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How to master Calculations & Approximations

 

Read the options along with the question. The extent to which you will do calculations, depend upon how far your options are. If the options are close, you need to do exact calculations. If options are far, then approximation will work. That is why you must glance through options after reading the question and then only start solving questions. This judgement is developed by practice and conscious effort to decide whether options are far or near.

 

 

Break-up Method of Subtraction

 

 

Example 1

Calculate 512 - 289
Solution

This can be broken up into (500 + 12) – (300 – 11)
Now it is easy to see that the sum is (500 – 300) + (12 + 11)
= 200 + 23 = 223.

 

 
 

Note that the above should be a mental process and thus calculations can be done easily.

 

Example 2
 

Solve 857 – 532

Solution

This can be written as (800 + 57) – (500 + 32)
= (800 – 500) + (57 – 32) = 300 + 25 = 325.

In case of both of the above discussed examples, the calculation was just to convey you the method.
But you have to put the same habit to do it mentally. If you start doing it by writing it would not help much.
 

 

 

Break-up Method of Multiplication

 

Example 3

 

Solution

 42 can be broken as (40 + 2). 
    = (40 + 2) × 7
    = 280 + 14
    = 294.

42 is broken in such way that multiplication becomes easy. We could have broken as 39 + 3 also but multiplying 39 with 7 is not easy. So, 42 was broken as 40 + 2 since doing 40 × 7 and 2 × 7 are easy.


 

Example 4
 

Solve 89 × 12
 

Solution

= (90 – 1) × 12
    = 1080 - 12
    = 1068
    So you can break any number, but note that you must get a round figure somewhere which makes the multiplication easy.
 

 

 

Example 5


Solve 236 × 23

Solution
 

= (240 – 4) × (20 + 3)     
= (4800 + 720) – (80 + 12)
= 5520 – 92
= 5428.
 

 

Calculating Percentages

 

The shortcut for percentages is to work with round figures, such as 10%, 1%, and so on, which are easy to calculate. Simply move the decimal point to calculate these: simply move the decimal one place for 10%, for 1% two places and for 0.1% three places and so on.
 

So, 10% of 2456 = 245.6 ; 1% of 2456 = 24.56 ; 0.1% of 2456 = 2.456.

 

Since percentages are easy to calculate, try to break any percentage given in the form of these three percentages.
 

Example 1
 

Solve 20 % of 2456.

Solution

20% can be broken as 2 × 10%
⇒ 10% is 245.6.
⇒ So, 20% = 2 × 245.6 = 491.2

 

 

 

Example 2
 

Solve 22% of 2456

Solution
 

22 can be broken as 2 × 10% + 2 × 1%
⇒ 2 × 245.6 + 2 × 24.5
= 491.2 + 49 = 540.2


 

Example 3
 

Solve 74% of 2456

Solution

In this case remember that 75% = ¾.
Hence the sum is ¾ × 2456 – 1%(2456)
        = 1842 – 24.5 = 1817.5
This method is very useful for approximations and works well when we keep the choices in mind.
 

Example 4
 

Solve 48% of 314
 

Solution


48% can be broken as 50% - 2%
= ½ (314) - 2 × 1%
⇒ 157 – (2 × 3.1)
⇒ 157 - 6.2
        = 150.8


 

Example
 

Solve 26% of 924.
 

Solution

26% = 25% + 1%
= ¼ (924) + 1%(924)
= 231 + 9.2 = 240.2.
 
Note that fractional values of the decimals and percentages are very important. Thus having calculative strengths i.e. tables, squares, cubes and fractions with you is very important.

 

When Percentage is required

 

 

Examples
 

15 is what % of 40?
 

Solution
 

In this question we have to calculate 15/40 × 100.
Here too, we use the basic break-up method, i.e., we use 10%, 1% and 0.1% of base.

We see that 10% of 40 is 4. So 40% of 40 is 16.
Since we need 15, the answer is slightly less than 40% (first approximation).
To refine the answer, we must reduce 16 to 15.
Since 1% of 40 is 0.4. Reduction of 1 is obtained is we reduce the answer by 2.5%.

 =>       So a reduction of 1 is obtained if we reduce 40% by 2.5%.
 =>       Hence answer is 40% - 2.5% = 37.5%

Note again that the above process should be mentally done in order to reduce time.
 

 

 

Note: Since the exam is a multiple choice exam, you do not have to write the exact answer.

 

As you have to choose the right answer from the choices given, you can simply see that 10% of 40 is 4, thus 40% of 40 will be 16, whereas you only require 15, thus it can very well be stated that the answer is close to 40%, but slightly less than that. Simply tick the answer without further calculation.

 

Example 7

26 is what % of 120.
 

Solution


We need 26/120 × 100
Observe that 10% of 120 = 12. So, 20% will be 24.
So the answer has to be slightly more than 20% (first approximation).
Further, 26 = 24 + 2

=>            1% of 120 = 1.2. So, 2% will be 2.4
=>            So, 26 = 24 (20%) + 2 (2% appx.) = 22% appx.

Here we knowingly took simple numbers to explain the method and we wrote too much to explain the method. But once you have understood the method, then you can calculate even for big & complex numbers and do it without writing much.
Alternately 10% is 12, and 20% will be 24. The required no. is 26, thus can be stated that the answer is close to 20% but slightly more than that.
 

 

 

Example 8
 

37.4 is what % of 237
 

Solution

We need 37.4/237 × 100
= 23.7 (10%) + 11.8 (5%) = 35.
So the answer is slightly more than 15%
Since 1% is 2.37, we can see that the answer is close to 16% but slightly less than that.
So the answer should be something between 15.7% and 15.8%.

 

Example
 

556.7 is what % of 797
 

Solution

We need 556.7/797 × 100.
        Taking approximation, we can use the base as 800.
        Then we see that 70% of 800 = 560.
        We need 556.7 but reducing by 1% would mean reducing by 8, which will make theapproximation too      wide.

        As we need 556.7 and also the base is slightly less than 800, we can see that we need to reduce our      answer very slightly.
        Hence the answer is around 69-70%.
 

   

 

Solving Fractions

 

 

Example 1
 

Solve 12/61
 

Solution

Here 61 can be written as 60.
So, 12/61 becomes 12/60 = 1/5 = 0.2.
But 0.2 isn't the actual answer. It is the approximate answer. The actual answer will be different from 0.2. But is it more than 0.2 or less than 0.2?
Since we decreased the denominator, the actual answer will be less than the approximate answer.

So the actual answer will be slightly less than 0.2, something like 0.19 or close to 0.20.

Knowing whether the actual answer is more or less than the approximate answer is knowing the direction. Having an idea about the direction is important for successful use of approximation in Quantitative Aptitude & DI.
 

 

 

Rules for knowing the direction of the answer

 

a)  If denominator is increased, then actual answer is more than appx. answer.

b)  If denominator is decreased, then actual answer is less than appx. answer.

c)   If numerator is increased, then actual answer is less than appx. answer.

d)  If numerator is decreased, then actual answer is more than appx. answer.

 

 

The 3 digit thumb rule:

In DI, you should never deal with more than 3 digits.

Always ignore digits and reduce it down to 3 digits. 

 

Example 1
 

Solve 10920000/455
 

Solution

Ignore four zeros in 10920000. So, we get
1092/455 = 910(200%) + 182 (40%) = 240% = 2.4
Bringing back the four zeros which we ignored earlier, we get 24000.

Example 2
 

Solve 527831/4567

Solution

Ignore three digits from numerator and one digit from denominator. We get 528 in numerator (after rounding off 527.8) and 457 in denominator (after rounding off 456.7)
⇒     528 /457 = 457 (100%) + 46(10%) + 25 (6% appx.)
                = 116%
                = 1.16.
Adjusting digits we ignored earlier, we bring back two digits, we get 116 as net we are ignoring two digits in numerator.
 

 

 

Converting ratios to decimals

 

For converting ratios into decimals, we use the break-up method discussed earlier.
 

Example

Convert 116:177 into decimals.
 

Solution

Here, we need to divide 116 by 177.
Rounding off by increasing both the numbers, we get: 120/180 = 2/3 = 0.66.

Change of units

 

1 lakh has five zeroes = 1,00,000

1 crore has seven zeroes = 1,00,00,000

 

1 thousand has 3 zeroes.

1 million has six zeroes = 1,000,000

1 billion has nine zeroes = 1,000,000,000

        So, 1 million = 10 lakhs

        So, 1 billion = 1000 million or 100 crores.

1 Km = 1000 m.

        1 m. = 100 cm.

1 inch = 2.5 cm.

1 foot = 12 inch = 30 cm.

1 yard = 3 foot = 36 inch = 90 cm.

 

 

Quick Calculations

 

For doing data questions the student must be able to do calculations quickly. This is helpful in maths as well.

We can illustrate this by some examples:

 

Example 1. 

Data is represented in a tabular form as given below.
 
Figures in Rs A B C D Total
Sales 24568 25468 23752 15782 89570
Operating costs 17198 19101 16151 10258 62708
Interest costs 2457 2292 2850 1578 9177
Profit 4914 4075 4750 3946 17684

 
Q. Which firm has the highest profitability?
1. A         2. B         3. C          4. D
 
Note that in this question, we need to know about profitability.
 
Solution

One is tempted to Tick A because the profits of A are the highest.
But we need is profitability, defined as Profit/Sales.
The answer would thus be (D) because for D, profitability = 3946/15782 = 25%, which is the highest.

Here one must know how to make a quick calculation to get 25%. The best approach is to round off the figures, so the fraction becomes 4000/16000 which is about ¼ or 25%.

 





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