**Directions:** Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(A) Quantity A is greater.

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.

A symbol that appears more than once in a question has the same meaning throughout the question.

__Quantity A____Quantity B__The least prime number greater than 24 The greatest prime number less than 28 (A) Quantity A is greater.

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.*Explanation*For the integers greater than 24, note that 25, 26, 27, and 28 are not prime numbers, but 29 is a prime number, as are 31 and many other greater integers. Thus, 29 is the least prime number greater than 24, and Quantity A is 29. For the integers less than 28, note that 27, 26, 25, and 24 are not prime numbers, but 23 is a prime number, as are 19 and several other lesser integers. Thus, 23 is the greatest prime number less than 28, and Quantity B is 23.

**Thus, the correct answer is Choice A, Quantity A is greater.**

Lionel is younger than Maria. __Quantity A____Quantity B__Twice Lionel's age Maria's age (A) Quantity A is greater.

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.*Explanation*If Lionel's age is 6 years and Maria's age is 10 years, then Quantity A is greater, but if Lionel's age is 4 years and Maria's age is 10 years, then Quantity B is greater. Thus, the relationship cannot be determined.

**The correct answer is Choice D, the relationship cannot be determined from the information given**.

__Quantity A____Quantity B__54% of 360 150

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.*Explanation*Without doing the exact computation, you can see that 54 percent of 360 is greater than of 360, which is 180, and 180 is greater than Quantity B, 150.

**Thus, the correct answer is Choice A, Quantity A is greater**.

##### Figure 1

__Quantity A____Quantity B__*PS**SR*

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.*Explanation*From Figure 1, you know that

*PQR*is a triangle and that point*S*is between points*P*and*R*, so and You are also given that However, this information is not sufficient to compare*PS*and*SR*. Furthermore, because the figure is not necessarily drawn to scale, you cannot determine the relative sizes of*PS*and*SR*visually from the figure, though they may appear to be equal. The position of*S*can vary along side*PR*anywhere between*P*and*R*. Below are two possible variations of Figure 1, each of which is drawn to be consistent with the information##### Figure 2

##### Figure 3

Note that Quantity A is greater in Figure 2 and Quantity B is greater in Figure 3.

**Thus, the correct answer is Choice D, the relationship cannot be determined from the information given**.

__Quantity A____Quantity B__*x**y*

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.*Explanation*If then so in this case, but if then so in that case,

**Thus, the correct answer is****Choice D**,**the relationship cannot be determined from the information given**.Note that plugging numbers into expressions

*may not*be conclusive. However, it*is*conclusive if you get different results after plugging in different numbers: the conclusion is that the relationship cannot be determined from the information given. It is also conclusive if there are only a small number of possible numbers to plug in and*all*of them yield the same result, say, that Quantity B is greater.Now suppose there are an infinite number of possible numbers to plug in. If you plug many of them in and each time the result is, for example, that Quantity A is greater, you still cannot conclude that Quantity A is greater for every possible number that could be plugged in. Further analysis would be necessary and should focus on whether Quantity A is greater for all possible numbers or whether there are numbers for which Quantity A is not greater.

**The following sample questions focus on simplifying the comparison.**

__Quantity A____Quantity B__*y*

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.*Explanation*Set up the initial comparison:

Then simplify:

Step 1: Multiply both sides by 5 to get

Step 2: Subtract 3

*y*from both sides to getStep 3: Divide both sides by 2 to get

The comparison is now simplified as much as possible. In order to compare 1 and

*y*, note that you are given the information (above Quantities A and B). It follows from that or so that in the comparison the placeholder represents*less than*(<): .However, the problem asks for a comparison between Quantity A and Quantity B, not a comparison between 1 and

*y*. To go from the comparison between 1 and*y*to a comparison between Quantities A and B, start with the last comparison, and carefully consider each simplification step in reverse order to determine what each comparison implies about the preceding comparison, all the way back to the comparison between Quantities A and B if possible. Since step 3 was "*divide*both sides by 2,"*multiplying*both sides of the comparison by 2 implies the preceding comparison thus reversing step 3. Each simplification step can be reversed as follows:- Reverse step 3:
*multiply*both sides by 2. - Reverse step 2:
*add*3*y*to both sides. - Reverse step 1:
*divide*both sides by 5.

When each step is reversed, the relationship remains

*less than*(<), so Quantity A is less than Quantity B.**Thus, the correct answer is****Choice B**,**Quantity B is greater**.While some simplification steps like subtracting 3 from both sides or dividing both sides by 10 are always reversible, it is important to note that some steps, like squaring both sides, may not be reversible.

Also, note that when you simplify an

*inequality*, the steps of multiplying or dividing both sides by a negative number change the direction of the inequality; for example, if then So the relationship in the final, simplified inequality may be the*opposite*of the relationship between Quantities A and B. This is another reason to consider the impact of each step carefully.- Reverse step 3:

__Quantity A____Quantity B__

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.*Explanation*Set up the initial comparison:

Then simplify:

Step 1: Multiply both sides by 2 to get

Step 2: Add to both sides to get

Step 3: Simplify the right-hand side using the fact that to get

The resulting relationship is

*equal to*(=). In reverse order, each simplification step implies*equal to*in the preceding comparison. So Quantities A and B are also equal.**Thus, the correct answer is Choice C**,**the two quantities are equal**.

__Quantity A____Quantity B__

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.*Explanation*Set up the initial comparison:

Then simplify by noting that the quadratic polynomial can be factored:Step 1: Subtract 2

*x*from both sides to getStep 2: Factor the left-hand side to get

The left-hand side of the comparison is the square of a number. Since the square of a number is always greater than or equal to 0, and 0 is greater than the simplified comparison is the inequality and the resulting relationship is

*greater than*(>). In reverse order, each simplification step implies the inequality*greater than*(>) in the preceding comparison. Therefore, Quantity A is greater than Quantity B.**The correct answer is Choice A, Quantity A is greater**.

__Quantity A____Quantity B__

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.*Explanation*Set up the initial comparison:

Then simplify:

Step 1: Subtract 2

*w*from both sides and add 4 to both sides to getStep 2: Divide both sides by 5 to get

The comparison cannot be simplified any further. Although you are given that you still don't know how

*w*compares to or 1.8. For example, if then but if then In other words, the relationship between*w*and cannot be determined. Note that each of these simplification steps is reversible, so in reverse order, each simplification step implies that the*relationship cannot be determined*in the preceding comparison. Thus, the relationship between Quantities A and B cannot be determined.**The correct answer is Choice D**,**the relationship cannot be determined from the information given**.The strategy of simplifying the comparison works most efficiently when you note that a simplification step is reversible while actually taking the step. Here are some common steps that are always reversible:

- Adding any number or expression to both sides of a comparison
- Subtracting any number or expression from both sides
- Multiplying both sides by any nonzero number or expression
- Dividing both sides by any nonzero number or expression

Remember that if the relationship is an

*inequality*, multiplying or dividing both sides by any*negative*number or expression will yield the*opposite*inequality. Be aware that some common operations like squaring both sides are generally not reversible and may require further analysis using other information given in the question in order to justify reversing such steps.

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