Percentages, profit and loss, and ratios

Most errors in percentage and profit questions come from dividing by the wrong number. If you always ask "percentage of what?", you will avoid the majority of traps in this topic.

Percentage change

Percentage change = (new - old) / old x 100. The base is always the original value.

Price 800 -> 920
Change = 120
120 / 800 x 100 = 15%

Successive changes do not simply add. A 10% rise followed by a 10% fall gives 100 -> 110 -> 99, a net 1% fall. For two changes of a% and b%, the net change is a + b + (a x b)/100, with falls written as negative numbers.

Profit, loss and discount

• Profit or loss % is on cost price: profit / CP x 100.
• Discount % is on marked price: discount / MP x 100.
• Selling price = marked price - discount.

Example: CP = 500, SP = 600. Profit = 100, profit % = 100 / 500 x 100 = 20%. Dividing by 600 gives 16.67%, which is the most common wrong option.

A typical harder question: "The cost of 12 items equals the selling price of 10." Assume each item costs 1. Ten items sell for 12, so one sells for 1.2. Profit = 0.2 on a cost of 1, which is 20%.

Ratios and sharing

To share an amount in a ratio, add the parts, find one part, then multiply.

Share 1,800 in ratio 4 : 5
Total parts = 9
One part    = 1,800 / 9 = 200
A = 800, B = 1,000

Check your answer by adding the shares. They must equal the total.

Fast percentage values

1. 10% is the number divided by 10; 5% is half of that.
2. 1% is the number divided by 100; build 15% as 10% + 5%.
3. Useful fractions: 25% = 1/4, 20% = 1/5, 12.5% = 1/8, 33.33% = 1/3.

Habit to build: before calculating, write the base value next to the question, such as "on CP" or "on old price". Then estimate the answer roughly, so you can reject options that are clearly too large or too small.