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Number System: Level 3 Test -4

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Question 1
Sam has forgotten his friend’s seven-digit telephone number. He remembers the following: the first three digits are either 635 or 674, the number is odd, and the number 9 appears once. If Sam were to use a trial and error process to reach his friend, what is the minimum number of trials he has to make before he can be certain to succeed?
A
10,000
B
2,430
C
3,402
D
3,006
Question 1 Explanation: 
In this type of question, one thing is very clear that there we have three positions reserved for 635,674.
. Now we are given that the number is odds, in the end there would be 1,3,5,7,9.
For nine we have a condition here that the position of nine appears once.
It’s clear that we have two cases where the number nine will either come at the end or it will come at the 4,5,6th position.
For the first case, the number would look like: 635 ...9/674....9 In both these cases,
the blanks can be occupied by any of the available 9 digits (0, 1, 2, ..., 8).
Thus, total possible numbers would be 2 × (9 × 9 × 9) = 1458. For the second case, the number 9 can occupy any of the given position 4, 5, or 6,
and there shall be an odd number at position 7.
Thus, the total number of ways shall be 2[3(9 × 9 × 4)] = 1944. Hence,  the correct answer is the sum total of the two: 3402.
Question 2
A lord got an order from a garment manufacturer for 480 Denim Shirts. He brought 12 sewing machines and appointed some expert tailors to do the job. However, many didn’t report to duty. As a result, each of those who did, had to stitch 32 more shirts than originally planned by A lord, with equal distribution of work. How many tailors had been appointed earlier and how many had not reported for work?
A
12, 4
B
10, 3
C
10, 4
D
None of these
Question 2 Explanation: 
The problem is that if we want to do this question with the help of an equation
then we will fail to understand the problem.
The best way to solve this question is by considering options so read carefully. 
Get the logic between question and the solution: if the lord had appointed 10 tailors,
each one would have had to do 48 shirts.
Now if 4 of them don’t report to the duty, each one of them would have to do 80,
which is 32 more than originally planned per head.
Hence, we solve this question with the help of options.
Question 3
Let x < 0.50, 0 < y < 1, z > 1. Given a set of numbers, the middle number, when they are arranged in ascending order, is called the median. So the median of the numbers x, y, and z would be
A
less than one
B
between 0 and 1
C
greater than 1
D
cannot say
Question 3 Explanation: 
We are given two numbers which are less than one and since they are less than one, it is clear that the median will be less than one.
Hence (c) cannot be the answer. As we said above,
we know that there are two numbers which are < 1 (viz. x & y),
it is obvious that the median will be less than 1 and extending the logic, we have another number which is 0.5.
So by plain and simple logic, the median of these number has to resided between) and 1.
It cannot be greater than 1 as then it will shift to z territory (that is the third number) and the median has to be the middle number.
The median cannot be less than one as in that case it will shift to x territory (that is the first number).
Hence, the number has to lie between 0 and 1.
Question 4
In Sivakasi, each boy’s quota of match sticks to fill into boxes is not more than 200 per session. If he reduces the number of sticks per box by 25, he can fill 3 more boxes with the total number of sticks assigned to him. Which of the following is the possible number of sticks assigned to each boy?
A
200
B
150
C
125
D
175
Question 4 Explanation: 
To find a integer value we will have to assume it.
Let x be the total number of sticks given to each boy and let y be the number of boxes in which they fill them. 
Now  as per the given condition if the boy reduces number of sticks per box by 25,
now he can fill three new boxes. Using this, the equations for the question are:
x = (x/y – 25)(y + 3) = x + 3x/y – 25y – 75
Rearranging we get 3x = (25y + 75) y or x = (25y2 + 75y)/3.
Now come to the options. If we go for the long method of calculations,
then there would be chances of wrong calculations so by logic.
For x to have an integer value, x has to be a multiple of 3. The only answer choice that supports this is 150.
Question 5
Shyama and Vyom walk up an escalator (moving stairway). The escalator moves at a constant speed. Shyama takes three steps for every two of Vyom’s steps. Shyama gets to the top of the escalator after having taken 25 steps, while Vyom (because his slower pace lets the escalator do a little more of the work) takes only 20 steps to reach the top. If the escalator were turned off, how many steps would they have to take to walk up?
A
40
B
50
C
60
D
80
Question 5 Explanation: 
In this question let,  Shyam take 1 min for every 3 steps.
Going by this logic, he takes 1/3 min for every step. 
So he will take 8.33 minutes for 25 steps. On the other hand,
Vyom takes1/2 min for every step. For 20 steps, he takes20/2 min, i.e. 10 min.
So Difference between their time = 1.66 min.
Escalator takes 5 steps in 1.66 min and difference in number of steps covered = 5,
Speed of escalator is 1 step for 0.33 min, i.e. 3 steps per minute.
If escalator is moving, then Shyam takes 25 steps and escalator also takes 25 steps.
Hence, total number of steps = 50.
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