## Approximations Part 1

1)49.6*5.08 + 485.987 +42.6+5.3 = ?

1. 740      2.780       3.720    4.760   5.800

50 * 5  +  486  +  $\frac{40}{5}$

250  +  486  +  8 = 746 it is near to 740

2)$(64.15)^{2}-(17.90)^{2}-(21.9)^{2}=?$

1.3600  2.3900  3.3300  4.4000   5.3000

$64^{2}-18^{2}-22^{2}$

4096 – 324  – 484 = 3288 it is near to 3300

3)$(\sqrt{900.021}-\sqrt{255.989})^{2}\div&space;(\sqrt{575.989}-\sqrt{289.006})^{2}=?$

$(\sqrt{900}-\sqrt{256})^{2}\div&space;(\sqrt{576}-\sqrt{289})^{2}$

$(30-16)^{2}\div&space;(24-17)^{2}$

$\frac{14\times&space;14}{7\times&space;7}=4$

4)9.09% of 1334 + 14.30% of 2400 =?

1.486  2.465  3.450  4.430  5.418

9.09% of 1334 +14.28% of 2400

$\frac{1}{11}\times&space;1334+\frac{1}{7}\times&space;2400$

121+343 = 464 it is near to 465

5)$33\frac{1}{3}$% of 768.9 + 25% of 161.2 – 58.12 = ?

1.250  2.220  3.230  4.200  5.240

$\frac{1}{3}\times&space;768+\frac{1}{4}\times&space;160-58$

256 + 40 -58 =238 which is near to 240

## Simplifications Part 2

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

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

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

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

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

## Number Series Part 4

Problem 1: 3   20   88   273   ?   559

3 * 5 + 5 = 20     (5=5*1)

20 * 4 + 8 = 88  (8=4*2)

88 * 3 + 9 = 273 (9=3*3)

273*2 + 2*4 = 554

554*1 + 1*5 = 559

Problem 2: ?   48  96   240   720   2520

48 * 2 = 96

96 * 2.5 = 240

240 * 3 = 720

720 * 3.5 = 2520

1.5=$\frac{3}{2}$

$\frac{48\times&space;2}{3}=32$

Problem 3: 1   ?   6   21   88   445

6 * 3 +3 = 21

21 * 4 + 4 = 88

88 * 5 + 5 = 445

1 * 1 + 1 = 2

2 * 2 + 2 = 6

Problem 4: 6   4   5   11   ?   361

6 * 0.5 + 1 = 4

4 * 1 + 1 = 5

5 * 2 + 1 = 11

11 * 4 + 1 =45

45 * 8 + 1 =361

Problem 5: 180   179  183   156   172   ?

180 – 179 = -1 = $1^{3}$

179 -183 = +4 = $2^{2}$

183 – 156 = -27 = $3^{3}$

156 – 172 = +16 = $4^{2}$

$-5^{3}=125$

172 – 125 = 47

## Number Series Part 3

Problem 1: 4   6   14   44   ?   892

4 * 1 + 2 = 6

6 * 2 + 2 = 14

14 * 3 + 2 = 44

44 * 4 + 2 = 178

Problem 2: 7   3   2   2   4   ?

7 * 0.5 – 0.5 = 3

3 * 1 -1 = 2

2 * 2 – 2 = 2

2 * 4 – 4 = 4

4 * 8 – 8 = 24

Problem 3: 5   6   14   23   87   ?

5 – 6 = 1 = $1^{2}$

6 – 14 = 8 = $2^{3}$

14 – 23 = 9 = $3^{2}$

87 – 23 = 64 = $4^{3}$

$5^{2}=25$

87+25 = 112

Problem 4: 5   6.2   9   13.4   19.4   ?

5 – 6.2 = 1.2

9 – 6.2 = 2.8

13.4 – 9 = 4.4

Second diference

2.8 – 1.2 = 1.6

4.4 – 2.8 = 1.6

1.6+4.4 = 6

1.6 + 6 + 19.4 = 27

Problem 5: 3   2   2   3   8   35   ?

3 * 1 – 1 = 2

2 * 2 – 2 = 2

2 * 3 – 3 = 3

3 * 4 – 4 = 8

8 * 5 – 5 =35

35 * 6 – 6 = 204