HSPT arithmetic

What the question looks like

Arithmetic questions are the bread and butter of the HSPT Mathematics section. Each one is a short problem with four answer choices. Most are pure computation with no story attached, such as "What is 3/8 of 64?" or "Which of the following is equal to 0.6?"

The topics you will see include whole numbers, fractions, decimals, percents, ratios, order of operations, exponents like 5², square roots of perfect squares like √49, rounding, and factors and multiples. Some questions blend two ideas, like a fraction of a percent or an exponent inside an order-of-operations chain.

Nothing here needs algebra. What the test is really checking is whether you can do 8th-grade arithmetic by hand, without a calculator, in well under a minute.

What you need to know

• To add or subtract fractions, rewrite them with a common denominator first, then add or subtract the numerators only.
• To multiply fractions, multiply straight across. To divide, flip the second fraction and multiply.
• A percent is a number out of 100, so 15% = 15/100 = 0.15.
• To find a percent of a number, multiply: 15% of 240 = 0.15 × 240.
• Order of operations is parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right.
• 3² means 3 × 3 = 9, and √49 = 7 because 7 × 7 = 49.
• A ratio of 3 to 5 means 8 total parts. Divide the total by 8 to find the size of one part.
• The greatest common factor is the largest number that divides into both. The least common multiple is the smallest number both divide into.
• To round, look at the digit one place to the right of where you are rounding. Five or more rounds up.
• When multiplying decimals, count the total decimal places in the factors and put that many in the answer.

How to solve it

1. Read the whole question and underline the operation words: "of" usually means multiply, "sum" means add, "difference" means subtract, "product" means multiply, "quotient" means divide.
2. Decide which rule applies before you write anything. Fraction problem? Find the common denominator. Percent problem? Turn it into a decimal.
3. Do the computation in the margin of your test booklet in neat, small steps. Do not do two steps in your head at once.
4. Look at your result and ask whether it is reasonable. Fifteen percent of 240 cannot be 360.
5. Find your answer among the choices. If it is not there, you likely skipped a step or misplaced a decimal, so recheck quickly.

Shortcuts that save the minute

Most arithmetic items can be finished, or at least narrowed to two choices, without doing the full calculation. Six moves cover nearly all of them.

Check the last digit. The last digit of a product depends only on the last digits of the factors. For 36 times 72, 6 times 2 ends in 2, so the answer ends in 2; usually only one choice does. The same trick works for sums: 0.76 plus 2.35 plus 1.6 ends in the digit from 6 plus 5 plus 0, which is 1.

Estimate first. Round each number and get a ballpark before touching the exact arithmetic. If 28 minus 9 and two thirds must be a little under 19, the choices near 18 and 19 survive and the rest go. Often the ballpark alone picks the answer.

Ten percent, then build. Ten percent of any number is the number with its decimal point moved one place left. Five percent is half of that, 1 percent is a tenth. Almost every percent on the test is a short sum of those three.

Scale, do not cross-multiply. In a proportion, look for the multiplier between the two numbers you know on the same side. If 3 mittens need 4 skeins, then 15 mittens is 5 times as many, so 20 skeins. Cross-multiplying gets there too, but with a division you did not need.

Test the choices from the middle. For a missing number, try the second-largest choice. Too big means the answer is smaller, too small means it is larger, and one trial has removed half the choices. Best on "of what number" percents, consecutive-integer problems, and small equations.

Reduce before you multiply. In a fraction product or division, cancel a common factor from a top and a bottom first. Three fifths divided by 2 fifteenths becomes 3 over 5 times 15 over 2, the 5 and 15 cancel to 1 and 3, and the answer is 9 over 2 with no large numbers along the way.

The properties lesson explains why the reordering and splitting moves are legal, and the strategies guide puts them in the context of the whole test.

Worked examples

Traps to watch for

• Decimal placement. When you multiply by a percent, count your decimal places. Getting 3.6 instead of 36 is the most common arithmetic slip on the test.
• Adding fractions straight across. The tops and bottoms are not separate numbers. Always find a common denominator before adding or subtracting.
• Order of operations. Exponents come before multiplication, and multiplication comes before subtraction, even when the subtraction sits to the left. Do not just read left to right.
• Answering the wrong part. Ratio and percent questions often ask for one specific group. If you found the number of boys, make sure the question did not ask for girls.

Pacing

You have 45 minutes for 64 questions, which works out to about 42 seconds each, so arithmetic questions are where you should bank time. Do the ones you can see immediately first, and if a problem stalls you, pick your best choice and move on, because there is no penalty for guessing. Never leave a bubble blank.

Practice

Try a timed set of arithmetic questions along with every other section in our free diagnostic.

Shaky on the skill itself? Start with the lessons on fractions, decimals, percents, integers and order of operations, and factors, multiples, and primes, then come back to the question format.