Gauss Contest (NMTC PRIMARY LEVEL- V and VI Grades) - Problems and Solution

Problem 1

The value of $\frac{999 \times 999 \times 999}{(999+999) \times 111 \times 111}$ , when simplified, is

a) $\frac{999}{2}$
b) $\frac{111}{3}$
c) $\frac{81}{2}$
d) 1000

Problem 2

When $4 \frac{1}{2}$ is divided by $3 \frac{1}{4}$ , the result is $x$ . When $3 \frac{3}{4}$ is divided by $2 \frac{1}{8}$ , the result is $y$ . Then the numerical value of $13 x+17 y$ is

a) 38
b) 40
c) 39
d) 48

Problem 3

There are 5 cards numbered as shown In the figure. The number of ways in which one can choose 3 or less cards which contain only odd numbers is

a) 8
b) 7
c) 3
d) 21

Problem 4

$A B C$ is a triangle. The bisector of $\angle C$ meets $A B$ at $D$ . The bisectors of $\angle A$ and $\angle B D C$ meet at $E$ .
Then the measure of $\angle A E D$ is

a) $\frac{\angle C}{2}$
b) $\frac{\angle C}{3}$
c) $\frac{\angle C}{4}$
d) $\frac{\angle C}{5}$

Problem 5

Samrud secures $20 \%$ of the marks but fails by 30 marks. Saket gets $32 \%$ marks which is 42 marks more than the minimum pass marks. The maximum marks in the test would be

a) 100
b) 200
c) 300
d) 600

Problem 6

The greatest number that divides 25,73 and 97 to leave the same remainder is

a) 19
b) 22
c) 24
d) 37

Problem 7

If $\frac{19}{7}=a+\frac{2}{a+\frac{b}{c}}$ where $a, b, c$ are natural numbers, then the numerical value of $(a+b+c)$ is

a) 8
b) 13
c) 12
d) 9

Problem 8

In the adjoining figure, two equilateral triangles $A B P, C D P$ are placed such that $A B$ is parallel to $C D$ . If $A B=3 \mathrm{~cm}$ , $C D=1 \mathrm{~cm}$ , the area $\left(\right.$ in $\left.\mathrm{cm}^2\right)$ of trapezium $A B C D$ is

a) $\sqrt{3}$
b) $\frac{2}{\sqrt{3}}$
c) $2 \sqrt{3}$
d) $4 \sqrt{3}$

Problem 9

Siva found the average of 5 numbers. He got an answer 40 which is wrong because, while listing, instead of writing the number 43 he wrote 48 . The correct average must be

a) 38
b) 39
c) 41
d) 31

Problem 10

In the adjoining figure, $A B C D$ is a square. Also $B E=B F$ . Then the value of $2 x$ (in degrees) is

a) 210
b) 220
c) 215
d) 225

Problem 11

If the numerator and diameter of a fraction are increased by $20 \%$ and $30 \%$ respectively, then the fraction becomes $\frac{9}{13}$ . If the original fraction is $\frac{p}{q}$ , where $p$ and $q$ have no common factors, then $p+q$ is

a) 3
b) 6
c) 7
d) 9

Problem 12

Gita divided 360 into 4 parts such that twice the first part, thrice the second part, five times the third part and six times the fourth part are all equal. Then the difference between the third and fourth parts is

a) 20
b) 15
c) 10
d) 7

Problem 13

In the adjoining figure, the degree measure of $\angle F A B$ is

a) $95^{\circ}$
b) $105^{\circ}$
c) $115^{\circ}$
d) $125^{\circ}$

Problem 14

Consider the following sequence:

$1,3,5,7,9,7,5,3,1,3,5,7,9,7,5,3,1,3,5,7,9,7,5,3,1, \ldots \ldots \ldots$

The digit in the $2023^{\text {rd }}$ place is

a) 3
b) 5
c) 7
d) 1

Problem 15

In the two figures, there is a pattern of numbers which are same. Then the number in the head of the second figure,

a) 6
b) 13
c) 8
d) 10

Problem 16

There are two cars $C_1$ and $C_2$ . The speed of $C_1$ is $20 \%$ less than that of $C_2$ . They travel a certain equal distance. The percentage of time does $\mathrm{C}_1$ need to travel than $\mathrm{C}_2$ is $x \%$ . Then $x$ is $=$ $\rule{1cm}{0.15mm}$ .

Problem 17

Six equal unit squares are arranged in different shapes as shown in the diagram below:

In diagram (1), the perimeter is $A B C D$ , which equals 10. Similarly the perimeters of the other shapes also are found out. Let the perimeters be denoted by $P_1, P_2, P_3$ and $P_4$ .
Then the value of $\left(P_1+P_4\right)-\left(P_2+P_3\right)$ is $\rule{1cm}{0.15mm}$ .

Problem 18

In a two digit number, the digit in the tens place is twice the digit in the units place. If we swap the places of these two digits, a new two-digit number is formed. The sum of these two numbers is 132 . The original number is $\rule{1cm}{0.15mm}$ .

Problem 19

An ant starts from $A$ and wants to go to $D$ . It is allowed to go along the lines and pass a line and a point only once. The number of different routes that it can take to go from $A$ to $D$ is $\rule{1cm}{0.15mm}$ .

Problem 20

Three consecutive natural numbers are taken from 1 to 6 . With these three numbers, three digit numbers are formed. The total number of such 3-digit numbers is $\rule{1cm}{0.15mm}$ .

Problem 21

In the adjoining figure,

$\angle B A C=20^{\circ}, \angle B C A=10^{\circ} \text {, }$
$\angle A C D=90^{\circ}$ and $\angle C D B=55^{\circ}$ .
If $\angle A B D=x^{\circ}$ , then $x=$ $\rule{1cm}{0.15mm}$ .

Problem 22

The number of 5-digit numbers of the form $34 a 5 b$ (where $a, b$ are digits), each of which is divisible by 36 is $\rule{1cm}{0.15mm}$ .

Problem 23

The units digit of the sum of all 2-digit numbers is $\rule{1cm}{0.15mm}$ .

Problem 24

A natural number is taken. One sixth of this number is subtracted from it. From the resulting number, half of the number is taken and from this number one fifth is taken. If the resulting number is 3 , then the original number taken is $\rule{1cm}{0.15mm}$ .

Problem 25

The least number that is added to 2716321 to make it exactly divisible by 3456 is $\rule{1cm}{0.15mm}$ .