Why it matters on the HSPT
Percents show up in every math-flavored part of the test. Mathematics uses them in word problems about discounts, tax, and change. Quantitative Skills uses them in comparisons, such as 25 percent of 80 against 50 percent of 40, and in manipulations, such as what number is 5 less than 40 percent of 60. There is no calculator, so the whole skill is knowing a few conversions and a few shortcuts cold.
The rules
Percent means per hundred. 35 percent is 35 out of 100, which is 35/100 as a fraction and 0.35 as a decimal. That one sentence is the whole conversion table.
Converting. Percent to decimal: move the decimal point two places left, so 8 percent is 0.08 and 150 percent is 1.5. Decimal to percent: two places right, so 0.6 is 60 percent. Fraction to percent: write it over 100 if you can, or divide. 3/4 is 75/100, which is 75 percent. 1/8 is 0.125, which is 12.5 percent.
The fractions worth memorizing. 50 percent is one half. 25 percent is one fourth. 75 percent is three fourths. 20 percent is one fifth. 10 percent is one tenth. 33 and a third percent is one third. 12.5 percent is one eighth. When a question uses one of these, use the fraction, not the decimal.
Percent of a number. Of means multiply. 30 percent of 50 is 0.3 times 50, which is 15. Faster in your head: 10 percent of 50 is 5, and 30 percent is three of those, 15. Build any percent from 10 percent, 5 percent, and 1 percent. 15 percent of 60 is 6 plus 3, which is 9. 12 percent of 200 is 20 plus 4, which is 24.
The ten percent rule. Almost every percent on the test can be built from 10 percent, which is the number with its decimal point moved one place left. Five percent is half of that, and 1 percent is a tenth of it. Fifteen percent of 24 dollars is 2.40 plus 1.20, which is 3.60. Twelve percent of 960 is 96 plus 9.60 plus 9.60, which is 115.20. Use it to estimate before you compute: 12 percent of 960 must be a little more than 96, which throws out every choice near 11 or 1,100 at a glance.
What percent is one number of another. Part divided by whole, then convert. 18 out of 24 is 18/24, which simplifies to 3/4, which is 75 percent. Simplify the fraction before converting; it is almost always one of the memorized ones.
Finding the whole. If 20 percent of a number is 12, the number is 12 divided by 0.2, which is 60. Easier: 20 percent is one fifth, so the number is 5 times 12. This is the reversed form the Quantitative section likes, of what number.
Increase and decrease. The percent change is the amount of change divided by the original amount. A shirt that goes from 40 dollars to 30 dollars dropped by 10, and 10 out of 40 is 25 percent off. To apply a change, find the percent of the original and add or subtract it: 40 dollars with 15 percent off is 40 minus 6, which is 34. A 5 percent tax on 60 dollars is 60 plus 3, which is 63.
The amount off versus the sale price. Read whether the question wants the discount or the price after the discount. A 20-dollar book at 8 percent off saves 1.60 and costs 18.40. Both numbers are always among the choices. Underline the words "how much is saved" or "what is the sale price" before you pick.
Interest for one period. Interest is a percent of the money in the account. Annual means per year. Compounded quarterly means the yearly rate is split into four equal payments, one each quarter; monthly means twelve. The test only asks for the first period, so divide the rate, then take that percent of the balance. One hundred dollars at 16 percent a year, compounded quarterly, earns 4 percent in the first quarter, which is 4 dollars. Dividing the balance by four instead of the rate gives 25, a trap answer.
Worked examples
Example 1
Which is equal to 3/8?
- A.3.8%
- B.37.5%
- C.38%
- D.0.38
Show answer
Answer: B: 37.5%
Three eighths is 0.375, because one eighth is 0.125 and three of those is 0.375. Moving the decimal two places right gives 37.5 percent. Choice C, 38 percent, is a rounding of the right idea, and the test does not round.
Example 2
What is 15% of 80?
- A.8
- B.10
- C.12
- D.15
Show answer
Answer: C: 12
Ten percent of 80 is 8. Five percent is half of that, 4. Fifteen percent is 8 plus 4, which is 12. Choice A, 8, stops at 10 percent.
Example 3
24 is what percent of 60?
- A.24%
- B.36%
- C.40%
- D.250%
Show answer
Answer: C: 40%
Part over whole is 24/60, which simplifies to 2/5, which is 40 percent. Choice D, 250 percent, comes from dividing the wrong way, 60 by 24.
Example 4
30% of what number is 21?
- A.6.3
- B.63
- C.70
- D.7
Show answer
Answer: C: 70
Thirty percent of the number is 21, so 10 percent is 7, and the whole number is 10 times 7, which is 70. Check: 30 percent of 70 is 21. Choice A, 6.3, is 30 percent of 21, the forward computation.
Example 5
A jacket priced at $60 is on sale for 25% off. What is the sale price?
- A.$15
- B.$35
- C.$45
- D.$48
Show answer
Answer: C: $45
Twenty-five percent is one fourth, and one fourth of 60 is 15. The sale price is 60 minus 15, which is 45. Choice A, 15, is the discount, not the price. Choice D, 48, is 20 percent off.
Example 6
A class had 20 students last year and has 25 this year. By what percent did it grow?
- A.5%
- B.20%
- C.25%
- D.125%
Show answer
Answer: C: 25%
The change is 5 students. Percent change is the change divided by the original: 5 out of 20 is one fourth, which is 25 percent. Choice B, 20 percent, comes from dividing by this year's number instead of last year's.
Example 7
What number is 8 less than 40% of 70?
- A.20
- B.28
- C.36
- D.48
Show answer
Answer: A: 20
Start with 40 percent of 70: 10 percent is 7, so 40 percent is 28. Eight less than 28 is 20. Choice B, 28, stops before subtracting; choice C, 36, adds 8 instead.
Example 8
A $40 sweater is marked 15% off. How much money does the discount save?
- A.$6
- B.$25
- C.$34
- D.$15
Show answer
Answer: A: $6
The question asks for the amount saved, not the sale price. Ten percent of 40 is 4, and 5 percent is 2, so 15 percent is 6 dollars. Choice C, 34 dollars, is the sale price, the trap for answering a different question. Choice D reads the percent as dollars.
Example 9
A savings account holds $500 and pays 8% annual interest, compounded quarterly. How much interest does it earn in the first quarter?
- A.$40
- B.$20
- C.$10
- D.$125
Show answer
Answer: C: $10
Quarterly means the 8 percent yearly rate is split into four payments of 2 percent. Two percent of 500 is 10 dollars. Choice A is a full year of interest. Choice D divides the balance by 4 instead of the rate.
Common mistakes
- Moving the decimal the wrong way. 0.05 is 5 percent, not 50 percent. Percent to decimal moves left; decimal to percent moves right.
- Dividing by the wrong whole. Percent change always uses the original number as the whole, never the new one.
- Confusing the discount with the price. Read the question twice: does it ask how much is taken off, or what you pay?
- Treating 12.5 percent as 12 percent. The test uses eighths on purpose. One eighth is 12.5 percent exactly.
- Doing the forward computation on a reversed question. Of what number means you are finding the whole, so your answer should be larger than the percent's result.
Answering the sale price when the discount was asked. Both numbers sit in the choices. Underline what the question wants before you compute.
Dividing the money instead of the rate. For quarterly interest, the 16 percent becomes 4 percent. The balance stays whole.
Practice
Percents feed arithmetic, non-geometric comparison, number manipulation, and the discount and tax items in word problems. The fractions lesson covers the fraction equivalents this page leans on.