Answers # 6 to 10

question:6=> AB + CD = AAA, where AB and CD are two-digit numbers and AAA is a three digit number; A, B, C, and D are distinct positive integers. In the addition problem above, what is the value of C?

(A) 1

(B)3

(C) 7

(D) 9

(E) Cannot be determined
Ans: AB + CD = AAA

Since AB and CD are two digit numbers, then AAA must be 111

case even if ab and cd are the biggest numbers possible even than 96 + 87 = 183.

Therefore 1B + CD = 111

THATS WHY C=9, FOR ALL VALUES OF B.

PROOF: B can assume any value between 3 and 9

If B = 3, then CD = 111-13 = 98 and C = 9

If B = 9, then CD = 111-19 = 92 and C = 9

So for all B between 3 & 9, C = 9

Therefore the correct answer is D (C = 9)

QUESTION:7

1. A certain quantity of 40% solution is replaced with 25% solution such that the new concentration is 35%. What is the fraction of the solution that was replaced?

(A) 1/4

(B) 1/3

(C) 1/2

(D) 2/3

(E) ¾
Ans: Let X be the fraction of solution that is replaced.

Then X*25% + (1-X)*40% = 35%

Solving, you get X = 1/3

QUESTION:8

A bag contains 3 red, 4 black and 2 white balls. What is the probability of drawing a red and a white ball in two successive draws, each ball being put back after it is drawn?

(A) 2/27

(B) 1/9

(C) 1/3

(D) 4/27

(E) 2/9

Ans:

Case I: Red ball first and then white ball P1 = 3/9*2/9= 2/27

Case 2: White ball first and then red ballP2 = 2/9*3/9 = 2/27

Therefore total probability: p1 + p2 = 4/27

QUESTION:9

What is the least possible distance between a point on the circle x^2 + y^2 = 1 and a point on the line y = 3/4*x - 3?

A) 1.4

B) sqrt(2)

C) 1.7

D) sqrt(3)

E) 2.0

Ans:

The equation of the line will be 3x - 4y - 12 = 0.

This crosses the x and y axis at (0,-3) and (4,0)

The circle has the origin at the center and has a radius of 1 unit.

So it is closest to the given line when, a perpendicular is drawn to the line, which passes through the origin.This distance of the line from the origin is 12 / sqrt (9 + 16) which is 2.4
[Length of perpendicular from origin to line ax +by + c = 0 ismod (c / sqrt (a^2 + b^2))]
The radius is 1 unit.

So the shortest distance is 2.4 - 1 unit = 1.4 units

QUESTION:10

In the square above, 12w = 3x = 4y. What fractional part of the square is shaded?

A) 2/3

B) 14/25

C) 5/9

D) 11/25

E) 3/7

ANSWER:

Since 12w=3x=4y,

w:x=3:12=1:4 and x:y=4:3

so, w = 1x = 4y = 3

the fractional part of the square is shaded:

{(w+x)^2 - [(1/2)wx + (1/2)wx +(1/2)xy + (1/2)w(2w)]}/(w+x)^2

= {(w+x)^2 - [wx + (1/2)xy + w^2)]}/[(w+x)^2]=[(5^2) -(4+6+1)]/(5^2)

= (25 - 11)/25

= 14/25.