Quant Boosters - Vikas Saini - Set 2
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Time taken by A = 25/3.
Time taken by B = 20/2 = 10.
Let speed of escalator = a.
(a+2)10 = (a+3)25/3
30a + 60 = 25a + 75.
a = 3.
Total steps = (3+2)10 = 50.
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Q24) On an upward moving escalator A, B, C take 10,8,5 steps respectively to reach the top. A takes 30 step to come down on same moving up escalator. Find the time ratio by B & C.
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Let speed of A = 1step/sec.
Speed of escalator = a.
10(1+a) = 30(1-a)
10 + 10a = 30 – 30a
a = 1/2
Total steps = 10 ( 1+1/2) = 15.
B takes time = 15-8 = 7.
C takes time = 15-5 = 10.
Time ratio = 7:10
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Q25) 677 has exactly 5 digits when converted into base 'n' from the decimal system. What is the minimum possible value of n.
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Let's start from base 3.
smallest 5 digit number in base 3 = (10000)_3 = 3^4.
Largest 5 digit number in base 3 = (22222)_3 = 2(11111)_3 = 3^5 - 1.
in base 3 not possible.
Now let's take base 4.
smallest number = (10000)_4 = 4^4 = 256.
Largest number = (33333)_4 = 4^5 - 1 = 1023.
256 < 677 < 1023.
Hence base 4.
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Q26) How many 3 digit numbers are there in base 9 system.
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Hundred place = 1 to 8 = 8 digit.
Ten's place = 0 to 8 = 9 digit.
Unit's place = 0 to 8 = 9 digit.
Total = 8 x 9 x 9 = 648 digits.
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Q27) How many base 10 numbers are three digit numbers in base 7 and four digit numbers in base 6.
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Smallest 3 digit number in base 7 = (100)_7 = 7^2 = (49)_10.
Largest 3 digit number in base 7 = (666)_7 = 7^3 - 1 = (342)_10.
[49,342] numbers are 3 digits numbers in base 7.
Smallest 4 digit number in base 6 = 6^3 = 216.
Largest 4 digit number in base 6 = 6^4 - 1 = 1295.
[216,1295] numbers are 4 digit numbers in base 6.
[216,342] common numbers.
Total decimal numbers = 342 - 216 + 1 = 127.
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Q28) N in base 10 = aaa in base 3. Here a is a single digit number, then N is a multiple of.
(a) 9
(b) 3a
(c) 13
(d) 26.
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(aaa)_3 = a (111)_3
a (3^2 + 3 + 1)
(13a)_10 = (N)_10.
Hence 13.
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Q29) How many 3 digits are there in decimal system which have exactly 3 digits when expressed in both base 7 & base 13.
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Smallest 3 digit number in base 7 = (100)_7 = 7^2 = 49.
Largest 3 digit number in base 7 = (666)_7 = 7^3 - 1 = 342.
[100,342] are the number which are exactly 3 digit number in base 7.
Smallest 3 digit number in base 13 = (100)_13 = 13^2 = 169.
Largest 3 digit numbers in base 13 = (CCC)_13 = 12^3 - 1 = 1727.
[169,999] are the three digit numbers which are exactly 3 digit numbers in base 13.
[ 169,342] is common in both.
Total = 342 - 169 + 1 = 174 numbers.
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Q30) In base x, 206 + 216 = 424. What is value of x ?
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2x^2 + 6 + 2x^2 + x + 6 = 4x^2 + 2x + 4
= > x + 12 = 2x + 4.
= > x = 8.