Percentages.
Choose the original whole, build the multiplier and reverse the change when needed.
Find and compare percentages · calculate changes, profit, discount and deposit · use simple, compound and reverse percentages
Percent means out of 100
Percent means “out of one hundred”. If 25% of a class walks to school, we are describing the share 25100 of that class, whatever its actual size.
To find a percentage of an amount, first understand 1% as one hundredth of the whole. You can multiply that by the required percentage, or use the equivalent decimal multiplier. Always identify the whole: a percentage is a comparison with a base amount.
Choose the whole before calculating. The same amount can be a different percentage when the comparison quantity changes.
Find 18% of $250, and express $45 as a percentage of $250.
18% of 250 = 0.18 × 250 = $45.
45250 × 100% = 18%. These calculations reverse one another.
A deposit is 15% of a $640 purchase. Find the deposit and the balance.
Deposit = 0.15 × 640 = $96. Balance = 640 − 96 = $544.
Equivalently, the balance is 85% of the price: 0.85 × 640 = 544.
Pause and explainWhat is 10% of $80, and why?
$8. Ten percent is one tenth, so divide 80 by 10.
Change the multiplier, then multiply
An increase of 20% keeps the original 100% and adds another 20%, giving 120% of the starting value. A decrease of 20% leaves 80%. This is where the multipliers 1.20 and 0.80 come from.
When calculating a percentage change, compare the change with the original amount, not the final amount. An increase followed by the same percentage decrease does not generally return to the starting value, because the second change uses a different base.
An increase of p% uses 1 + p100. A decrease of p% uses 1 − p100.
20% increase: the new value is 120% of the original.
A $75 jacket is discounted by 12%. Find the sale price.
The discount is $9; the amount paid is $66. Read which quantity is requested.
A price rises from $80 to $92. Find the percentage increase.
Percentage change uses the original value as the denominator. A fall from 92 back to 80 would instead be 1292 × 100% ≈ 13.0%.
Pause and explainWhy does increasing $100 by 10% and then decreasing by 10% give $99?
The increase gives $110. Ten percent of $110 is $11, so the decrease leaves $99.
Profit, loss and reverse percentages
A reverse-percentage question starts with the result of a change and asks for the amount before it. You are undoing a multiplication, so the reverse operation is division by the original multiplier.
For example, a selling price after a 25% profit is 125% of cost price. Label cost price as 100% before writing the equation. This prevents the common error of subtracting 25% of the selling price, which would use the wrong base.
Profit = selling price − cost price. Loss = cost price − selling price. Unless the question states a different basis, percentage profit or loss is measured against the cost price.
An item costs $80 and sells for $100. Find the profit percentage.
The $20 profit is 25% of cost. Dividing by the selling price would answer a different question.
After a 20% discount, a bag costs $72. Find its original price.
The sale price is 80% of the original, so 0.80 × original = 72.
Check forward: 90 × 0.80 = 72. Adding 20% to 72 gives the wrong original value.
A trader sells for $156 at a profit of 30%. Find the cost price.
The selling price represents 130% of cost.
Pause and explainA discounted price is 80% of the original. How do you recover the original?
Divide the discounted price by 0.8. The discount operation multiplied the original by 0.8.
Simple and compound interest
Interest is money added for using or saving money over time. Simple interest calculates each period’s interest from the original principal. Compound interest calculates each new period’s interest from the balance then present.
Follow the balance one year at a time before using a power. At 5% growth, each year multiplies the current balance by 1.05. Repeating that multiplier produces the compound formula. For simple interest, the same fixed interest amount is added each year instead.
Simple: same amount each year
P is principal, r is the annual percentage rate, t is time in years. Total amount = P + I. Each year’s interest uses the original principal.
Compound: same multiplier each period
n is the number of compounding periods, and r must be the percentage rate per period. Interest earned = A − P.
Compare $1000 at 5% annually for 3 years.
Simple: I = 1000 × 5 × 3 ÷ 100 = $150. Final amount = $1150.
Compound: A = 1000 × 1.05³ = $1157.625, or $1157.63 to the nearest cent. Interest = $157.63.
Pause and explainWhy does compound interest grow by larger amounts over equal years at a fixed positive rate?
Later interest is calculated on a larger balance, including earlier interest, so the amount added becomes larger.
Your turn. Identify the original whole.
For every percentage calculation, write what represents 100%. Decide whether you are finding a share, changing an amount, comparing a change, or undoing a change.
In repeated-change questions, show the multiplier for every stage. Keep the calculator’s full precision until the final money rounding, and check whether a recovered original should be larger or smaller than the given final amount.
Find 7.5% of 480 and express 54 as a percentage of 45.
Reveal worked answer
A $240 item is reduced by 15%, then the reduced price is increased by 10%. Find the final price.
Reveal worked answer
An item sells for $68 at a loss of 15%. Find its cost price.
Reveal worked answer
Find the simple interest on $750 at 4% per year for 212 years.
Reveal worked answer
Find the final amount and interest on $800 compounded annually at 3% for 2 years.