# Sample Data Sufficiency Questions

**Question 1.**

In the * xy *- coordinate plane, what is the

*- intercept of line*

**x***L*?

_{1}1. The *y-intercept* of the line *L _{1}* is 3.

2. The slope of line *L _{1}* is 5 times its

*y-intercept*.

(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, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Explanation:**

The equation of a straight line in the *xy-plane* is generally given by *y = mx + c, *where

*m*is the slope and c is the

*y-intercept*of the line. The question asks you to find the

*x-intercept*of line

*L*, that is, the value of

_{1}*x*for which

*y = 0*. If

*y = 0*, then

*mx + c = 0*and x = .

So, to find the *x - intercept*, you need to know the *y - intercept*, *c*, and the slope of the line, *m*.

**Statement (1):**

This tells you the *y - intercept*, *c*, only but not the slope of the line, *m*.

Statement (1) ALONE is NOT sufficient.

Statement (2):

This tells you the slope of the line, *m*, only but not the *y - intercept*.

Statement (2) ALONE is NOT sufficient.

BOTH statements TOGETHER:

Together, the two statements tell you both the slope *m* and the *y - intercept*, *c*. So the two statements provide all the required information to calculate the *x - intercept*.

Note, that you do not actually have to calculate the *x - intercept*, as soon as you know that you have all the information needed to do so you can answer the question.

**BOTH statements TOGETHER are sufficient.**

**The correct answer is C.**

**Question 2.**

It is known that *b* and *c* are positive integers. What does *b − c* equal?

(1) 2^{b}5^{c} = 800

(2) 5^{b} = 125(5^{c})

(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, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Remember that (x^{m})(x^{n}) = x^{m+n} and = x^{m−n} for any integers *m* and *n*.

Statement (1):

Expressing 800 as a product of its prime factors yields: 800 = 2 × 2 × 2 × 2 × 2 × 5 × 5 = 2^{5} × 5^{2}.Thus, 800 = 2^{5} × 5^{2} = **2 ^{b} 5^{c}**. From here you can infer that

*b = 5*and

*c = 2*. Remember that prime factorization of any integer is unique when all the factors are arranged in increasing (or decreasing) order.

As you know the values of both *b* and *c*, you can evaluate *b − c*. Statement (1) ALONE is sufficient.

Statement (1) ALONE is sufficient.

Statement (2):

You are given that 5^{b} = 125(5^{c}).

Dividing both sides of the equation by 5^{c} is a valid operation as 5^{c} does not equal 0 for any positive integer value of *c*. This operation results in the following equation:

= 125

Expressing 125 in its prime factorization form, you find that: 125 = 5 × 5 × 5 = 5^{3}. Thus, you may write that: = 5^{3}. You know that, according to the rules of Exponents, = 5* ^{b − c}*.

Thus, you can deduce that *5 ^{b − c} = 5^{3}. *So

*b − c = 3.*

This answers the question.

Statement (2) ALONE is sufficient.

**EACH statement ALONE is sufficient.**

**The correct answer is D.**

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