Problem Solving - 3

There are two buses A and B. On Monday, bus A departs at 3pm and bus B at 4pm. After this, bus A departs every 10 hours and bus B every 15 hours. What is the earliest day they will depart at the same time?

A). Tuesday

B). Wednesday

C). Thursday

D). Sunday

E). so long as they continue operating in this manner, they will never leave at the same time.

Answer - E is the right choice.

The LCM of 10 hours and 15 hours = 30 hours,
30 hours = 6 hours mod 24 hour (day)

After that period they will just repeat the cycle.

Problem Solving - 2

If the positive integers m and n have the same two digits, but in reverse order, then the difference between m and n cannot be:

a) 24

b) 36

c) 54

d) 63

e) 72

Answer - (a)

Let m consists of digits x & y,

thus m = 10x+y and

n = 10y+x

m-n = 9x-9y = m-n = 9(x-y)

=> m-n has to be a multiple of 9....

DS Question - 7

A building has two types of apartments, big and small. 65 percent of the apartments are small. The number of occupied big apartments is twice the number of unoccupied small apartments. What percent of the apartments in the building are occupied ?

(1) The number of occupied big apartments is six times the number of unoccupied big apartments.

(2) The building has a total of 160 apartments

Answer - A

Statement (1 ) -- Sufficient.

Percentage occupied = (big & occupied + small & occupied) / total (say, n)

(35/100)n * (6/7) = big & occupied

(small ) - (small & unoccupied) = (small & occupied)

(65/100)n - [(1/2) big & occupied] = small & occupied

(65/100)n - [(1/2) (35/100)n * (6/7)] = small & occupied

Total = [(35/100)n * (6/7) + (65/100)n - (35/100)n * (3/7) ] / n

(multiplied (6/7) by (1/2) here ).

n is removed from the numerator and divisor

(35/100) * (6/7) + (65/100) - (35/100) * (3/7) --- This is approx equal to 0.8

so whatever n is, it is the answer.

Statement (2) -- Insufficient -- 160 total apartments does not give any information regarding any number of occupied apartments -- there could be 2 big and 1 small, or 40 big and 20 small.

Manhattan challenge problem of the week -- June 26

If set S = {7, y, 12, 8, x, 9}, is x + y less than 18?

(1) The range of set S is less than 9.

(2) The average of x and y is less than the average of set S.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer - B

(1) INSUFFICIENT: Statement (1) tells us that the range of S is less than 9. The range of a set is the positive difference between the smallest term and the largest term of the set. In this case, knowing that the range of set S is less than 9, we can answer only MAYBE to the question "Is (x + y) <>

Consider the following two examples:Let x = 7 and y = 7. The range of S is less than 9 and x + y <>

Let x = 10 and y = 10. The range of S is less than 9 and x + y > 18, so we conclude NO.

Because this statement does not allow us to answer definitively Yes or No, it is insufficient.

(2) SUFFICIENT: Statement (2) tells us that the average of x and y is less than the average of the set S. Writing this as an inequality:

(x + y)/2 < (7 + 8 + 9 + 12 + x + y)/6

(x + y)/2 < (36 + x + y)/6

3(x + y) <>

2(x + y) <>

x + y <>

Therefore, statement (2) is SUFFICIENT to determine whether x + y <>

DS Question - 6

Give that n is an integer, is n-1 divisible by 3?

1. n^2 +n is not divisible by 3

2. 3n+5>=k+8, where k is a positive multiple of 3

Answer - A

Sttement (1) Sufficient -- It can be simplified as n(n+1) is not divisible by 3

i.e to satisfy the above condition n can take a value immediately following 3 or multiples of 3.

Such as n = 4, 7, 10 .....

Thus case n-1 is always divisible by 3.

Statement (2) Insufficient -- 3n >= k + 3

This implies n >= (k+3)/3

the possible values of k includes 3,6,9,12 .....

So this can be either multiple of 3 or any other number..

Problem solving - 1

Problem Solving - Numbers

If (3^4)(5^6)(7^3)=(35^n)(x), where x and n are both positive integers, how many different positive values of n are there?

A) 1

B) 2

C) 3

D) 4

E) 6

Answer - C

(3^4)(5^6)(7^3)=(35^n)(x)

Now 35 = 5 * 7

=> 35^n = 5^n * 7^n

3^4 * 5^3 * (5^3 * 7^3) = (35^n) * x

So x is a multiple of 3^4 * 5^3 n => 3 n is a positive integer So n=1, 2, 3

Manhattan challenge problem of the week - June 19

If x is a non-zero integer, what is the value of x ^ y?

(1) x = 2
(2) (128 ^ x)[6 ^ (x + y)] = (48 ^ 2x)(3 ^ -x)

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer : must be B

Statement (1) - not sufficient - value of y is still not known.

Statement (2)- Sufficient.

(128 ^ x)(6 ^ x)(6 ^ y) = [(48 ^x) (48 ^ x)] / (3 ^ x)

=> (16.8 ^ x)(6 ^ x)(6 ^ y) = (16 ^ x) (48 ^ x)

=> (16 ^ x)(8 ^ x)(6 ^ x)(6 ^ y) = (16 ^ x) (8 ^ x)(6 ^ x)
=> (6 ^ y) = 1
=> y = 0
=> (x^y) = 1 , because any number raised to the power zero is equal to 1.

Hence B must be the answer.

Official Answer and Explanation to the above problem

One of the most effective ways to begin solving problems involving exponential equations is to break down bases of the exponents into prime factors and combine exponents with the same base. Following this approach, be sure to simplify each statement as much as possible before arriving at the conclusion, since difficult problems with exponents often result in unobvious outcomes.

(1) INSUFFICIENT: While this statement gives us the value of x, we know nothing about y and cannot determine the value of x^y.

(2) SUFFICIENT: (128^x)[6^(x + y)] = (48^2x)(3^-x)

[(2^7)^x][(2 × 3)^(x + y)] = {(2^4 ) 3]^2x}(3^-x)

[(2^7)^x][2^(x + y)][3^(x + y)] = (2^8x)(3^2x)(3^-x)

[2^(8x + y)][3^(x + y)] = (2^8x)[3^(2x - x)]

(2^8x)( 2^y)(3^x)(3^y) = (2^8x)(3^x)

( 2^y)(3^y) = 1

(2 × 3)^y = 1

6^y = 1

y = 0

Since y = 0 and x is not equal to zero (as stated in the problem stem), this information is sufficient to conclude that x^y = x^0 = 1.

The correct answer is B.

DS Question - 5

What is the first term of an arithmetic progression of positive integers ?

a)Sum of the squares of the first and second term is 116.
b)The seventh term is divisible by 10.

Answer - A

Explanation - let x be the first and y be the second term.
Hence 0 less than x less than y

From Statement 1: x^2 + y^2 = 116 only when x=2 and y=10.
Hence the first term is 4. This is the only combination that works - hence sufficient

Statement 2: let 10z be the 7th term (z is an integer)
Thus x+6(y-x)=10z => y=(10z+5x)/6. Hence the last digit of the numerator can be 5 or 0. Now y is also an integer => x must be even => least possible value of y =10.
Assuming different values of z we get
z=1; x=12 This is impossible (x should be less than y)
z=2; x=8; 8 10 12 14 16 18 20
z=3; x=6; 6 10 14 18 22 26 30 and so on -- hence insufficient

DS Question - 4

Is the integer x divisible by 3?

1) The last digit in x is 3.
2) x+5 is divisible by 6.

Answer - B

Statement (1) is not sufficient.
If x is 33, then x is divisible by 3, and (1) holds, but if x=43, then (1) is true but x is not divisible by 3.

Statement (2) is sufficient. According to it , there exists an integer k such that x+5 = 6k => x = 6k-5 => x/3 = (6k-5)/3 = x/3 = (2k) - 5/3 => x is not divisible by 3.

Hence B is the answer i.e the (2) statement alone is sufficient to answer the question but (1) statement alone is not sufficient to answer the question.

Manhattan challenge problem of the week! - June 12th

If the reciprocals of two consecutive integers are added to one another, what is the sum in terms of the greater integer x?

(A) 3/x
(B) x^2 – x
(C) 2x – 1
(D) 2x – 1 /x^2 + x
(E) 2x – 1 /x^2 – x

Solution:Official answer will be known next week - i.e June 19th

Answer must be E

Greater integer = x
Consecutive smaller integer number to x = x-1
Sum of reciprocals
=1/x +1/x-1 = x-1+x/x(x-1) = 2x - 1/(x^2 - x)

Thus E is the answer.

Data Sufficiency -- Q No - 3

Question

Is x^2 greater than x?

1)x^2 is greater than 1.
2)x is greater than -1.

Answer- A

1) implies either x<-1 or x>1 (key is its being compared with '1' not '0' to get rid of the fractions that are <1.>

2) not sufficient. Take values 1/2 and 2 to verify.

Manhattan challenge problem of the week! - June 5th

All Squares --- Data sufficiency Question!

Is x^2 + y^2 > 4a?

(1) (x + y)^2 = 9a
(2) (x – y)^2 = a

(A) Statement (1) alone is sufficient, but statement(2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement(1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, butNEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOTsufficient.

Answer - A

Statement (1) alone is not sufficient to answer the question as - depends upon sign of x and y.
Statement (2) alone is not sufficient to answer the question as - depends upon sign of x and y.
Together, statement (1) and (2) ---

On solving (1)

(x+y)^2 = 9a

x+y = + - 3 (a^1/2)

For x^2 + y^2 to be minimum x = y = + - 1.5 (a^1/2)

x^2 + y^2 = 2.25 a + 2.25 a

=4.5a > 4a

Thus answer is A

However if

a=0 then x=y=0 also, so this inequality will not hold true...
In that case it is very easy for a=x=y=0 ,no way x^2 + y^2 > 4a can be satisfied for any condition.

So answer will straight away be E.

This is assumption that x,y and a are different numbers.

Thus answer will be E in this case.

Official Answer to the above problem.

(1) INSUFFICIENT: If we multiply this equation out, we get:

x2 + 2xy + y2 = 9a

If we try to solve this expression for

x2 + y2, we getx2 + y2 = 9a – 2xy

Since the value of this expression depends on the value of x and y, we don't have enough information.

(2) INSUFFICIENT: If we multiply this equation out, we get:

x2 – 2xy + y2 = a

If we try to solve this expression for x2 + y2,
we getx2 + y2 = a + 2xy

Since the value of this expression depends on the value of x and y, we don't have enough information.

(1) AND (2) INSUFFICIENT: We can combine the two expanded forms of the equations from the two statements by adding them:

x2 + 2xy + y2 = 9ax2 – 2xy + y2 = a----- 2x2 + 2y2 = 10ax2 + y2 = 5a

If we substitute this back into the original question, the question becomes: "Is 5a > 4a?"If a > 0, the answer is yes.We know from the question stem that a is nonnegative.However, if a = 0 the answer is no.

The correct answer is E.