What Is a Percentage?
Percent means per hundred. The sign x percent stands for x divided by 100.
Basic Rules
The p percent of a number a equals a times p divided by 100. To find what percent x is of y, divide x by y and multiply by 100. Percent increase or decrease is the change divided by the original value, times 100. If p percent of a number equals k, the number is k times 100 divided by p.
Useful fractions to memorise: 50 percent is one half, 25 percent is one quarter, 20 percent is one fifth, 12.5 percent is one eighth, 10 percent is one tenth, 33 and one third percent is one third, and 16 and two thirds percent is one sixth.
Successive Changes
For two successive changes of a percent and b percent, the net change is a plus b plus a times b divided by 100. Treat a decrease as negative. A 20 percent rise followed by a 20 percent fall gives a net 4 percent fall. If both the length and breadth of a rectangle rise by 10 percent, the area rises by 21 percent.
When a price rises by x percent, consumption must fall by x divided by 100 plus x, times 100 percent, to keep spending the same.
Profit and Loss
Profit is selling price minus cost price when the selling price is higher. Loss is cost price minus selling price when the cost price is higher. Percent profit or loss is always calculated on the cost price. Selling price equals cost price times 100 plus or minus p, divided by 100. Cost price equals selling price times 100 divided by 100 plus or minus p.
Discount
Selling price equals the marked price times 100 minus d, divided by 100. For two successive discounts d1 and d2, the net discount is d1 plus d2 minus d1 times d2 divided by 100.
Shortcuts
If two articles are sold at the same price, one at p percent profit and the other at p percent loss, there is always a loss of p squared over 100 percent. Use multipliers: a 20 percent rise is multiplication by 1.2 and a 15 percent fall is multiplication by 0.85. Percentages can be swapped: 8 percent of 25 equals 25 percent of 8.
Worked Examples
Example 1: What is 12 percent of 450?
Solution: 450 times 12 divided by 100 equals 54.
Example 2: Cost price is 250 taka. Find the selling price at 20 percent profit.
Solution: Selling price equals 250 times 120 divided by 100, which is 300 taka.
Example 3: Two successive discounts of 20 percent and 10 percent equal what single discount?
Solution: 20 plus 10 minus 20 times 10 divided by 100 equals 28 percent.
Quick Check
Question 1: On what quantity is profit percent calculated?
Answer: The cost price.
Question 2: What is the net effect of a 20 percent rise followed by a 20 percent fall?
Answer: A 4 percent fall.
Question 3: If the price rises 25 percent, by how much must consumption fall to keep expense equal?
Answer: 20 percent.