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Illustration 1

To find difference of 635 and 279. 
 

Note that 635 is 35 more than 600 and 279 is 21 less than 300.

Thus 635 – 279   = (600 + 35) – (300 – 21)

                    = (600 – 300) + (35 + 21)

                    = 300 + 56 = 356.
 

The only difference here is that one has to carry out the operations indicated in brackets mentally rather than resorting to writing.

 

Illustration 2:

To find difference of 279 and 135. 

279 – 135         = (200 + 79) – (100 + 35)

                = (200 – 100) + (79 – 35)

                = 144.

Both the above-discussed examples, the writing part has to be reduced to minimum one must learn how to do these calculations mentally.

 

To multiply or divide by 5 or multiples of 5

 

Illustration 3:

To find 436 × 5:

 

Whenever we have to multiply a number by 5, divide the number by 2 and multiply by 10. 

So the above can be written as 436 × 10/2 = 4360/2 = 2180.

 

To find 436 ÷ 5

We can write the above as 436 × 2/10 = 43.6 × 2 = 97.2.

 

This principle can be extended for all the integral powers of 5:

 

To find 436 × 25

We can write this as: 436 × 100/4

= 43600/4 = 10900.

 

In general, to multiply by powers of 5, add as many zeroes to the number as the power of 5, and divide by the power of 5.

 

This principle can be extended to other numbers also, as explained below:

 

Illustration 4:

Calculate: 379 × 9

 

This can be broken as 379 × 9 = 379 × (10 – 1) = 3790 – 379 = 411

Also 379 × 11 = 379 × (10 + 1) = 3790 + 379 = 4169

 

Note that we have converted multiplication into subtraction / addition.

 

Finding Percentages

 

Illustration 5:

Find 20% of 546

This is = 546 × 20/100 or 546 × 1/5 = 109.2

We can generally by saving formula for %age is % =  × 100

 

During the calculation of percentage, use of formulae is lengthy.  The following text will tell us how to avoid formulae and calculate percentage in a manner that is easier and faster.

 

Finding percentage of integral powers of 10:

Calculating these percentages is simple; it just involves moving the decimal place.

E.g. 10% of 786 = 78.6 and 1% of 786 = 7.86, 1000% of 786 = 7860.0

 

We can calculate the required percentage by converting in the form of these percentages:
 

(i)        9% of 385 = 10% of 385 – 1% of 385 = 38.5 – 3.85 = 34.65

(ii)      11% of 385 = 10% of 385 + 1% of 385 = 38.5 + 3.85 = 42.35

(iii)    20% of 1372 = 2 × [10% of 1372] = 2 × [137.2] = 274.4

(iv)    22% of 3476 = 2 × 10% of 3476 + 2 × 1% of 3476 = 2 × 347.36 + 2 × 34.76 = 6956.2 + 69.52 = 764.72.

(v)      27.5% of 476 = 25% of 476 + 2.5% of 476 = ¼ × 476 + 10% of (1/4 × 476) = 119 + 11.9 = 130.9

 

Calculating percentages

 

(i) 45 is what % of 120?

To reduce the denominator from 120 to 100, we have to reduce it by 1/6.

So, to get the correct figure, we can reduce both numerator and denominator by 1/6.

Reducing 45 by 1/6 i.e. 7.5, we get 45 – 7.5 = 37.5
 

(ii) 35 is what % of 160?

Note that ¼ of 160 is 40.

So the answer has to be less than 25%.

To reduce to 35, we see that 1% of 160 is 1.6.

To reduce by 5, we must reduce by approx 1.6 × 3 = 4.8.

So the answer is close to 25 – 3 = 22%.
 

(iii)      Find 59.4% of 378?

We can reduce the sum to 60% of 380 and then reduce the answer somewhat.

Since 10% is 38, 60% will be 38 × 6 = 228.

So the answer would be approx 225.

Alternately, 378 = (50% of 378) + (10% of 378) = 226.8; reduce slightly to get 59.4%.
 

(iv)      236 is what % 5347?

We see that 1% of 5347 is 53.4

So 4% of 5347 = 214 approx

To add another 21 to get the numerator, we should add 5.3 × 4 or about 0.4%

Hence the answer is approx 4.4%.
 

(v)       816 is what % of 65?

65 × 12 = 780 780 is 1200 % of 65.

65 × 13 = 845 is 1300% of 65


So the answer must lie somewhere in between, i.e. 1255%.

Fractions

Solve 13/63:
 

Using approximation, we can write 13/63 = 13/65 = 0.2

But 0.2 is not the actual answer. 
 

We have to find out whether actual answer is more than 0.2 or less than 0.2

Since the denominator must be decreased from 65 to 63, the actual answer must be increased.


Since 13/65 = 0.2 and  
 > 0.2.

The determine how much increase must be made, see that the denominator is decreased by 2/65 or about 3%. So the actual answer is around 0.206. In general,


(i)        
If denominator is decreased, then the actual answer is less the approx. answer

(ii)        If denominator is increased, then the actual answer is more than approx. answer.

(iii)      If numerator is decreased, then actual answer is more than approx. answer.

(iv)      If numerator is increased, then actual answer is less than approx. answer.

 

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