a fraction and as a decimal 1.6D Percentages as operatorsunderstand the multiplicative nature of percentages as operators 1.6E Simple percentage problemssolve simple percentage problems, including percentage increase and decrease 1.6F Reverse percentagesuse reverse percentages In a sale, prices were reduced by 30%. The sale price of an item was £17.50 Calculate the original price of the item 1.6G Compound interest and depreciationuse compound interest and depreciation 1.6HA Repeated percentage changeuse repeated percentage change Calculate the total percentage increase when an increase of 30% is followed by a decrease of 20%1.6HB Compound interest problemssolve compound interest problems

A percentage tells how many equal parts out of 100 are being considered. The symbol $\%$ means ‘per 100’, so $37\%=37/100$ .

Percentage	Per-100 meaning
$8\%$	8 parts in every 100
$100\%$	the whole amount
$125\%$	one whole and 25 extra parts per 100

The actual whole need not contain 100 objects. If 25% of 60 students travel by bus, the same proportion is $25/100=1/4$ , so 15 students travel by bus.

Percentages put different-sized groups on the same per-100 scale, which makes proportions comparable.

A percentage can exceed 100% and can be below 1%. It is a proportion, not automatically an amount.

To express an amount as a percentage of a reference amount, divide by the reference amount and multiply by 100%.

Question	Calculation
31 500 as a percentage of 42 000	$\frac{31500}{42000}\times100\%=75\%$
18 as a percentage of 24	$\frac{18}{24}\times100\%=75\%$

The words ‘of another number’ identify the denominator. Ask: percentage of which whole or reference value?

If the first number is smaller than the reference, the answer is below 100%; if it is larger, the answer is above 100%.

Reversing the fraction changes the comparison. ‘ $a$ as a percentage of $b$ ’ uses $a/b$ , not $b/a$ .

Because percent means per 100, divide the percentage number by 100. This gives both a fraction and, after division, a decimal.

Percentage	Fraction	Decimal
$23\%$	$23/100$	$0.23$
$45\%$	$45/100=9/20$	$0.45$
$2.5\%$	$2.5/100=1/40$	$0.025$
$125\%$	$125/100=5/4$	$1.25$

Dividing by 100 moves the decimal point two places left; multiplying a decimal by 100 converts it back to a percentage.

Write the percentage over 100, remove any decimal in the numerator if needed, then simplify the fraction fully.

$0.6\%=0.006$ , not $0.6$ . The percent sign already includes division by 100.

A percentage of an amount is multiplication by its decimal or fractional equivalent: $p\%$ of $Q$ is $(p/100)Q$ .

Task	Operator	Result
$45\%$ of 800	$0.45\times800$	360
$12.5\%$ of 64	$0.125\times64$	8
$150\%$ of 40	$1.5\times40$	60

Use whichever equivalent operator is easiest: $25\%=0.25=1/4$ and $10\%=0.1$ .

Successive percentage operators multiply. This multiplicative idea underpins percentage change, interest and depreciation.

‘15% of 120’ means $0.15\times120$ , not $120-15$ and not $120\div15$ .

Increase by $p\%$ using multiplier $1+p/100$ ; decrease by $p\%$ using multiplier $1-p/100$ .

Change	Multiplier	Example from 240
increase 15%	$1.15$	$240\times1.15=276$
decrease 15%	$0.85$	$240\times0.85=204$

A percentage change may also be found from $\frac{\text{new}-\text{original}}{\text{original}}\times100\%$ . Use the original value as the denominator.

In a word problem, calculate each required percentage amount, keep units, and round only when the context or question requires it.

Adding 15% means adding 15% of the original amount, not adding the number 15. An increase and an equal percentage decrease do not cancel.

A final amount after a percentage change equals the original amount multiplied by a change multiplier. Reverse the change by dividing by that multiplier.

Information	Equation	Original
sale price £17.50 after 30% off	$0.70x=17.50$	$x=17.50/0.70=£25$
price 9.45 after 8% rise	$1.08x=9.45$	$x=9.45/1.08=8.75$

Identify what percentage the final value represents: after 17% off it is 83%; after a 12% rise it is 112%.

Apply the stated change to the recovered original to check that it returns the given final value.

Do not undo a 30% decrease by increasing the final value by 30%. Divide by $0.70$ because the final value has a different base.

Compound change applies each period to the current value, so the multiplier is applied repeatedly.

Situation	Value after $n$ periods
compound interest at $r\%$	$P(1+r/100)^n$
depreciation at $r\%$	$P(1-r/100)^n$

6000 dirham at 1.5% compound interest for four years becomes $6000(1.015)^4=6368.18\ldots$ ; the interest earned is $6368.18\ldots-6000=368.18\ldots$ .

Keep full calculator precision through the powers, then round the final money value as instructed.

Compound interest is not $P+nrP/100$ ; that adds the same simple-interest amount every period and ignores growth on earlier interest.

Represent each percentage change by a multiplier and multiply the multipliers in time order.

Sequence	Combined multiplier	Overall change
increase 30%, then decrease 20%	$1.30\times0.80=1.04$	4% increase
depreciate 15% for two years	$0.85^2=0.7225$	27.75% decrease

After finding the combined multiplier $m$ , the total percentage change is $(m-1)\times100\%$ ; a negative result indicates a decrease.

If the final value is known, divide by the product of all change multipliers to recover the starting value.

Do not add signed percentage changes. A 30% rise followed by a 20% fall acts on different base values.

Model a compound-interest account with $A=P(1+r/100)^n$ , where $P$ is principal, $r$ is the annual percentage rate, $n$ is the number of compounding periods and $A$ is the final amount.

Unknown	Rearrangement
final amount	$A=P(1+r/100)^n$
principal	$P=A/(1+r/100)^n$
rate	$r=100[(A/P)^{1/n}-1]$

If 6000 grows to 6311.16 after two years at 1.5% and a third year at rate $r$ , then $6000(1.015)^2(1+r/100)=6311.16$ , giving $r=2.1\%$ .

Match the exponent to the number of compounding periods. When rates change, use a separate multiplier for each rate interval.

Interest earned is $A-P$ , whereas the account balance is $A$ . Read which quantity the question asks for.

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—1.7 Ratio and proportionRead

1.6 Percentages

Learning objectives

Understand percentage as parts per 100

Express one number as a percentage of another

Convert percentages to fractions and decimals

Use percentages as multiplicative operators

Solve percentage increase and decrease problems

Use reverse percentages

Calculate compound interest and depreciation

Combine repeated percentage changes

Solve compound interest problems

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