### Important Formulas and Facts

Decimal Fraction:

A fraction where the bottom number (denominator) is a power of ten.
e.g. 10, 100, 1000 etc

Decimal fraction is written with a decimal point (without denominator), which make it easier to do calculations like addition and multiplication on fractions.
e.g. $\frac{55}{100} = .55$

Vulgar fraction:

A number expressed with numerator and denominator ia known as simple or vulgar fraction.
e.g $\frac{3}{4}$

Convert decimal into vulgar fraction:

1)Calculate the total numbers after the decimal point
2)Write the number without decimal point
3)In the denominator put 1 and annex with as many as '0' in the step 1
4)Simplify the fraction to its lowest terms

e.g 0.50 = $\frac{50}{100} = \frac{1}{20}$

Recurring decimal:

Recurring decimal means, a decimal with a recurring value.
e.g 0.3333...., where 3 is the recurring number.

Convert a recurring decimal to vulgar fraction:

1)find the recurring number in the decimal fraction
2)In the denominator put as many as '9' as the length of the recurring number
3)Simplify the fraction to its lowest terms

e.g 0.3434343434

Recurring number = 34
Length of recurring number = 2
$\therefore$ vulgar fraction = $\frac{34}{99}$

Mixed-recurring decimal:
a demcimal with recurring and non recurring numbers is known as mixed-recurring decimal.
e.g 1.119999...., where 11 is the non-recurring and 9 is the recurring numbers.

Convert mixed-recurring decimal to vulgar decimal:

1)Seperate the recurring and non-recurring numbers in the decimal number.
2)In the numerator, take the difference between the number formed by all the digits after decimal point (only take repeate digits once)and that formed by the digits which are not repeated.
3)In the denominator, take the number formed by as many '9' as the lengh of recurring number followed by as many '0' as the length of non-recurring number.
4)Add the fraction with the number before decimal point.

e.g. 1.11999999...

Step 1)

Non-recurring number = 11 and recurring number = 9

Step 2)

119-11 = 108 (Numerator)

Step 3)

900 (denominator)

Step 4)

$1 +\frac{108}{900} = 900 +\frac{108}{900} = \frac{1008}{900} = \frac{28}{25}$

1)

The value of $[35.7-(3+\frac{1}{3+\frac{1}{3}})-(2+\frac{1}{2+\frac{1}{2}})$ is :

A) 30

B) 34.8

C) 36.6

D) 41.4

2)

If  $1^{3}+2^{3}+..........+9^{3}=2025 then the value (0.11)^{3}+(0.22)^{3}+......+(0.99^{3})$ is close to

A) 0.2695

B) 0.3696

C) 2.695

D) 3.695

3)

Simplify $\frac{0.05\times0.05\times0.05+0.04\times0.04\times0.04}{0.05\times0.05-0.05\times0.04+0.04\times0.04}$

A) 0.09

B) 0.009

C) 0.08

D) 0.008

4)

Convert the 3.004 into vulgar fraction ?

A) 3/4

B) 75/50

C) 751/250

D) 751/1000

5)

.00625 of $\frac{23}{5}$ when expressed as a vulgar fraction equals :

A) 23/80

B) 23/800

C) 23/8000

D) 125/23

6)

The sum of the first 20 terms of the series

$\frac{1}{5 \times 6} + \frac{1}{6 \times 7}+\frac{1}{7 \times 8}+....$ is

A) 0.16

B) 1.6

C) 16

D) None of these

7)

Find the value of the following expression upto four places of decimals.

$\left[1+\frac{1}{1 \times 2}+\frac{1}{1 \times 2 \times 4}+\frac{1}{1 \times 2 \times 4 \times 8} + \frac{1}{1 \times 2 \times 4 \times 8 \times 16}\right]$

A) 1.6414

B) 1.6415

C) 1.6416

D) 1.6428

8)

$\frac{1}{4}+\frac{1}{4 \times 5}+\frac{1}{4 \times 5 \times 6}$ correct to 4 decimal place is

A) 0.3075

B) 0.3082

C) 0.3083

D) 0.3085

9)

$\frac{96.54-89.63}{96.54+89.63}+\frac{965.4-896.3}{9.654+8.963}=?$

A) $10^{-2}$

B) $10^{2}$

C) $10^{-1}$

D) $10^{1}$

E) None of these

10)

If 213 x 16 = 3408, then 1.6 x 21.3 is equal to :

A) 3,408

B) 1.72

C) 34.08

D) 340.8

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