Solutions and explanations to GMAT problems in sample 1 .
Given a problem and two statements numbered (1) and (2), The data sufficiency problems are answered as follows
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Find the length of the rectangle.
(1) The perimeter of the rectangle is 120
(2) The width of the rectangle is 20
Solution
If L is the length and W the width of the rectangle, the perimeter P is given by
P = 2 L + 2 W = 120
The equation 2 L + 2 W = 120 has two unknown and can only be solve using the width W = 20.
Both the perimeter and the width are needed to answer the question. Answer C

Given the following
x = 2 t  z + 5w, y = 2 t  1, z = 2w + 1,
find the value of x.
(1) y = 3
(2) z =  5
Solution
To find the value of x, the values of t, z and w are needed. Z is given. Also z can be used to find w using z = 2w + 1 as follows
 5 = 2w + 1
y is needed to find t using y = 2 t  1 as follows
3 = 2 t  1
Both statements (1) and (2), together, given above are needed to answer the question. Answer C

Find the area of an isosceles right triangle.
(1) The length of one of its legs is 10 cm
(2) The length of its hypotenuse is 10√2
Solution
The area of a right triangle is given by
(1/2) × length of leg1 × length of leg2
Being an isosceles triangle the length of the two legs are equal to 10 and the area may be calculated using statement (1) only.
If h is the length of the hypotenuse and x is the length of leg of the isosceles right triangle, by Pythagora's theorem we have
x^{2} + x^{2} = h^{2}
The above equation may be used to find x^{2} which in turn may be used to find the area of the triangle. Hence statement (2) may also be used, on its own, to answer the given question.
Since each statement, on its own, may be used to answer the question, the answer is D

Is x = y?
(1)  x  =  y 
(2) x y > 0
Solution
Statement (1) means either x = y or x =  y and cannot be used to answer the question. Statement (2) means that x and y have the same sign, since their product is positive, but cannot be used to answer the question. If we now use statement (1) and (2) together we can answer the question and say x = y. Answer C

In a library there are 3 kinds of books: Math, Biology and Geography. There are 150 more Geography than Math books. How many books are there in the library?
(1) The ratio of Math to Biology books is 5:7
(2) The ratio of Biology to Geography books is 7:8
Solution
Let x, y and z be the number of Math, Biology and Geography books respectively. We need to find the total number of books which is given by
x+ y + z
"There are 150 more Geography than Math books" is translated as follows
z = x + 150
The ratio of Math to Biology books is 5:7 could be written as
x / y = 5 / 7
It is clear that the two equations found above are not enough to find the sum x + y + z
The ratio of Biology to Geography 7:8 could be written as
y / z = 7 / 8
We now have 3 equations with 3 unknown and the question can be answered. So both statement (1) and (2) are necessary and no one on its own is sufficient to answer the question.Answer C

What is the value of x^{2} + y^{2}?
(1) x + y = 4
(2) x = 2 / y
Solution
Statement (1) could be used to write
(x + y)^{2} = 4^{2}
Expand the left side
x^{2} + y^{2} + 2xy = 16
At this point we cannot find x^{2} + y^{2} because the product xy is unknown. Statement (2) can be used to write
xy = 2
which can therefore be used in the equation x^{2} + y^{2} + 2xy = 16 to find x^{2} + y^{2}. So both statements (1) and (2) are necessary to answer the question but none on its own is sufficient to answer the question.Answer C

What is the value x?
(1) 2x + 0.5 y = 7
(2) 0.4 x + 0.2 y = 1.4
Solution
x is one of two unknown in a given system of equations and therefore both statements are necessary to answer the question and none on its own is enough to answer the question.Answer C

What is the perimeter of the quadrilateral?
(1) The area of the quadrilateral is 600
(2) One side of the quadrilateral is 30
Solution
No information is given as to what kind of quadrilateral is the question about. Statements (1) and (2) together are not sufficient to answer the question asked, additional information is needed.Answer E

m and n are both positive and m is less than n. What is the value of m^{2}  n^{2}?
(1) m + n = 14
(2) m  n = 2
Solution
Since m and n are positive, we can write
m + n = m + n = 14
Since m is less than n,
m  n < 0
Hence
m  n = (m  n) = 2
Multiply m + n by (m  n)
(m + n)((m  n)) =  (m^{2}  n^{2}) = 14 * 2
Both statements are necessary to answer the question and none on its own is sufficient to answer the question.Answer C

What is the volume of the sphere?
(1) The surface area of the sphere is 400 pi
(2) The radius of the sphere is 10
Solution
Statement (1) gives the value of the surface area S which can used to find the radius r using the formula
S = 4 Pi r^{2}
and r can be used to find the volume V of the sphere using the formula
V = (4/3) Pi r^{3}
statement (2) can be used to find the volume using again the above formula. Hence each statement alone is sufficient to answer the question asked. Answer D
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