## Simplifications Part 2

Problem 1: $13\frac{5}{7}+18\frac{3}{14}-16\frac{7}{8}+26\frac{5}{28}-22\frac{13}{56}=?$

First, add individual numbers

13+18-16+26-22 = 19

$\frac{40+12-49+10-13}{56}=\frac{0}{56}=0$

19+0 =19

Problem 2: $\frac{1}{(11.56)^{2}}\times&space;(39.304)^{2}\div&space;\frac{1}{133.6336}=3.4^{?}$

$(11.56)^{2}=((3.4)^{2})^{2}=(3.4)^{4}$

$(39.304)^{2}=((3.4)^{3})^{2}=(3.4)^{6}$

$133.6336=(3.4)^{4}$

-4+6-4 = 6

Problem 3: $(7)^{2}+(11)^{2}-(9)^{2}+(12)^{2}-(14)^{2}+(16)^{2}=?$

49+121-81+144-196+256 = 293

Problem 4: $3\frac{1}{2}\times&space;5\frac{2}{3}\div&space;12\frac{3}{4}\times&space;5\frac{2}{5}=?$

$\frac{7}{2}\times&space;\frac{17}{3}\times&space;\frac{4}{51}\times&space;\frac{27}{5}=\frac{42}{5}=8.4$

## Simplifications Part 1

Problem 1: $112\frac{1}{2}$% of $12\frac{1}{2}$% of $116\frac{2}{3}$% of 12.8 = ?

$12\frac{1}{2}=\frac{1}{8}$

100% = 1

112$\frac{1}{2}=1+\frac{1}{8}=\frac{9}{8}$

100+$16\frac{2}{3}=1+\frac{1}{6}=\frac{7}{6}$

$\frac{9}{8}\times&space;\frac{1}{8}\times&space;\frac{7}{6}\times&space;12.8=x$

x = 2.1

Problem 2: $(81\times&space;9)^{3}\div&space;(9)^{5}\times&space;(3\times&space;27)^{2}=(9)^{?}$

Convert all terms into 9th power

$(81\times&space;9)^{3}=9^{9}$

$(3\times&space;27)^{2}=9^{4}$

9 – 5 + 4 = 8

Problem 3: $\frac{?}{18}of\sqrt{361}=\frac{171}{?}of\sqrt[3]{5832}$

$\sqrt{361}$= 19

Square ends  with 1 is 1 and 9

Delete one’s place and tens place

The number 3 is between $1^{2}$ and $2^{2}$

$\sqrt[3]{5832}=$ 18

Cube ends with 2 is 8

Delete ones, tens, hundred place

5 is between $1^{3}$ and  $2^{3}$

Select the least number i.e 1 so answer is 18

$\frac{x}{18}\times&space;9=\frac{171}{x}\times&space;18$

$x^{2}=9\times&space;18\times&space;18$

$x=&space;3\times&space;18=54$

## Simplifications Part 3

#### Problem 1: Simplify (33076161)^⅓ + (279841)^¼ + (243)^⅕ = ?

(33076161)^1/3 + (279841)^1/4 + (243)^1/5 = ?

Given options are

• 344
• 345
• 346
• 347
• None of these

The solution involves four steps

• Finding the cube root of the number 33076161
• Finding the fourth root of 279841
• Find the fifth root of 243
• Adding all the three numbers
1. Finding the cube root of the number 33076161

TP SP FP – We will be finding the numbers in these three places as below.

• Divide the number into three parts
• First number (FN) : 33
• Second number (SN): 076
• The third number (TN): 161

To find the cube root, we will use TN, then FN and finally SN.

• TN ends with 1, so the number in the one’s place (FP) is 1. (Refer Table 1 below)

TP SP 1

• FN is 33, which in between the cube of 3 and 4, so we have to choose the small number which is 3 will be in third place (TP). (Refer table 1 below)

3 SP 1

• To find the number in second place, we can use the formula – 3*(TP^2)*SP= second digit of (TN-FP)
• 3*(1^2)*SP= Second digit of (161-1) = Second digit of 160 = 6
• 3*(1^2)*SP=6
• SP=6/3=2

3 2 1

• Finally, the cube root of (33076161)⅓ is 321

2. Finding the fourth root of 279841

• First, find the square root of the number 279841
• The square root of 279841 is 529
• Again find the square root of 529
• The square root of 529 is 23
• (23^4)¼=23

3. Find the fifth root of 243

• Rewrite 243 as 3^5
• Cancel the common factor of 5
• And the answer is 3

4. Then, the final step is adding all the three numbers 321+23+3=347

Table 1: Square, Cube, Numbers Ranging 0 – 10

 Number x Square x2 Cube x3 1 1 1 2 4 8 3 9 27 4 16 64 5 25 125 6 36 216 7 49 343 8 64 512 9 81 729 10 100 1000

#### Problem2:  55555+5555+555+55+5+=?

To find the solution in an easy way first we have to count the no.of digits in a number and we have to choose the highest digit number

Step1: In this number 55555, the no.of digits are 5

5555=4

555=3

55=2

5=1

Step 2: Multiply as shown in the below

5*5=25 here 5 is the number and 2 is carried

5*4=20 add the previous carry to the number 20 then 20+2=22 2 is the number and 2 is carried

5*3=15 add the previous carry to the number 15 then 15+2=17 7 is the number and 1 is carried

5*2=10 add the previous carry to the number 10 then 10+1=11 1 is the number and 1 is carried

5*1=5 add the previous carry to the number 10 then 5+1=6 so 6 is the number

And finally, the answer is 61725.

 Number digit* no.of digits sum Carry Forward (CF) sum+CF Final Digit 55555 5*5 25 0 25 5 5555 5*4 20 2 22 2 555 5*3 15 2 17 7 55 5*2 10 1 11 1 5 5*1 5 1 6 6

Final Digit = second digit of (sum+CF)

Carry Forward = first digit of previous (sum+CF)

So the Final sum is 61725

#### Problem 3: 0.4444+0.444+0.44+0.4=?

To find the solution for this there is a shortcut which is similar to the above problem

Here we have to choose the smallest digit number

The smallest digit is 0.4 the no.of digits are 1

0.44=2

0.444=3

0.4444=4

Step 2: multiply as shown in the below

4*1=4

4*2=8

4*3=12 here 2 is the number and 1 is carried

4*4=16 add the previous carry to the number 16 then 16+1=17

And finally, we get 17284

After four digits there is a decimal point so the answer is 1.7284

 Number digit* no.of digits sum Carry Forward (CF) sum+CF Final Digit 0.4 4*1 4 0 4 4 0.44 4*2 8 0 8 8 0.444 4*3 12 1 2 2 0.4444 4*4 16 1 17 17

Final Digit = Second digit of (sum+CF)

Carry Forward = First digit of previous (sum+CF)

So the Final sum is 1.7284

#### Problem 4: 3333.666+33.6666+333.66+3.6=?

This can be divided into two parts

(3333+33+333+3)+(0.666+0.6666+0.66+0.6)

4            2      3      1          3             4             2        1

Part-1 and 2 can solve by using the above method

Choose highest digit number choose the lowest digit number

3*4=12 2 is number 1 is carry 6*1=6

3*3=9 9+1=10 0 is number 1 is carry 6*2=12 2 is number and 1 is carry

3*2=6 6+1=7 6*3=18 18+1=19 9 is the number 1 is carry

3*1=3 6*4=24 24+1=25

Ans is 3702

ans is 2.5926

3702+2.5926=3704.5926

Trick: 61+16

When it is addition we have to multiply with 11

11*(1+6) or 11*(6+1)

11*7=77

Trick: 76-67

When it is subtraction we have to multiply with 9

9(7-6)

9*1=9

Problem: 53+35-56-65+63+36-98-89=?

53+35-(56+65)+63+36-(98+89)

11(8-11+9-17)

11*-11

-121

Problem: 53-35+83-38+68-86=?

9(2+5-2)

9*5=45

Trick: 65*75

1.7+1=8

2.6*8=48

3.difference between 6 and 7 is 1

4.last digit is 25

5.4825+1*50

=4825+50

=4875

Example: 25*95

9+1=10

2*10=20

2025

Difference between 2 and 9 is 7

2025+7*50

2025+350

2375