CBSE Class 10 Mathematics Percentage Notes Set 09

Percentage

Percent means per hundred. x% means x/100, so 25% = 1/4 and 12 1/2% = 1/8. To find x% of a quantity, multiply the quantity by x/100. To express a as a percentage of b, calculate (a/b) × 100.
Useful equivalents: 50% = 1/2, 33 1/3% = 1/3, 25% = 1/4, 20% = 1/5, 16 2/3% = 1/6, 12 1/2% = 1/8, 10% = 1/10.

Percentage Change

Percentage increase = (increase / original value) × 100 and percentage decrease = (decrease / original value) × 100; the base is always the original value.
If a value rises by r%, the new value is original × (1 + r/100); if it falls by r%, it is original × (1 − r/100).
If a price rises by r%, the consumption must fall by r/(100 + r) × 100 % to keep the expenditure constant. If a price falls by r%, consumption can rise by r/(100 − r) × 100 %.

Successive Changes

Successive percentage changes multiply: increases of a% and b% together give a factor (1 + a/100)(1 + b/100), a net change of a + b + ab/100 %.
Area of a rectangle changes by the product of the factors of the length and the breadth. A side increased by r% increases the area of a square by (2r + r²/100)%.
Population growth and depreciation: after n years the value is \( P(1 \pm r/100)^n \).

Percentage in Solutions and Marks

In a mixture, the amount of a pure substance is (percentage/100) × total. When water is added, the amount of the other ingredient does not change, so write the new percentage using the unchanged quantity.
In marks problems, find the maximum marks first from the information given, then compute every other percentage on that maximum.

Worked Examples

Illustration 1. A number decreased by 20% becomes 136. Find the number.
Answer:
80% of the number = 136, so the number = 136 × 100/80.
Answer: 170.

Illustration 2. The price of a machine falls by 5% every year. Find its value after 2 years if it cost Rs. 32000.
Answer:
Value = 32000 × 0.95 × 0.95 = Rs. 28880.

Illustration 3. By what percent does the area of a square change if its side is increased by 10%?
Answer:
New area = 1.1² = 1.21 times the old area.
Answer: 21% increase.