# 3 HSPT Example Problems and Solutions

Here are a few examples of the kinds of problems you can expect to see on the HSPT along with their solutions.

VERBAL
John runs faster than Carol. Frank runs more slowly than John or Beth. Carol runs faster than Beth. If the two statements are true, the third is
(A) true
(B) false
(C) uncertain

This problem is an example of verbal logic. It tests understanding of how well a student understands how logical statements can be combined to draw conclusions.

The problem tells us to assume that the first two statements are true. Thus, we know that John is faster than Carol, which we will denote as (faster people are on the left):

J <== C

We also know that Frank does not run as quickly as either John or Beth, which we will denote as (faster people are on the left):

J <== F
B <== F

The third statement is making the claim that Carol is faster than Beth. Can we draw this conclusion based on the statements given? Can we chain statements together to show that Carol is, in fact, faster than Beth? Let’s give a visual representation of some possible conclusions that we can draw from the information given. Here’s one in which we show that Carol and Beth run at the same speed.

J <== C
J <== B <== F

Here’s one in which we show that Carol is faster than Beth.

J <== C
J <===== B <== F

Here’s one in which we show that Beth is faster than Carol.

J <====== C
J <== B <== F

All of these visual representations adhere to the first two statements, but they also show that there is not enough information to make any sort of definitive conclusion about the relationship between Carol’s speed and Beth’s speed. Therefore, the answer is (C) uncertain.

MATH
Xavier and Yvonne both try independently to solve a problem. The probability that Xavier will get a correct answer is 1/4, and the probability that Yvonne will get a correct answer is 5/8. What is the probability that Xavier, but not Yvonne, will solve the problem?
(A) 7/8
(B) 3/8
(C) 5/32
(D) 3/32

This is an advanced probability problem that tests a student’s understanding of how to combine probabilities.

When two events are independent, the probability of both of them occurring together (event A AND event B) is simply P(A) * P(B), where P(A) represents the probability of A and P(B) represents the probability of B. The probability that Xavier will solve the problem is still 1/4, and the probability that Yvonne will NOT get the problem is 3/8 (which is 1 – 5/8). Thus, the probability of both events occurring together is simply 1/4 * 3/8 = 3/32, which is answer choice (D).

LANGUAGE
b) He forgot everything accept his keys.
c) We will neither cry nor laugh tonight.
d) No mistakes.

This problem tests a student’s understanding of vocabulary and idioms. In particular, this question tests whether students can identify commonly confused words.

The error in this problem is found in sentence b). The words “accept” and “except” are homophones (they sound the same) in English and, as a result, are commonly confused. “Accept” is a verb meaning “to take or receive”; “except” is a preposition meaning “but” or “excluding.” In the context of this sentence, accept does not make sense. Try replacing the word with the definition:

He forgot everything take his keys.
vs.
He forgot everything but his keys.

Clearly the first sentence makes no sense, and the second sentence makes perfect sense. Therefore, “accept” is incorrect.

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