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Explanation of Basic Concepts

Concept 1: Conversion from fraction to percent and vice versa

1. Fraction to Percent: Multiply the fraction by 100 to convert it into a percent.

Illustration 3:
0.2 = 0.2 × 100 = 20%
3/8=3/8*100=37.5%

 

2. Percent to Fraction: Reversing the earlier operation will convert a percent to a fraction - i.e. divide the percent by 100.

Illustration 4:
40%=40/100=0.4,
55%=55/100=0.55=55:100

 

Concept 2: Increase or Decrease of a Quantity

Remember that the increase or the decrease is always on the original quantity.

% increase/decrease = (Quantity increase or decrease/original quantity) × 100.

Illustration 5:

The salary of a man goes up from Rs 100 to Rs 125. What is the percentage increase in his salary?

Solution

Increase = 125 – 100 = Rs. 25.
∴%increase=25/100 *100%=25%


 

Illustration 6:

If the salary of the man had been reduced from Rs 125 to Rs 100, what is the percentage decrease in his salary?

Solution
Decrease=125-100=Rs25.
∴%decrease = 25/125 * 100% = 20%

 

 
Note: that for the same quantity of increase or decrease the % increase and % decrease have two different answers. The change in the denominator - which is the original value changes in the above two situations and hence the difference.

 

 

To Increase a Number by x%

If a number is increased by 10%, then it becomes 1.1 times of itself. We can increase the number by 10% by multiplying by 1.1 or 110/100. If a number is increased by 20%, then it becomes 1.2 times of itself.If a number is increased by 30%, then it becomes 1.3 times of itself... and so on.

To Decrease a Number by x%:

If a number is decreased by 10%, then it becomes 0.90 times of itself. To decrease a number by 10%, we can multiply the number by 0.9 or 90/100.If a number is decreased by 20%, then it becomes 0.80 times of itself.If a number is decreased by 30%, then it becomes 0.70 times of itself... and so on.

Equivalent Percentages of some commonly used Fractions

It is useful to learn these by heart

Fraction

Equivalent%

  

50%

33.33%

25%

20%

16.67%

40%

60%

66.67%

 

Fraction

Equivalent%

75%

80%

12.5%

8.33%

37.5%

62.5%

87.5%

 

11.11%

 

 

Fraction

Equivalent%

22.22%

6.67%

5%

4%

2%

133.33%

125%

120%

 

 

 Percentage Increase/Decrease

If A’s income is R% more than B, then B is income is less than that of A by 100 × R/(100 + R)%.

If A is 25% more than B, then B is 20% less than A

If A is 20% more than B, then B is 16.67% less than A

If A is 33(1/3)% more than B, then B is 25%  less than A

If A is 50% more than B, then B is 33(1/3)% less than A.

If A’s income is R% less than B, then B’s income is more than that of A by 100 × R/(100 – R)%.

If A is 16 2/3% less than B, then B is 20% more than A.

If A is 20% less than B, then B is 25% more than A.

If A is 25% less than B, then B is 33 1/3% more than A.

 Successive increase and decrease

Example : If price increase by 10 % ( R %)and then decrease by 10% (R %), actual reduction is 1% (R2/100) %

Concept 3: Increase and Decrease By the Same % Age

If a number is increased by R%, then this number is decreased by R%, then in total there would be a decrease of (R2/100)%. If a price goes up by 10% and then decreases by 10%, we do not come back to the same figure but a different one given by the formula. Students often mistake in this.

Illustration 7:

A shopkeeper marks up his goods by 20% but then to encourage sales, reduces the price by 20%. By doing so, he makes a profit/loss of:
1. 0%
2. 4% profit
3. 4% loss
4. none of these

Solution

Start with 100. After the first increase his price goes up to 120.
Reducing 20% of 120, he will have to reduce Rs 24 and the new price is 120 – 24 = 96.
So after increasing the price by 20% and reducing it by the same percentage, he will make a 4% loss.
Some students make the mistake of marking 0%, but increasing and decreasing by the same percentage will not result in the original figure.


Concept 4: Successive Discounts

If successive discounts are made, then each successive discount must be calculated on the discounted price. Do not make the mistake of adding the discounts.

Illustration 8:

A retail chain gives a discount of 50% and then to increase sales offers another 40% off. By doing this, it has effectively reduced prices by:
1. 90% 
2. 10% 
3. 20% 
4. 70%

Solution

Start with 100, we arrive at 50 after the first discount.
Then another 40% discount is given, so we discount 50 by 40% and that gives us 20.
So the effective price is 50 – 20 = 30, so the shopkeeper has effectively reduced prices by 70%.


Concept 5: Compound Growth

Typically compound growths are used in investment growth analysis (compound interest) or in population growth (things like cattle population, production output, growth).

If P is the population of a country and if it grows at r% per annum, then the population after n years will be

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Illustration 9:

A bank offers fixed deposit at 10% per annum. What will be the value of Rs 10,000 after 3 years?

Solution

The value will be: 10,000 × (110/100)3.
= 10,000 × 1.1 × 1.1 × 1.1 = 13,310
More details of solving compound growth problems will be found in the explanation of Compound Interest.

 
Time saver: Approximating Percentages

It is useful, in many DI sums, to arrive at an approximation instead of getting into calculations. In this section we illustrate how approximations can be made and how time can be saved.

Illustration 10:

Find the percentage represented by the fraction 53/81.

Solution

The fraction can be written as 53/80. Note that by reducing the denominator, we have increased the value, so we must reduce the answer that we will get.
The denominator is 80, and it must be increased to 100 to find percentage.
This represents an increase of 20 and in percentage terms it means that the denominator must increase by 20/80 or by 1/4th.
To make the fraction equal, we must increase the numerator by 1/4th as well.
Increasing 53 by 1/4th, we get 53 + 13 = 66%.
Since we reduced the denominator, we get a higher approximation and this must be reduced.
So 53/81 represents the percentage of around 64-65%.The process explained above is a mental process and can be done in seconds, thus saving time.

 
Illustration 11:

A student obtained 82.5% marks in a certain examination. If the maximum marks is 600 find the total marks obtained by her.

Solution

Since the maximum marks are 100, we know that 10% = 60 and 1% = 6. This is easily done no matter how big the figure is.
Then 82.5% = 80% + 2.5%
= 8(10%) + 2.5(1%)
= 8(60) + 2.5(6)

= 480 + 15 = 495.
The student will appreciate how much time can be saved by this approach.
The conventional way is to multiply 82.5/100 × 600 = 495, but this will take longer.

 

Illustration 12:

The annual sales of company X were Rs 72,534 in 2004-05 and Rs 84,875 in 2005-06. What was the % increase in sales turnover?

Solution

Round off the figures to 72,000 and 84,000.
Increase = 84,000 - 72,000 = 12,000.
∴% Increase = 12000/72000 or 1/6th = 16.67%.

 

Illustration 13:

The sales of a company increased by 13% to Rs 54,578. What was its sales in the previous year?

Solution

The method to do this sum is 54,578 × 100/113. However, this calls for a lengthy division. Thus we have to look for a simpler way of doing this.
If the sales were 100, the figure increased by 13% to 113.
To reverse the process, we must decrease the figure by a figure less than 13%, say 11 or 12%.
So the problem becomes 54,578 – 11%(54,578).
We know that 10% is 5457 and 1% is 2 × 545 = 5457 + 545 = 6002.
Subtracting, we get 54,578 – 6000 = 48,500 approx.
We can work backwards to see whether our calculation is correct. 48,500 + 13%(48,500) = 48,500 + 4850 + 3(485) = 54,800 approx.
So we see our answer is approximately correct.
This method is very useful in solving sums.





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