Question 1 :
If x and y are integers, which is always even ?
(A) (x + y)/2 (B) 2(x + y)/x (C) x - y
(D) 2(x + y) (E) x^{2} + y^{2}
Solution :
Since x and y are integers, the sum of those integers will be odd or even. The multiple of 2 will not be odd. So the answer is 2 (x + y).
Question 2 :
What is the probability that a point chosen will be in the shaded region ?
(A) y^{2}/x^{2 }(B) (x + y)^{2}/y^{2 }(C) x^{2}/(y^{2}-x^{2})
(D) (y^{2}-x^{2})/y^{2 }(E) π
Solution :
Sample space = n(S) = πy^{2}
Let "a" be the event that a point chosen in a shaded region.
Area of shaded region = πy^{2 }- πx^{2}
n(A) = π(y^{2 }- x^{2})
P(A) = n (A) / n (S)
= π(y^{2 }- x^{2})/πy^{2}
= (y^{2 }- x^{2})/y^{2}
Let us look into the next example on "PSAT Practice Questions in Math".
Question 3 :
If q = x + y and x = y + z, what is z in terms of y and q?
(A) q - 2x (B) q - 2y (C) 2x - q (D) 2y + q (E) 2q + x
Solution :
q = x + y -----(1)
x = y + z ------(2)
z = x - y
From the first equation, let us find the value of x in terms of y and q
x = q - y
z = (q - y) - y
z = q - y - y
z = q - 2y
Question 4 :
What is (5 !)!/ 5! ?
(A) 1 (B) 5 (C) 120 (D) 720 (E) None of these
Solution :
Let x be 5 !
(5 !)!/ 5! = x!/x
= x (x- 1)!/x
= (x - 1)!
= (5! - 1)!
= (120 - 1)!
= 119!
Hence none of the above is the answer.
Question 5 :
What is (a^{2} + 2ab + b^{2})/(a + b)^{3} ?
(A) a + b (B) a^{2} + b^{2} (C) 1/(a + b)
(D) 1/(a+ b)^{2 }(E) (a + b)^{2}
Solution :
(a^{2} + 2ab + b^{2})/(a + b)^{3}
= (a + b)^{2}/(a + b)^{3}
= 1/(a + b)
Question 6 :
It takes 3 cats 3 minutes to catch 3 mice. How many cats are needed to catch 99 mice in 99 minutes ?
(A) 3 (B) 6 (C) 11 (D) 33 (E) 99
Solution :
Given : It takes 3 cats 3 minutes to catch 3 mice.
That is,
3 cats -----> 3 minutes -----> 3 mice
Let us assume,
3 cats = 1 man
Then, we have
1 man -----> 3 minutes -----> 3 mice
Based on the above statement, we can consider the following situations also.
1 man -----> 1 minute -----> 1 mouse
1 man -----> 5 minutes -----> 5 mice
1 man -----> 10 minutes -----> 10 mice
In this way, we can have
1 man -----> 99 minutes -----> 99 mice
But, we know that,
1 man = 3 cats
Then, we have
3 cats -----> 99 minutes -----> 99 mice
Therefore, 3 cats are needed to catch 99 mice in 99 minutes.
Question 7 :
The sum of seven consecutive odd integers is 749. What is the largest of the seven integers ?
(A) 99 (B) 103 (C) 11 (D) 113 (E) 115
Solution :
Let x be the odd integer.
6 consecutive odd integers will be (x+2), (x+4), (x+6), (x+8), (x+10) and (x+12).
Sum of seven consecutive integers = 749
x + x+2 + x+4 + x+6 + x+8 + x+10 + x+12 = 749
7x + 42 = 749
7x = 749 - 42
7x = 707
x = 707/7 ==> 101
Largest number of seven consecutive odd numbers
= x + 12
= 101 + 12
= 113
Question 8 :
V = πr^{2}h Using the formula, if r is doubled and h is divided by 2, what is the ratio of the original volume to the new volume ?
(A) 1 : 4 (B) 1 : 2 (C) 1 : 1 (D) 2 : 1 (E) 4 : 1
Solution :
r is doubled means r = 2r
h is divided by 2 means h = h/2
New volume :
V_{1} = π(2r)^{2} (h/2)
V_{1 }= π(4r^{2}) (h/2)
V_{1} = 2πr^{2} h
Ratio of the original volume to the new volume :
= V : V_{1}
= πr^{2} h : 2πr^{2} h
= 1 : 2
Hence the required ratio is 1 : 2.
Question 9 :
A yellow cab has the base fare of $3.50 per ride plus $0.20 for each 1/4 of mile ridden. If a yellow cab costs $ 22.50, how many miles long was the ride ?
(A) 23.75 miles (B) 42.5 miles (C) 47.5 miles
(D) 95 miles (E) 112.5 miles
Solution :
Let "x" be the number of miles ridden
To find the (1/4)^{th} miles of x miles, we have to multiply 4 by x.
So, there are 4x (1/4)^{th} miles in x miles.
To understand this, let us consider the following example in the picture given below.
Cost of yellow cab = $22.50
3.50 + 0.20 (4x) = 22.50
0.80x = 22.50 - 3.50
0.80x = 19
(80x/100) = 19
x = 19(100)/80
x = 23.75 miles
Hence he traveled 23.75 miles.
Question 10 :
John works 40 hours a week, and his monthly salary in June was $4000. In the month July, John got 4% raise on his monthly salary. In the month of July, what was John's hourly rate ?
(A) $25 (B) $26 (C) $40 (D) $100 (E) $104
Solution :
June month salary of John = $4000
Percentage of salary raised = 4%
New salary in the month July = 4000 + 4% of 4000
= 4000 + 160
= 4160
John's 1 week salary = 4160/4 = 1040
He works 40 hours per week.
Hourly rate for the month July = 1040/40 = $26
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