Why it matters on the HSPT
Decimals are how the test writes money, measurements, and percents once they have been converted, so they show up all through the Mathematics section and in Quantitative comparisons that mix 0.35 with 3/8 and 34 percent. The arithmetic is the same as with whole numbers. The only new skill is knowing where the decimal point lands, and the test builds its wrong answers around that one point being off by one place.
The rules
Place value. The places to the right of the decimal point are tenths, hundredths, thousandths. So 0.372 is 3 tenths, 7 hundredths, and 2 thousandths. Reading a decimal as a fraction is the same thing: 0.372 is 372 thousandths, or 372/1000.
Comparing. Line up the decimal points and add zeros on the right until every number has the same number of places. Then compare like whole numbers. To compare 0.9, 0.09, and 0.099, write 0.900, 0.090, and 0.099. The order from smallest is 0.090, 0.099, 0.900. More digits does not mean a bigger number.
Rounding. Look at the digit to the right of the place you are rounding to. Five or more rounds up, four or less leaves it. 3.746 rounded to the nearest tenth is 3.7, because the hundredths digit is 4. Rounded to the nearest hundredth it is 3.75.
Adding and subtracting. Line up the decimal points, fill in zeros, and add or subtract as usual. 4.5 plus 0.35 is 4.50 plus 0.35, which is 4.85. The decimal point in the answer sits directly under the others.
Multiplying. Ignore the decimal points and multiply. Then count the decimal places in both factors and give the answer that many places. 1.2 times 0.4: 12 times 4 is 48, and there are two decimal places in total, so the answer is 0.48. Multiplying by 10, 100, or 1,000 moves the point right one, two, or three places: 2.75 times 100 is 275.
Dividing. If the divisor is a whole number, divide as usual and keep the point in the same place: 7.2 divided by 3 is 2.4. If the divisor has a decimal, slide its point right until it is whole, and slide the dividend's point the same number of places: 0.84 divided by 0.4 becomes 8.4 divided by 4, which is 2.1. Dividing by 10 or 100 moves the point left.
Decimals and fractions. A few conversions should be automatic: one half is 0.5, one fourth is 0.25, three fourths is 0.75, one fifth is 0.2, one eighth is 0.125, one third is 0.333 repeating. For anything else, divide the top by the bottom.
Worked examples
Example 1
Which of the following is the largest?
- A.0.8
- B.0.75
- C.0.085
- D.0.79
Show answer
Answer: A: 0.8
Give every number two places: 0.80, 0.75, 0.08, 0.79. The largest is 0.80. Choice C looks large because it has the most digits, but 0.085 is less than a tenth.
Example 2
6.4 + 0.58 =
- A.6.98
- B.7.02
- C.12.2
- D.6.122
Show answer
Answer: A: 6.98
Line up the points: 6.40 plus 0.58. Tenths and hundredths add to 98 hundredths with no carry, so the sum is 6.98. Choice C, 12.2, comes from lining up the right-hand digits instead of the decimal points.
Example 3
0.6 × 0.3 =
- A.0.018
- B.0.18
- C.1.8
- D.18
Show answer
Answer: B: 0.18
Six times 3 is 18. There is one decimal place in each factor, two in total, so the answer has two places: 0.18. Choice C, 1.8, gives only one place. A quick check: a bit more than half of 0.3 should be a bit less than 0.2, and 0.18 fits.
Example 4
8.4 ÷ 0.2 =
- A.0.42
- B.4.2
- C.42
- D.420
Show answer
Answer: C: 42
Slide both decimal points one place right: 84 divided by 2, which is 42. Choice B, 4.2, forgets to slide the dividend's point. Dividing by a number less than 1 should make the answer bigger than 8.4, which rules out A and B on sight.
Example 5
Round 14.096 to the nearest hundredth.
- A.14.09
- B.14.1
- C.14.10
- D.14.0
Show answer
Answer: C: 14.10
The hundredths digit is 9 and the digit after it is 6, so the 9 rounds up to 10, which carries: 14.096 becomes 14.10. Choice B, 14.1, is the same value but rounded to tenths, and the question asks for hundredths, so the zero stays.
Example 6
Which decimal is equal to 3/8?
- A.0.38
- B.0.375
- C.0.35
- D.0.3
Show answer
Answer: B: 0.375
One eighth is 0.125, so three eighths is 3 times 0.125, which is 0.375. Choice A, 0.38, is a rounding, and the test's correct choice is always exact.
Example 7
A pencil costs $0.35. How much do 12 pencils cost?
- A.$3.60
- B.$4.20
- C.$4.80
- D.$42.00
Show answer
Answer: B: $4.20
Thirty-five times 12 is 420. Two decimal places in 0.35 and none in 12, so the answer has two places: 4.20. Choice D, 42.00, leaves the point in the wrong place; a dozen pencils at less than half a dollar each cannot cost forty dollars.
Common mistakes
- Comparing by counting digits. 0.5 is more than 0.499, no matter how long the second number is.
- Lining up the right edge instead of the decimal points when adding. 6.4 plus 0.58 is not 12.2.
- Forgetting the place count when multiplying. Count places in both factors, then place the point.
- Sliding only one decimal point when dividing. Both the divisor and the dividend move the same number of places.
- Rounding in two steps. 14.096 rounds to 14.10, not first to 14.1 and then somewhere else. Look only at the next digit.
Practice
Decimals run through arithmetic, money and measurement word problems, non-geometric comparison, and number series that use them. The percents lesson covers the conversions this page relies on.