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Number System: Level 2 Test - 8

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Question 1
ABCDEFGH is a regular octagon. A and E are opposite vertices of the octagon. A frog starts jumping from vertex to vertex, beginning from A. From any vertex of the octagon except E, it may jump to either of the two adjacent vertices. When it reaches E, the frog stops and stays there. Let an be the number of distinct paths of exactly n jumps ending in E. Then what is the value of a2n – 1
A
0
B
4
C
2n – 1
D
Cannot be determined
Question 1 Explanation: 
In order to reach E from A, it can walk clockwise as well as anticlockwise.
In all cases, it will have to take odd number of jumps from one vertex to another.
But the sum will be even.
In simple case, if n = 4, then an = 2.
For a2n–1 = 7 (odd), we cannot reach the point E.
Question 2
In a four-digit number, the sum of the first 2 digits is equal to sum of the last 2 digits. The sum of the first and last digits is equal to the third digit of the number. Finally, the sum of the second and fourth digits is twice the sum of the other 2 digits. What is the third digit of the number?
A
5
B
8
C
1
D
4
Question 2 Explanation: 
Let the four-digit number be abcd.
a + b = c + d ... (i)
b + d = 2(a + c) ... (ii)
a + d = c ...(iii)
From (i) and (iii), b = 2d
From (i) and (ii), 3b = 4c + d
5d=4c
c=5/4d
Now d can be 4 or 8. But if d = 8, then c = 10 not possible. So d = 4 which gives c = 5.
Question 3
You can collect as many rubies and emeralds as you can. Each ruby is worth Rs. 4 crore and each emerald is worth Rs. 5 crore. Each ruby weighs 300grams and each emerald weighs 400 grams Your bag can carry at the most 12 kg. What should you collect to get the maximum wealth?
A
20 rubies and 15 emeralds
B
40 rubies
C
28 rubies and 9 emeralds
D
None of these
Question 3 Explanation: 
To maximise the value of the wealth, we must carry more of the one whose value per kilogram is more.
Value per kilogram of ruby =4/0.3 = Rs. 13.33 crore,
and value per rupee of each emerald 5/0.4 = Rs. 12.5 crore.
It is obvious that we should carry entire 12 kg of ruby.
This would amount to 12/0.3= 40
Question 4
A money changing machine contains one-rupee, two-rupee and five-rupee coins. The total number of coins is 300. The amount of total number of coins is Rs. 960. If the numbers of one-rupee coins and two-rupee coins are interchanged, the total value comes down by Rs. 40. Find the total number of five-rupee coins is
A
100
B
140
C
60
D
150
Question 4 Explanation: 
Let the number of five-rupee, two-rupee and one-rupee coins be x, y and z respectively.
x + y + z = 300
5x + 2y + z = 960
5x + y + 2z = 920
y – z = 40
And x + 2y = 340
Use the answer choices now. If x = 140, y = 100 and z = 60, this satisfies all the given conditions
Question 5
A company has three machines A, B and C  to complete a job to prepare can . Machine A can complete the job in 3 days, Machine B can complete the job in 4 days, and C can complete the job in 6 days. How many days will the company take to complete the job if all the machines are used simultaneously?
A
4 days
B
4/3 days
C
3 days
D
12 days
Question 5 Explanation: 
In one day, A would do1/3  of the job,
B would do1/4 of the job and C would do 1/6 of the job.
Hence, if all three of them work simultaneously,
in one day they would do1/3+1/4+1/6=3/4 of the job
Hence, to complete the entire job together they would Take 4/3  days.
Shortcut: A can complete the job in 3 days.
So A, B and combined will take less days than A alone to finish the job.
So straightway option (b).
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