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## 3 HSPT Example Problems and Solutions

Here are some examples of the types of problems you can expect to see on HSPT along with their solutions.

__VERBAL__**John runs faster than Carol. Frank runs slower than John or Beth. Carol runs faster than Beth. If both statements are true, the third is(A truly(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 note (the fastest people are on the left):

J <== C

We also know that Frank doesn’t run as fast as John or Beth, which we’ll refer to as (the fastest people are on the left):

J <== F

B <== F

The third statement claims that Carol is faster than Beth. Can we draw this conclusion based on the given statements? Can we string the 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 provided. Here’s one where we show Carol and Beth running at the same speed.

J <== C

J <== B <== F

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

J <== C

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

Here’s one where 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 isn’t enough information to draw a definitive conclusion about the relationship between Carol’s speed and Beth’s speed. The answer is therefore (C) uncertain.

__MATH__**Xavier and Yvonne each try to solve a problem on their own. The probability that Xavier gets a correct answer is 1/4, and the probability that Yvonne gets a correct answer is 5/8. What is the probability that Xavier, but not Yvonne, solves 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 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 solves the problem is still 1/4, and the probability that Yvonne does NOT solve the problem is 3/8 (i.e. 1 – 5/8). So the probability of the two events happening together is simply 1/4 * 3/8 = 3/32, which is answer choice (D).

__LANGUAGE__**a) No, I can’t help you tonight.b) He forgot everything to accept his keys.c) We are not going to cry or laugh tonight.d) No errors.**

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 in sentence b). The words “accept” and “except” are homophones (they sound the same) in English and, therefore, are often 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 to take his keys.

vs.

He forgot everything except his keys.

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

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