SHORTCUTS IN SIMPLIFICATION

Rule of Fractions:

Proper fraction:
$\dfrac{a}{b}$ is said to be proper where the value of a is less than b.
Improper fraction:
$\dfrac{a}{b}$ is said to be improper where the value of b is less than a.

(1)Proper Fraction:

$2+\dfrac{3}{5}$
We can simply say that it is equal to $2\dfrac{3}{5}$ since it is a proper fraction addition of an integer with a fraction is always equal to multiplication of those integer and fraction values.
Normal method: $2+\dfrac{3}{5}$=$\frac{13}{5}$=$2\dfrac{3}{5}$

(2)Improper Fraction:

$2+\dfrac{5}{3}$
=$2+1 \dfrac{2}{3}$=$3+\dfrac{2}{3}$
(3)$8-\dfrac{3}{5}$
=$7+(1-\dfrac{3}{5})$
=$7+\dfrac{2}{5}$
=$7\dfrac{2}{5}$

Question:

Solve :$27\dfrac{1}{2}+15\dfrac{3}{4}-12\dfrac{2}{5}+18\dfrac{4}{5}$

Solution:

$(27+15-12+18)+(\dfrac{1}{2}+\dfrac{3}{4}-\dfrac{2}{5}+\dfrac{4}{5})$
=$48 +\dfrac{10+15-8+16}{20}$
=$48+(\dfrac{5}{4}+\dfrac{2}{5}$)
=$48+(1\dfrac{33}{20}$)
=$49\dfrac{33}{20}$

Question:

Solve: $13\dfrac{3}{4}+17\dfrac{2}{7}+31\dfrac{1}{4}+15\dfrac{5}{7}+12\dfrac {2}{3}$

=(13+17+31+15+12)+($\dfrac{3}{4}+\dfrac{2}{7}+\dfrac{1}{4}+\dfrac{5}{7}+ \dfrac{2}{3}$) =$88+(1+1+\dfrac{2}{3})$ =$90+\dfrac{2}{3}$=>$90\dfrac{2}{3}$

Question:

Solve :What will be value of c,if a(a+b+c)=85;
b(a+b+c)=96;
c(a+b+c)=108

Solution:

a(a+b+c)+b(a+b+c)+c(a+b+c)=85+96+108
(a+b+c)(a+b+c)=289
(a+b+c)^2=289
a+b+c=17
c(a+b+c)=108
c * 17 =108
c=$\dfrac{108}{17}$

Question:

What is value of ($13.8 \times 1.9 \div 5.7 +11.2 of \dfrac{1}{16}-\dfrac{1}{20}$) ?

Solution:

Using BODMAS,
step 1: 11.2 of $\dfrac{1}{16}=>11.2 \times \dfrac{1}{16}$=0.7
step 2:($13.8 \times 1.9 \div 5.7 +0.7-\dfrac{1}{20}$)
step 3:($13.8 \times \dfrac{1.9}{5.7}+0.7-0.05$)
step 4:$13.8 \times \dfrac{1}{3} +0.7-0.05$
step 5:$4.6+0.7-0.05=>5.25$

Question:

Solve: $\dfrac{(147*147*147)+(143*143*143)}{(147*147-147*143+143*143}$

Solution:

The above question is in the form of
=$\dfrac{a^3+b^3}{a^2-ab+b^2}$
=$\dfrac{(a+b)(a^2-ab+b^2)}{a^2-ab+b^2}$
=(a+b)
=147+143=290

Question:

Find the value of : 88888+ 8888+ 888+ 88+ 8=?

Solution:

88888 + 8888 + 888 + 88 + 8

 5           4         3       2      1    ->Frequency values (No of 8's present in every number from left to right is said to be frequency values)                                           carry Step 1: 5 * 8 = 40       0       4   
 Step 2: 4 * 8 = 32+4     6       3
 Step 3: 3 * 8 = 24+3     7       2
 Step 4: 2 * 8 = 16+2     8       1
 Step 5: 1 * 8 =   8+1    9  
 So the answer is 98760
 

Question:

Solve :1035/[(3/4) of (71+65)-$15\dfrac{3}{4}$]=?

solution:

Step 1:First,solve in the square bracket.so,
71+65=136
Step 2: 3/4 * 136 =102
Step 3:$102-\dfrac{60}{3}$
Step 4:102-20=82
Step 5:$\dfrac{1035}{82}$=12.6

Question:

Solve:$(7\times7)^3 /(49 \times 7)^3 \times (2401)^2$=7^x

Solution:

Step 1:We can rewritten as$(\dfrac{(7^{2})^3}{(7^{3})^3})^3 \times (7^{4})^2=7^ {x}$
Step 2:$(\dfrac{1}{7})^3\times (7^{4})^2=7^{x}$
Step 3:$7^{-3} \times 7^{8}=7^{x}$
Step 4:$7^{5}=7^{x}$
Therefore,x=5

Question:

Solve:$\sqrt{5 +\sqrt{11+\sqrt{19+\sqrt{29+\sqrt{49}}}}}$

Solution:

Step1:Solve from right to left
$\sqrt{5 +\sqrt{11+\sqrt{19+\sqrt{29+7}}}}$
Step 2:$\sqrt{5 +\sqrt{11+\sqrt{19+\sqrt{36}}}}$
Step 3:$\sqrt{5 +\sqrt{11+\sqrt{19+6}}}$
Step 4:$\sqrt{5 +\sqrt{11+\sqrt{25}}}$
Step 5:$\sqrt{5 +\sqrt{11+5}}$
Step 6:$\sqrt{5 +\sqrt{16}}$
Step 7:$\sqrt{5 +4}$
Step 8:$\sqrt{9}$=3

Question:

If a*b=2a+3b,then the value of 2 *3+3 *4 is ?

Solution:

Step 1:Put a=2 and b=4 in given expression,
=>2*2 + 3*4=13
Step 2:Put a=3 and b=4
=>2*3 + 4*4=18
Step 3:By adding 13+18 we get a*b=31