Percentages, ratios and proportional reasoning

A percentage expresses a value out of 100. Before calculating a percentage change, identify the original value. The original amount is the base for both a percentage increase and a percentage decrease.

Worked example: a price changes from 800 naira to 1000 naira. The increase is 200 naira. Percentage increase = 200/800 x 100 = 25%. Dividing by the new price would answer a different question and give the wrong percentage increase.

For a discount, calculate the reduction first and subtract it from the marked price. A 15% discount on 2000 naira is 300 naira, so the amount paid is 1700 naira. Check that a discounted price is below the original price.

A ratio divides a total into relative shares. To share 600 in the ratio 2:3, add the parts to get five. One part is 600/5 = 120. The shares are 240 and 360. They add back to 600, providing a useful check.

Direct proportion has the form y = kx. If y = 18 when x = 3, then k = 6. When x = 5, y = 30. This method is appropriate only when the stated relationship is direct proportion; do not assume every pair of quantities behaves this way.

Practice:
1. Find 12% of 250.
2. A value rises from 400 to 500. Find the percentage increase.
3. Share 840 in the ratio 3:4.
4. If y is directly proportional to x and y = 20 when x = 4, find y when x = 7.

Answers: 1 30; 2 25%; 3 360 and 480; 4 35.

Before selecting an option, inspect its size and units. A larger ratio part must receive the larger share. A reduction of less than 100% cannot produce a negative price. Record mistakes involving the wrong base separately from arithmetic slips so you know which idea to revise.