# Useful Illustrations

**Illustration 1:**

The price of A is Rs 25,000, which is 20% lesser than that of B. What is the price of B?

Use the formula of percentage increase/decrease given above. If B is 100, A must be 80. But since A = 25000, B will be 25000 × 100/80 = 31250.

# Short cut 1

Refer to the box given above under the section “Percentage Increase/Decrease.” We find: “If A is 20% less than B, then B is 25% more than A.

Using this fact, we can easily do this sum by increasing Rs 25,000 by 25%, or by 1/4th. Since 1/4th of Rs 25000 is 6250, the answer is 25000 + 6250 = 31250. This line of reasoning avoids a division which could take time.

**Illustration 2:**

The price of a commodity increases by 25%. In order to keep the expenses on it constant as before, by what percentage should a person cut down his consumption?

Apply the formula 100 × R/(100 + R). Hence consumption should be reduced by 100 × 25/125 = 20%.

# Short cut 2

Remember the fact as in the previous illustration: “If A is 25% more than B, then B is 20% less than A.” Hence the answer is 20%.

**Illustration 3:**

If the population of a town is 231525 and it has been growing annually at 5% - what was the population 3 years ago?

Let the population three years earlier be X. Apply the formula for compounding.

# Short cut 3

To avoid the long calculation, we can use an approximation.

If we see that the population increases 5% every year, it should increase by 15% in 3 years, ignoring the compounding.

If we add compounding, we population increases by slightly more than 15% in 3 years.

To arrive at the previous figure, we thus have to reduce the final figure of 2,31,525 by a percentage less than 15%, say 13%.

Reducing by 13% we get: [using approximations] = 2,31,000 – 10%(2,31,000) – 3%(2,31,000) = 2,31,000 – 23,100 – 3 × 2310 = 2,31,000 – 30,000 (approx) = 2,01,000.

Rounding off (or by looking at the choices) we can see that the answer must be 2,00,000.

**Illustration 4**:

A square is converted into a rectangle by increasing one of its sides by 5% and reducing the other by 5%. What will be the % change in the area of the two figures?

See the formula above.
Percentage change will be (R2/100)%, *h*ence a reduction of 0.25% will take place. To find the new area, we reduce the original area by 0.25%

**Illustration 5**:

If A’s salary is 50% more than that of B, then how much per cent is B’s salary less than that of A?

Let B = 100, then A = 150. Difference = 50. Required percentage = 50/150 × 100 = 33%.

**Illustration 6**:

The tax on a commodity is diminished by 20% and its consumption increases by 15%. Find the effect on the revenue.

Let price and consumption both be 100 each. Old Revenue = 100 × 100 = 10,000. New revenue = 80 × 115 = 9200. We see that revenue has decreased by 800/10000, or 8%.

**Illustration 7:**

The value of a machine depreciates at the rate of 10% per annum. If its present value is Rs. 100000, what will be its worth after 3 years?

Use the formula above. The machine becomes 90/100 of itself in every year, hence 100000 × (90/100)3 = 72,900.

# Short cut 4

Ignoring compounding, we see that the value becomes 90,000 after first year, 81,000 after second year and approx 73,000 after third year. By subtracting consecutively, we can arrive at the answer without a lengthy multiplication.

**☺****Time Savers: Using Percentages in Multiplication**

Large multiplications can be avoided by using our knowledge of percentages. The following examples will make the method clear.

**1. If we have to multiply 1.23 **× **58745****
**This means that the number must be increased by 23%.
Either we can use 10% + 10% + 3% as explained above, or
(BETTER) Calculate 25% of the number, which is 1⁄4 × 58745, reduce it and get to an approximation.

**2. If we have to multiply 123 **× **76589****
**Simply use any of the above methods and add to 0s to the answer.

**3. If we have to multiply 0.87 **× **46789****
**This means that the number must be reduced by 13%.
Simply subtract 10% + 3% from the number, or
Calculate 12.5% of the number, which is 1/8th of the number and subtract. Reduce the answer slightly to get a better approximation

**4. If we have to multiply 83 **× **673425****
**Calculate the approximation as above and add two 0s to the answer. The above methods are of great help in Data Interpretation.