# Amcat model questions - 4

1. Manav drives from his home to the nearest metro station at an average speed of 40km. From station, he boards a train that takes him to his office at a speed of 80km. the entire distance covered by him is 52km and the entire journey took him 1 hour. Find the distance between the station from where he boards the train and his office.
Explanation:
Let the distance from his home to station = x
Therefore, $\dfrac{x}{{40}} + \dfrac{{52 - x}}{{80}} = 1$
$\Rightarrow \dfrac{{2x + 52 - x}}{{80}} = 1$
x + 52 = 80
x = 28
Hence, the distance from office to station = 52 - x = 52 - 28 = 24 km

2.  Santa Singh was driving a car at a uniform speed. He saw a milestone showing number x0y. After travelling for an hour, he saw another milestone showing number yx. After another hour, he saw the next milestone showing the same two digits, but in the reverse order, i.e. xy.  What was the average speed of the car?
Explanation:
The first milestone xoy in the decimal format = 100x + y
Second mile stone yx in the decimal format = 10y + x
Third mile stone xy in the decimal format = 10x + y
As the speed of the car is constant, difference between the two consecutive mile stones should be constant.
Therefore, 100x + y - (10y + x) = (10y + x) - (10x + y)
99x - 9y = 9y - 9x
11x - y = y - x
12x = 2y
$\dfrac{x}{y} = \dfrac{1}{6}$
So the first mile stone = 106 and second mile stone = 61.  Therefore, speed of the car = 106 - 61 = 45 kmph

3.   If a and b are natural numbers and a-b is divisible by 3, then a3-b3 is divisible by:
1. 3 but not by 9
2. 9
3. 6
4. 27
Explanation: ${a^3} - {b^3} = {(a - b)^3} + 3ab(a - b)$
Given that a - b is divisible by 3. So let a - b = 3k
Then, ${(3k)^3} + 3ab(3k)$ = $27{k^3} + 9abk$
So the given expression is divisible by 9.

4.  A number becomes a perfect square when we subtract 1 from it.which of the following cannot be the last digit of that number
1. 2
2. 4
3. 5
4. 0
Explanation:
Square of any number should end with 0, 1, 4, 5, 6, 9 only.
So the given number should end with, 1, 2, 5, 6, 7, 0. It cannot be 4.

5.  20 Men can do a job in 10 days, working 8 hrs a day if women is 33.33% more efficient than men. How many women will it take to finish the same job in 10 days working 6 hrs a day?
1) 10
2) 12
3) 15
4) 16
5) 20
Explanation:
Let the capacity of each man = 3 units/hour, then capacity of woman = 4 units/hour.  ($\because$ woman is 33.33% more efficient than man)
Total work done by man = 20 × 8 × 10 × 3 = 4800 units.
Now Let $x$ woman required to complete the job. Total work done by $x$ woman = $x$ × 4 × 6 × 10 = 240x
Equating the work, 240x = 4800 ⇒ x = 20.

6.  Sujitha invests 7% i.e. 2170, of her monthly salary in mutual funds.  Later she invests 18% of her monthly salary in recurring deposits, Also she invests 6% of her salary on NSC's . What is the total annual income invested by Sujitha ?
Explanation:
Let her monthly salary = x  and she invests 7% of her salary in mutual funds which equal to Rs.2170
Therefore, 7%(x) = 2170 ⇒ x = 31000
Her total investments per month =  7%+18%+6% =31% of her monthly salary.
So per month she invests =  31% (31000) = Rs.9610
Her total investments per annum = 9610 × 12 = 1,15,320

7.  Ishan spent 35,645 on buying a Bike, 24,355 on buying a Television ,and the remaining 20% of the total amount he had as cash with him. What was the total amount ?
Explanation:
He has 20% left. So he spent 80% of his money.
⇒80% (x) = 35,645 + 24,355 = 60,000
⇒ x = 75,000

8.  Manoj sold an article for 15,000. Had he offered a discount of 10 % on the selling price, he would have earned a profit of 8 %. What is the Cost price?
Explanation:
If he offered 10%, then new selling price = 90% (15000) = Rs.13500
At this selling price, he got a profit of 8%.
So cost price = $\dfrac{{13500}}{{108\% }}$ = Rs.12,500.

9.  A man bought oranges at the rate of 8 for Rs.34 and sold them at the rate of 12 for Rs.57.  How many oranges should be sold to earn a net profit of Rs.45 ?
Explanation:
To solve these type of questions, equate oranges in both cases. Let us say, he bought 24 oranges.
Then, his cost price = Rs.102
Selling price = Rs.114
Profit = Rs.12.
So while selling 24 oranges he made a profit of Rs.12. i.e., For each rupee, he has to sell 2 oranges.
To make a profit of Rs.45, he has to sell 90 oranges.

10. Naresh purchased a TV set for Rs.11,250 after getting discount of 10% on the labeled price. He spent Rs.150 on transport and Rs.800 on installation. At what price should it be sold so that the profit earned would be 15% ?
Explanation:
Here labelled price is irrelevant.  His total costprice = 11,250 + 150 + 800 = 12,200
Selling price to get 15% profit = 115% (12200) = 14030