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Quick Calculations

It becomes obvious here that one must be fairly good at quick calculations. Suppose we have to find the percentage of the fraction 295/397. We round off to get 300/400 = ¾ or 75%. Hence knowledge of fractions and decimals is required so that one can quickly convert from one to the other. Students should learn these conversions:

 

Fraction

Decimal

Percentage

1/2

0.5

50%

1/3

0.33

33%

¼

0.25

25%

1/5

0.20

20%

1/6

0.16

16.66%

…and so on.

 

From this table, we can find almost any decimal. For instance 1/18 will be 1/3 of 1/6 i.e. 1/3 × 16.66% or almost 5.5%. Then, find any fraction such as 4/18 or 7/18 is also a simple task as we multiply 5.5 by the required numerator.

 

Further, keep the multiple of the denominator in mind that is close to 100. For instance to find 1/17, we know that 17 × 6 = 102, so 1/17 will be equal to 0.06 or about 6%. A closer approximation will yield a figure slightly less than 6%, since the multiple is not exactly 100.

 

Similarly, to find 1/37, we see that 37 × 3 = 111 so 1/37 will be less than 0.3 and the approximation would be 0.27 or so.

 

Once we know these basic figures we can find out just about any percentage.

 

Illustration 1

 

 

Example
 

Find the percentage equivalent of 15/17.

Solution

We find 15/18, which is close to our fraction.

Since 15/18 = 5/6, it must be 16.66 × 5 (from the above table) = 83% approx.
Since the denominator is less than the required number, note that our approximation is less and must be increased.

We must thus increase our approximation by 1/17 or about 5% to get the answer as 88%.

The student will appreciate that this is a mental process. Any fraction can be converted by this method.
 
Alternately, we know that 1/17 is about 0.06, so 15/17 will be 0.9 or 90% and the answer must be reduced by a small decimal as explained above.
 

 

 

Illustration 2


 

Example

Find the percentage increase from 5867 to 9764.
 

Solution

Increase both the figures by approximately similar quantities.
Our job then is to find the percentage increase from 6000 to 10000. Now the sum becomes quite easy: the percentage increase is 4000/6000 × 100 = 66% approx.

Note: Either both quantities should be increased, or both quantities should be decreased to find approximations.
 

 

 

Illustration 3

If the stock market moves from 9876 to 12580 points, how much percentage does it move?

To calculate, we have to do (12580 – 9876)/9876 × 100 which is 2704/9876 or about 2700/9900 or about 27%.

 

This is a three step calculation and cannot be done without pen and paper. But by approximation, we can do it visually:

 

First, to calculate the difference we can round off the figures as 12700 – 10000 = 2700. Note that we must increase similar quantities. If the student increases the quantities blindly, a mistake might result. Now calculate percentage as 2700/10000 × 100 = 27% approx.

 

Note that here we increased both quantities. The student must remember to increase both or decrease both, otherwise the answer will be wrong. Also note that we increased approximately similar amounts: we increased 12580 to 12700, an increase of 120, and we increased 9876 to 10000, an increase of 124. Always remember to increase or decrease similar amounts.

 

Now we can try some approximations. Complete the table below in which if quantity A increases to quantity B, then what would be the percentage increase?

Remember to do these sums mentally, or with very little calculation. Allow yourself not more than 30 seconds for each sum. Use your knowledge of tables and approximation to do the sums mentally. Answers are given at the end of chapter.

 

Exercise 1:Percentage


 

S. No

A

B

% increase/decrease

1

257

395

 

2

232

647

 

3

75

750

 

4

1.3

3.4

 

5

538

522

 

6

116

221

 

7

8638

9475

 

8

12832

11689

 

9

721

936

 

10

45

53

 

 

 

(i)         Why approximate: Lengthy calculations are avoided and time is saved.

(ii)        How to approximate: Learn the techniques of approximation.

(iii)      How much to approximate: Look at the choices. If the given answers are wide apart, a rough approximation will work, otherwise one will have to be more accurate.

 

 

 





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