**Quantitative Aptitude Simplification and Approximation Tutorial (Study Material)**

Exponents play a large role in mathematical calculations. This chapter provides an introduction to the meaning of exponents and the calculations associated with them. Since exponents are used abundantly in all of mathematics, the basics taught in this chapter will become important building blocks for future knowledge.Quantitative Aptitude Simplification and Approximation Tutorial (Study Material)

**Operation Order Sequence**

For simplifying an expression containing various types of fractions, the order of various operations involved should be strictly maintained. A simple technique for arranging the expression in the proper sequence, is by placing them in the order of the first letter appearing in VBODMAS where.

- V Stands for vinculum or bar as ( )
- B stands for bracket and operation of brackets in the order (). {} and then ||
- O stands for *of
- D stands for division ( /)
- M stands for multiplication ( x )
- A stands for addition (+)
- S stands for subtraction (-)

**Application for Algebraic Formula**

Some algebraic formulae are used in solving the problems on simplification. Following important formula arc to be memorised

**Square Root and Square**

When a number is multiplied by itself, the product obtained is called the square of the number since 6×6 = 36.

.’. 36 is the square of 6 or 6^{2} = 36

Also, – 3 x – 3 = 9 =* (- 3)^{2} = 9 9 is the square of – 3

The square root of a given number is equal to the value whose square is the given number and the sign for square root is √

Since 6^{2 }= 36 then

= 6. i.e. square root of 36 is 6

= 5. i.e. square root of 25 is 5

= an imaginary number

**Division Method for Finding the Square Root**

√ 64009 = ?

**Step 1** Pairing the digits from right to left, we get

- 6 40 09

**Step 2** Then take the first pair, here it is only ’6′ and find the largest whole number whose square is equal to 6 or less than 6. Such a whole number is 2.

**Step 3** Hence, write 2 in the quotient and also in the divisor, (see next page)

**Step 4** Subtract 2×2 = 4 from 6. The remainder is then 2.

**Step 5** Bring down the second pair of digits (i.e. 40) double the quotient (i.e. 2 x 2 = 4) and write the result on the left of 240 and then repeat Step-2 till the remainder is zero. This whole process can be enumerated step-by-step as shown in the following table.

**A number Properties of a Perfect Square Number**

A number whose exact square root (which must be a whole number can be obtained, is called a perfect square:

- ending with 2. 3, 7 or 8 cannot be a perfect square.
- The last digit of a perfect square must be 0. 1. 4, 5. 6 or 9.
- A number ending with odd number of zeroes cannot be a perfect square, e.g. 9000, 25000, 16000, etc. are not perfect squares.
- A perfect square number is either exactly divisible by 3 or leaves a remainder of 1. when divided by 3.

e.g. 64 if divided by 3, will leave a remainder of I 36 is exactly divisible by 3.

- A perfect square number is either exactly divisible by 4 or leaves a remainder of I, when divided by 4

e.g. 81 if divided by 4. will leave a remainder of 1.

100 is exactly divisible by 4.

Note: Above properties are very useful to check if a given number is a perfect square or not.

**Square Root of Vulgar Fraction**

**Step 2** Check the number of decimal places.

If it is odd. then affix a zero on the extreme right of decimal pan to make the even number of decimal places.

Here. no. of decimal places = 5. so. after placing a zero, it becomes .091260

**Step 3** Pair the decimal part accordingly

.09 12 60

**Step 4** Stan finding the square root by the division method as explained in 4.2.2 and put the decimal point in the square root as soon as the integer pan is exhausted.

**Simplification of Decimal Fraction**

The number of digits which arc present on the RIGHT OF A DECIMAL POINT is called the number of decimal places.

That is. 32.0075 has four digits on the right of the decimal point. Therefore, the number is expressed to four decimal places.

A WHOLE NUMBER can also be written as a decimal fraction by putting a decimal after its LAST DIGIT and adding as many zeros as are required, c.g. 12 = 12.0 = 12.000 and so on.

**Addition**

For addition of a decimal number with another decimal number or with another whole number write the given number in such a way that the number of decimal places are equal for all the numbers, e.g. I + 0.59 + 0.008

Here maximum number of decimal places = 3 (three) in 0.008.

Convert all the number so that they have 3 decimal places.

1 + 0.59 + 0.008 = 1.000 + 0.590 + 0.008 = 1.598

**Subtraction**

In subtraction also, the given numbers arc to be wirtten in such a way that the number of decimal places become equal for all the numbers (empty places are filled up with zeroes).

e.g. 2 – 0.283

In 0.283, number of decimal places = 3 In 2, number of decimal places = 0 So. make 2 as having 3 decimal places, i.e. 2.000 2 – 0.283 = 2.000 – 0.283 =1.717.

**Multiplication**

005 x 0.08 x 0.4= ?

**Step 1** Multiply the number only. i.e. 5 x 8 x 4 = 160

**Step 2** Add the total number of decimal places in the given number, i.e. 3 + 2 + 1=6

Step 3 Write the result of Step l and convert it to a number with decimal places as obtained in Step 2 by shifting the decimal point to the left, i.e. by six decimal places, we then get 0.003 x 0.08 x 0.4 = 0.000160 = 0.00016 Similarly 0.03 x 0.7 x 2 = 0.042

Total of 3 decimal places

**Division of Decimals**

- When the Divisor (or Denominator) is a Whole Number

3.0056 /7

**Step 1** Simply divide the number without considering the decimal points given i.e. 7) 30056 (4293.7

**Step 2** Count the no. of decimal places in the given number. Here it has 4 decimal places in 3.0056.

**Step 3** Shift the decimal point in the quotient obtained to the same no. of decimal places as in Step-2

Shift

Hence the result becomes. 42937 = 0.42937

**When the Divisor (Denominator) is also a Decimal Number**

**Step 1** Shift the decimal point to the right of the numerator and of the denominator such that

- total decimal point shift in numerator = total decimal point shift in denominator.
- there is no decimal place left after the shift.

Here. no. of decimal place in numerator (in 12.598) = 3

no of decimal place in denominator (in 27.08 and 1.417) = 2 + 3 = 5 since 5 > 3, so, total shift in decimal point to be made (in numerator and denominator) = 5 Now. 5 decimal point shifts are made.

**Simplification of Mixed Fraction**

A mixed fraction consists of two pans, the integer pan and the fractional pan.

eg. 2 7/18 has 2 as an integer and 7/18 as a fraction.

In fact 2 7/18 = 2+7/18

**Continued Fractions and its Simplification**

**Recurring Decimals**

A decimal fraction in which a digit or set of digits is repeated continually is called a Recurring or Periodic decimal.

e.g. 1/7 = 0.333…

Here, on performing the division, it is found that the remainder is always 1 and in the quotient, the digit

3 is continually repeated. Hence it is written as 0.3, where the dot over 3 indicates that the 3 has to be continually repeated.

Similarly, 1/7 is

7) 1.000000 (0.142857 142857…

So, if we continue the division, we shall get the same set of figures 142857 again and again and in the same order.

Therefore 1/7 = 0442857 or 0.142857 7

The repeated digits or repeated set of digits is called the period of the recurring decimal.

There are two types of Recurring Decimals,

- Pure recurring decimal: Such a decimal in which all the decimal digits recur, e.g. 0.142857
- Mixed recurring decimal: Such a decimal in which all the decimal digits do not recur, e.g. 0.7167

**Conversion of a Pure Recurring Decimal to the form p/q**

- Write the decimal part without the decimal point as the numerator.
- Write as many 9s as there are different repeating digits for the denominator.

**Conversion of a Mixed Recurring Decimal to the form p/q **

Step (1) First, write the decimal part without the decimal point and subtract the non-repeating part from it and write the result in the numerator

Step (2) Write a number in the denominator with as many 9s as there arc repeating digits in the decimal part and followed by as many zeroes as there are non-repeating digits in the decimal part.

**Important Derivations**

**Approximate Value**

In this type of questions, candidates do not have to find out the exact value, but all they have to do is

Step 1 To round off the numbers given in the question

Step 2 To simplify

Step 3 To round off the result obtained in Step-2

Very Important: In some of the questions, the choices given are very close to each other. In such case,

Step-1 is to be avoided, and we should go directly to Step-2.

**Rounding Off Numbers**

On some occasions for case in simplification, we require only a rough estimation and not the exact value. In such cases we round off the values to the nearest tens, or hundreds or thousands.

Rounding off a number to the nearest ten. hundred or thousand means finding the multiple of 10, 100 or 1000 which is closest to (or approximate) the original number. It can be done by the following procedure, (a) rounding off to the nearest 10: Replace the digit at unit’s place by 0. If the replaced digit is 5 or more, then add 1 to the digit at tens place, otherwise digit at tens place remains unchanged.

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