- 30% of the men are more than 25 years old and 80% of the men are less than or equal to 50 years old. 20% of all men play football. If 20% of the men above the age of 50 play football, what percentage of the football players are less than or equal to 50 years?
- 15%
- 20%
- 80%
- 70%
Correct Answer
80%. Choice (3)
Explanatory Answers
20% of the men are above the age of 50 years. 20% of these men play football. Therefore, 20% of 20% of 4% of the total men are football players above the age of 50 years.
20% of the men are football players. Therefore, 16% of the men are football players below the age of 50 years.
Therefore, the % of men who are football players and below the age of 50 = = 80%
Correct Answer (3)
2. A shepherd has 1 million sheeps at the beginning of Year 2000. The numbers grow by x% (x > 0) during the year. A famine hits his village in the next year and many of his sheeps die. The sheep population decreases by y% during 2001 and at the beginning of 2002 the shepherd finds that he is left with 1 million sheeps. Which of the following is correct?
- x > y
- y > x
- x = y
- Cannot be determined
Correct Answer
(x > y). Choice (1)
Explanatory Answers:
Let us assume the value of x to be 10%.
Therefore, the number of sheep in the herd at the beginning of year 2001 (end of 2000) will be 1 million + 10% of 1 million = 1.1 million
In 2001, the numbers decrease by y% and at the end of the year the number sheep in the herd = 1 million.
i.e., 0.1 million sheep have died in 2001.
In terms of the percentage of the number of sheep alive at the beginning of 2001, it will be (0.1/1.1)*100 % = 9.09%.
From the above illustration it is clear that x > y.
Correct Answer (1)
3. The Maximum Retail Price (MRP) of a product is 55% above its manufacturing cost. The product is sold through a retailer, who earns 23% profit on his purchase price. What is the profit percentage (expressed in nearest integer) for the manufacturer who sells his product to the retailer? The retailer gives 10% discount on MRP.
- 31%
- 22%
- 15%
- 13%
- 11%
Correct Answer Choice 4. The profit percentage is 13%.
Explanatory Answer
Key Data
- The MRP of the product is 55% above its manufacturing cost.
- The retailer sells the product after offering a discount of 10% on the MRP.
- The retailer makes a 23% profit on his purchase price.
Useful Assumption
In Profit Loss questions where all data is given in percentage terms, assume cost price to be 100.
Let the manufacturing cost = 100
The MRP of the product is 55% above its manufacturing cost
The MRP of the product = 100 + 55% of 100 = 155.
The retailer sells the product after offering a discount of 10% on the MRP
So, the retailer sells the product at 155 – 10% of 155 = 155 – 15.5 = 139.5
The retailer makes a 23% profit on his purchase price
Let the purchase price for the retailer be x.
So, the retailer sells the product at x + 23% of x = 123% of x.
Step to retailer sells the product @ 139.5 = 123% of x
1.23x=139.5
(or)x= 139.5/1.23
Therefore, x = 113.4
x is the purchase price for the retailer.
So, x has to be selling price for the manufacturer.
The manufacturer sold the product at 113.4.
Cost to the manufacturer is 100.
So, profit made by the manufacturer is 13.4.
Because we assumed the cost price to be 100, the manufacturer makes a 13.4% profit.
Rounded to the nearest integer, it is 13%
The correct answer is Choice 4.
4. A trader buys goods at a 19% discount on the label price. If he wants to make a profit of 20% after allowing a discount of 10%, by what % should his marked price be greater than the original label price?
- +8%
- -3.8%
- +33.33%
- None of these
Correct Answer –
8% profit. Choice (1)
Explanatory Answer
Let the label price be = Rs.100. The trader buys at a discount of 19%.
Hence, his cost = 100 – 19 = 81.
He wants to make a profit of 20%. Hence his selling price = 1.2 (81) = 97.2
However, he wants to get this Rs.97.2 after providing for a discount of 10%. i.e. he will be selling at 90% of his marked price.
Hence, his marked price M = = 108 which is 8% more than the original label price.
Correct answer choice (1).
5. On a certain sum of money, compound interest earned at the end of three years = Rs. 1456. Compound interest at the end of two years is Rs. 880. Compute the principal invested.
- 2,400
- 2,800
- 2,000
- 1,600
DETAILED SOLUTION
Let principal = P, rate of interest = r%
CI earned at the end of three years = P(1 + r)
3 – P = 1456
=> P(3r
2 + 3r + r
3) = 1456
CI earned at the end of two years = P(1 + r)
2 – P = 880
=> P(r
2 + 2r) = 880
Dividing one by the other we get: 3r2+3r+r3r2+2r=1456880
We can cancel ‘r’ and solve the resulting quadratic. However, let us see if we can spot something in the numbers.
1456880=728440
Remember that the amounts at the end of three years and two years are linked to (1 + r)
3and (1 + r)
2 respectively.
Now observe that 1728 = 12
3 and 144 is 12
2.
So,
728440=1728−10001440−1000=1.728−11.44−1
=>r=20%
CI at the end of 2 years = (1.2
2 – 1)P => 880 = 0.44P => P = Rs. 2000.
Alternatively:
(r
2 + 3r + 3) x 440 = (r + 2) x 728
(r
2 + 3r + 3) x 55 = (r + 2) x 91
55r
2 + 165r + 165 = 91r + 182
55r
2 + 74r -17 = 0
55r
2 + 85r – 11r – 17 = 0
5r (11r + 17) -1 (11r + 17) = 0
r = 0.2 or a negative number. Or, r has to be 20%.
Correct Answer: Rs. 2,000
6. A sum of money invested for a certain number of years at 8% p.a. simple interest grows to Rs.180. The same sum of money invested for the same number of years at 4% p.a. simple interest grows to Rs.120 only. For how many years was the sum invested?
A) 25 yrs
B) 15 yrs
C) 20 yrs
D) 22 yrs
Answer & Explanation
: A) 25yrs
Explanation:
From the information provided we know that,
Principal + 8% p.a. interest on principal for n years = 180 …….. (1)
Principal + 4% p.a. interest on principal for n years = 120 ……… (2)
Subtracting equation (2) from equation (1), we get
4% p.a. interest on principal for n years = Rs.60.
Now, we can substitute this value in equation (2),
i.e Principal + 60 = 120
= Principal = Rs.60.
We know that SI = , where p is the principal, n the number of years and r the rate percent of interest.
In equation (2), p = Rs.60, r = 4% p.a. and the simple interest = Rs.60.
Therefore, 60 =(60*n*4)/100
=> n = 100/4 = 25 years.
7. x, y, z are integer that are side of an obtuse-angled triangle. If xy = 4, find z.
- 2
- 3
- 1
- More than one possible value of z exists.
Correct Answer:- Choice (B). 3
DETAILED
SOLUTION
xy = 4
xy could be 2 x 2 or 4 x 1
221
222 These are the possible triangles.
223
441
22x will be a triangle if x is 1, 2 or 3 (trial and error).
44x is a triangle only if x is 1.
- 221 is acute. 12+ 22 > 22
- 222 is equilateral. So acute.
- 223 is obtuse. 22+ 22 < 32
- 144 is acute. 12+ 42 > 42
Only triangle 223 is obtuse. Hence, the third side has to be 3.
Correct Answer: 3
8. Two circles are placed in an equilateral triangle as shown in the figure. What is the ratio of the area of the smaller circle to that of the equilateral triangle?
- π : 36√3
- π : 18√3
- π : 27√3
- π : 42√3
DETAILED SOLUTION
In-radius of equilateral triangle of side a = a23√
Diameter of larger circle = a23√
Let us say common tangent PQ touches the two circle at R,center of smaller circle is I.
Now, PQ is parallel to BC. AR is perpendicular to PQ. Triangle PQR is also an equilateral triangle and AORID is a straight line. (Try to establish each of these observations. Just to maintain the rigour.)
AD= √3/2a
RD= a/√3
AR= √3/2a−a/√3
= 3a−2a/2√3=a2√3
AR = 13 AD.
Radius of smaller circle = 13 radius of larger circle
Radius of smaller circle = 1/3∗a/2√3=a/6√3
Area of smaller circle = πr
2
π(a/6√3)
2 = πa
2/108
Area of △ = (√3/4)a
2
Ratio = πa
2/108 : (√3/4)a
2
π : 273√. Answer choice (c)
Correct Answer: π : 27√3
