What the question looks like
Problem-solving questions are the word problems of the HSPT. STS calls this category "concepts and problem solving." Instead of a bare computation, you get two or three sentences describing a situation, and you have to figure out which operations to use and in what order.
The common scenarios are money and change, rate, time, and distance at a simple level, averages, meaning mean, median, and mode, percent increase or decrease, unit conversion, and reading a small table or list of values described in words. Most of these problems have two steps, and a few have three.
The math inside each problem is nothing harder than the arithmetic and pre-algebra you have already seen. The challenge is translation: reading the story, deciding what to do, and not stopping early.
What you need to know
- Mean is the sum of the values divided by how many values there are.
- Median is the middle value after you put the numbers in order. With an even count, it is the average of the two middle values.
- Mode is the value that appears most often.
- Distance = rate × time. Rate = distance ÷ time. Time = distance ÷ rate.
- Percent change = amount of change ÷ original amount, then turn the decimal into a percent.
- For a discount, subtract the percent of the price from the price. A 20% discount on 50 dollars is 50 − 10 = 40 dollars.
- To find change from a purchase, multiply to get the total cost first, then subtract from the amount paid.
- Unit conversions to know: 12 inches in a foot, 3 feet in a yard, 16 ounces in a pound, 60 minutes in an hour, 100 centimeters in a meter, 1,000 meters in a kilometer.
- When a table is described in words, jot the values down as a short list before doing anything else.
How to solve it
- Read the last sentence first. It tells you exactly what the question wants, and it keeps you from stopping at an intermediate number.
- Read the whole problem and write down each number with a one-word label, such as "3 notebooks, 2.45 each, paid 10."
- Plan the steps in order before computing: first find the total, then subtract, and so on.
- Do each step in the margin, one line at a time, and label the result so you know which step you are on.
- Compare your final number to the question. If the question asked for change and you found the total cost, you are not done yet.
Worked examples
Example 1
Maya buys 3 notebooks that cost $2.45 each. If she pays with a $10 bill, how much change should she receive?
- A.$2.65
- B.$3.65
- C.$7.35
- D.$7.55
Show answer
Answer: A: $2.65
First find the total cost: 3 × 2.45 = 7.35. Then subtract from what she paid: 10.00 − 7.35 = 2.65. Choice C, $7.35, is the total cost, not the change, so it is the answer of someone who stopped one step early. Choice D, $7.55, is 10 − 2.45, which forgets that she bought three notebooks.
Example 2
A train travels 180 miles in 3 hours. At the same speed, how many miles does it travel in 5 hours?
- A.240
- B.300
- C.360
- D.540
Show answer
Answer: B: 300
Find the rate first: 180 ÷ 3 = 60 miles per hour. Then multiply by the new time: 60 × 5 = 300 miles. Choice A, 240, adds one more hour of travel to 180 instead of two, and choice D, 540, multiplies 180 by 3, which mixes up the numbers in the problem.
Example 3
Jordan scored 82, 90, 75, and 93 on four quizzes. What is the mean of the four scores?
- A.85
- B.86
- C.88
- D.90
Show answer
Answer: A: 85
Add the scores: 82 + 90 = 172, then 172 + 75 = 247, then 247 + 93 = 340. Divide by the number of scores: 340 ÷ 4 = 85. Choice B, 86, is the median, because the two middle scores in order are 82 and 90 and their average is 86. Mean and median are different questions, so check which word is used.
Example 4
A jacket cost $80 last year and costs $92 this year. By what percent did the price increase?
- A.12%
- B.13%
- C.15%
- D.20%
Show answer
Answer: C: 15%
Find the amount of change: 92 − 80 = 12 dollars. Divide by the original price: 12 ÷ 80 = 0.15, which is 15%. Choice A, 12%, treats the dollar increase as if it were already a percent. Choice B, 13%, comes from dividing by the new price of 92 instead of the original price of 80.
Traps to watch for
- Stopping one step early. The total cost, the sum of the scores, or the rate per hour will always appear among the choices. Reread the last sentence before you bubble.
- Dividing by the wrong number. Percent change always divides by the original amount. Mean always divides by the count of values, not by the largest value or by 100.
- Mismatched units. If a problem gives minutes and asks about hours, or gives inches and asks for feet, convert before you compute. A mixed-unit answer is a wrong answer.
- Mean versus median versus mode. These three words look alike in a hurry. Underline the one the question uses.
Pacing
Word problems take longer than plain arithmetic, so expect some of these to run past the 42-second average and make up the time on easier questions. If you cannot see the first step within about 15 seconds, pick your best choice, mark the question, and come back if there is time. There is no guessing penalty, so every question should have an answer when the section ends.
Practice
Our free diagnostic includes multi-step problems like these and shows how your pace compares to the real test.
Shaky on the skill itself? Start with the lessons on fractions, percents, ratios, rates, and proportions, elementary algebra, measurement and units, and data and probability, then come back to the question format.