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# Kaprekar Contest (NMTC Sub-Junior 2018 - VII and VIII Grades) - Stage I- Problems and Solution

#### Part A

###### Problem 1

The fraction greater than $8 \frac{4}{9}$ is
(A) $8 \frac{1}{3}$
(B) $\frac{150}{18}$
(C) $8 \frac{2}{3}$
(D) $\frac{216}{27}$

###### Problem 2

A car is slowly driven in a road full of fog. The car passes a man who was walking at the rate of 3 $\mathrm{km}$ an hour in the same direction. He could see the car for 4 minutes and was visible for up to a distance of 100 meters. The speed of the car is (in $\mathrm{km}$ per hours)
(A) $4 \frac{1}{2}$
(B) $4$
(C) $3 \frac{1}{2}$
(D) $3$

###### Problem 3

Kiran sells pens at a profit of $20 \%$ for Rs. 60 . But due to lack of demand he reduced its price to Rs. 55. Then
(A) He gets a profit of $10 \%$
(B) He gets a profit of $12 \%$
(C) He incurs a loss of $10 \%$
(D) He incurs a loss of $8 \%$

###### Problem 4

If $40 \%$ of a number is added to another number then it becomes $125 \%$ of itself. The ratio of the second to the first number is
(A) $5: 8$
(B) $7: 5$
(C) $8: 5$
(D) None of these

###### Problem 5

The length of a rectangular sheet of paper is $33 \mathrm{~cm}$. It is rolled along its length into a cylinder so that width becomes height of the cylinder. The volume is 1386 cubic cms. The width of the rectangular sheet (in $\mathrm{cm}$ ) is
(A) 14
(B) 15
(C) 16
(D) 18

###### Problem 6

If $\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\ldots \ldots . .+\frac{1}{\mathrm{n} \times(\mathrm{n}+1)}=\frac{19}{20}$ then $\mathrm{n}=$
(A) 18
(B) 19
(C) 20
(D) 25

###### Problem 7

$a, b$ are natural numbers. If $9 a^2=12 a+96$ and $b^2=2 b+3$, the value of $2018(a+b)$ is
(A) 14226
(B) 14128
(C) 14126
(D) 14246

###### Problem 8

Shanti has three daughters. The average age of them is 15 years. Their ages are in the ratio $3: 5$ : 7. The age of the youngest daughter is (in years)
(A) 8
(B) 9
(C) 10
(D) 12

###### Problem 9

In the adjoining figure, $\mathrm{ABCD}$ is a quadrilateral. The bisectors of $\angle \mathrm{B}$ and the exterior angle at $\mathrm{D}$ meet at $\mathrm{P}$. Given $\angle \mathrm{C}=80^{\circ}, \angle \mathrm{ADC}=\frac{1}{2} \angle \mathrm{A}$ and $\angle \mathrm{A}=\angle \mathrm{C}+40^{\circ}$. Then $\angle \mathrm{DPB}$ is

(A) $50^{\circ}$
(B) $60^{\circ}$
(C) $70^{\circ}$
(D) $80^{\circ}$

###### Problem 10

The number of 3-digit number which contain 6 and 7 is
(A) 52
(B) 60
(C) 62
(D) 64

###### Problem 11

The difference between the biggest and the smallest three digit number each of which has different digits is
(A) 864
(B) 875
(C) 885
(D) 895

###### Problem 12

If $3 x+1=2 y-1=5 z+3=7 w+1=15$, the value of $6 x-3 y+5 z-8 w$ is
(A) 1
(B) 2
(C) 3
(D) None of these

###### Problem 13

Five years ago the average age of Aruna, Roy, David and salman is 45 years. Sita joins them now,. The average age of all the five now is 49 years. The present age of sita is (in years)
(A) 45
(B) 43
(C) 51
(D) 48

###### Problem 14

The fraction $\frac{B}{3 x-1}$ is subtracted from the fraction $\frac{A}{2 x+3}$. The resulting fraction is $\frac{-11}{(2 x+3)(3 x-1)}$. Then $A+B=$
(A) 11
(B) -11
(C) 5
(D) -5

###### Problem 15

There are some cows and ducks. The total number of legs is equal to 14 more than twice the number of heads. The number of cows is
(A) 5
(B) 6
(C) 7
(D) 8

###### Problem 16

The sum of $5 \%$ of a number and $9 \%$ another number is equal to sum of the $8 \%$ first number and $7 \%$ of the second number. The ratio between the numbers is
(A) $3: 2$
(B) $5: 7$
(C) $7: 9$
(D) $2: 3$

###### Problem 17

The length of two sides of an isosceles triangle are $8 \mathrm{~cm}$ and $14 \mathrm{~cm}$. The perimeter of the triangle (in $\mathrm{cm}$ ) is
(A) 30
(B) 36
(C) 19
(D) 30 or 36

###### Problem 18

There are three cell phones A, B, C. A is $50 \%$ costlier than C and B is $25 \%$ costlier than C.A is a $\%$ costlier than $\mathrm{B}$. Then $a=$
(A) 25
(B) 20
(C) 15
(D) 10

###### Problem 19

Sushant wrote a two digit number. He added 5 to the tens digit and subtracted 3 from the unit digit of the number and got a number equal to twice the original number. The original number is
(A) 47
(B) 74
(C) 37
(D) 73

###### Problem 20

The units digit of $5^{2018}-3^{2018}$ is
(A) 5
(B) 6
(C) 7
(D) 4

#### Part B

###### Problem 21

The smallest natural number that has to be added to 803642 to get a number which is divisible by 9 is $\rule{2cm}{0.15mm}$.

###### Problem 22

The greatest two digit number that will divided 398, 436, and 542 leaving respectively 7, 11 and 15 as remainders is $\rule{2cm}{0.15mm}$.

###### Problem 23

$\frac{2}{3}$ is $\rule{2cm}{0.15mm}$ of $\frac{1}{3}$.

###### Problem 24

The sum of 5 positive integers is 280. The average of the first 2 number is 40. The average of the third and fourth number is 60. The fifth number is $\rule{2cm}{0.15mm}$.

###### Problem 25

If $a: b=3: 4$ and $\frac{p}{q}=\frac{a^2+b^2+a b}{a^2+b^2-a b}$, where $p$, $q$ have no common divisors other than $1, p+q$ is $\rule{2cm}{0.15mm}$.

###### Problem 26

$a$ is a natural number such that a has exactly two divisors and $(a+1)$ has exactly three divisors. The number of divisors of $a+2$ is $\rule{2cm}{0.15mm}$.

###### Problem 27

The first term of a series is $\frac{2}{5}$. If $x$ is a term of this series, the next term is $\frac{1-x}{1+x}$. If $t_n$ denotes the $n$ th term and $t_{2018}-t_{2017}=\frac{p}{q}$, where $p$, $q$ are integers having no common factors other than $1, p+q$ is $\rule{2cm}{0.15mm}$.

###### Problem 28

In the adjoining figure, the side of the square is $\sqrt{\frac{2018}{\pi}} \mathrm{cm}$. The area of the unshaded region is $\left(\frac{\pi-2}{\pi}\right)$ A sq. cms. The value of $A$ is $\rule{2cm}{0.15mm}$.

###### Problem 29

$n$ is a natural number. The square root of the sum of the square of $n$ and 19 is equal to the next natural number to $n$. The value of $n$ is $\rule{2cm}{0.15mm}$.

###### Problem 30

Using only the digits 1,2, 4, 5, two- digit numbers are formed. The digits of the two digit number may be the same or different. The number of such two-digit number is $\rule{2cm}{0.15mm}$.

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