E1.13 Percentages

Learning objectives

E1.13.1Calculate a given percentage of a quantity.• Calculate a given percentage of a quantity.E1.13.2Express one quantity as a percentage of another.• Express one quantity as a percentage of another.E1.13.3Calculate percentage increase or decrease.• Calculate percentage increase or decrease.E1.13.4Calculate with simple and compound interest.• Calculate with simple and compound interest.E1.13.5Calculate using reverse percentages• Calculate using reverse percentages. Problems may include repeated percentage change. Formulas are not given. e.g. find the cost price given the selling price and the percentage profit. Percentage calculations may include: • deposit • discount • profit and loss (as an amount or a percentage) • earnings • percentages over 100%.

A percentage is a number of hundredths. To calculate $p$ percent of a quantity $Q$ , convert $p\%$ to the multiplier $p/100$ and multiply.

$p\%\text{ of }Q=\frac{p}{100}\times Q$

To find 37% of 640 dollars, calculate $0.37\times640=236.8$ , so the percentage amount is 236.80 dollars. For 135% of 80, $1.35\times80=108$ ; percentages above 100% are valid.

Percentage	Efficient route	Example
10%	divide by 10	10% of 470 is 47
1%	divide by 100	1% of 470 is 4.7
5%	half of 10%	5% of 470 is 23.5
25%	divide by 4	25% of 360 is 90

The percentage amount is not automatically the final value. A tax or increase amount is added to the original; a discount or decrease amount is subtracted. Keep money to cents only when reporting the final monetary result.

To express one quantity as a percentage of another, divide the quantity being described by the reference whole. The wording after 'of' usually identifies that whole.

$\text{percentage}=\frac{\text{part}}{\text{reference whole}}\times100\%$

Identify the part and whole, convert them to matching units, divide in that order, multiply by 100 and round only as requested.

In a box of 96 items, 18 are damaged. The damaged percentage is $18\div96\times100=18.75\%$ . To compare 1.92 with 1.60, $1.92\div1.60\times100=120\%$ .

A percentage has no physical unit, but the compared quantities must use the same unit. For example, 450 g as a percentage of 2 kg is $450\div2000\times100=22.5\%$ .

Do not reverse the fraction. A result above 100% simply means the part exceeds the chosen reference whole; it is not an error.

Percentage change compares the change with the original value. The original value is the denominator whether the result is an increase, decrease, profit or loss.

$\text{percentage change}=\frac{|\text{new}-\text{original}|}{\text{original}}\times100\%$

Required result	Multiplier or comparison
increase by $p$ %	multiply by $1+p/100$
decrease by $p$ %	multiply by $1-p/100$
percentage profit	profit $\div$ cost price $\times100$
percentage loss	loss $\div$ cost price $\times100$

A price rises from 72 dollars to 90 dollars. The increase is 18 dollars, so $18\div72\times100=25\%$ . Reducing 560 dollars by 18% uses multiplier $0.82$ , giving $560\times0.82=459.2$ dollars.

For repeated changes, multiply the factors rather than adding the percentages. A 12% rise followed by a 5% fall gives factor $1.12\times0.95=1.064$ , an overall 6.4% increase.

Equal percentage increases and decreases do not cancel because the second change uses a different base. Keep the change amount, change percentage and final value distinct.

Simple interest is calculated from the original principal every period, so it adds a constant amount. Compound interest is calculated from the current balance, so each period multiplies the balance and previous interest also earns interest.

Type	Formula	Pattern
simple	$I=P(r/100)n$ , then $A=P+I$	equal interest added each period
compound	$A=P(1+r/100)^n$	balance multiplied each period

$P$ is principal, $r$ is the percentage rate per period, $n$ is the number of matching periods, $I$ is total interest and $A$ is final amount. These formulas are not supplied, and monthly or daily rates require months or days in the exponent.

For a principal of 2400 dollars at 3% simple interest for 5 years, $I=2400\times0.03\times5=360$ and $A=2760$ . At 3% compound interest, $A=2400(1.03)^5=2782.58\ldots$ .

For compound interest with known $P$ , $A$ and $n$ , the period multiplier is $(A/P)^{1/n}$ , so $r=100[(A/P)^{1/n}-1]$ . For a minimum number of complete periods, test integer powers until the threshold is first reached.

Do not use $P(r/100)n$ for compound interest. Distinguish total interest $A-P$ from the final balance $A$ , and keep full precision until the requested final rounding.

A reverse-percentage problem gives a final amount and asks for the original. Write the forward multiplier first, then divide the final amount by it.

$\text{original}=\frac{\text{final}}{\text{forward multiplier}}$

Final amount described as	Forward multiplier	Reverse calculation
after a $p$ % increase or profit	$1+p/100$	final $\div(1+p/100)$
after a $p$ % decrease or discount	$1-p/100$	final $\div(1-p/100)$
$p$ % of the original	$p/100$	final $\div(p/100)$

A sale price of 44.80 dollars follows a 20% discount, so it is 80% of the original: $44.80\div0.80=56$ . A selling price of 270 dollars after a 35% profit gives cost price $270\div1.35=200$ dollars.

Reverse repeated changes by dividing by their product. If a value rises 10% and then falls 20% to 440, the original is $440\div(1.10\times0.80)=500$ .

Do not subtract the stated percentage from the final amount: that percentage was taken from the unknown original. Tax included, discount, profit and capacity questions all use the same multiplier logic.

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—E1.14 Using a calculatorRead

E1.13 Percentages

Learning objectives

Find a percentage amount with a multiplier

Choose the reference whole before finding a percentage

Use the original value to measure percentage change

Distinguish simple interest from compound growth

Reverse a percentage change by dividing by its multiplier

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