Number and Operations on Numbers

This category is about 19 percent of the Mathematics subtest^[1]. Its descriptors cover properties of real numbers, radicals and exponents, scientific notation, reasoning with units, choosing an appropriate level of accuracy, and multistep problems with rational numbers — "money, rate, percentage, average, estimation/rounding" among the settings^[2]. Much of the rest of the test is built on these skills.

7.1 The real numbers

• Integers: …, −2, −1, 0, 1, 2, …
• Rational numbers: any number that can be written as a fraction of two integers — including all integers, terminating decimals (0.75 = 3/4) and repeating decimals (0.333… = 1/3).
• Irrational numbers: numbers whose decimals never end or repeat, such as √2, √7 and π. The square root of any whole number that is not a perfect square is irrational.

Useful facts the test checks:

• rational + rational = rational; rational × rational = rational.
• A nonzero rational number times an irrational number is irrational (3 × √2 = 3√2).
• An irrational plus an irrational can be rational: √2 + (−√2) = 0.
• To place √20 on a number line: 4² = 16 and 5² = 25, so √20 is between 4 and 5, a little closer to 4 (4.47…).

Order of operations: parentheses, exponents, multiplication and division from left to right, addition and subtraction from left to right. 8 − 2 × 3 = 8 − 6 = 2, not 18.

Negative numbers: subtracting a negative adds (5 − (−3) = 8); a negative times a negative is positive; an even power of a negative is positive ((−2)⁴ = 16), an odd power is negative ((−2)³ = −8). Watch −2⁴: the exponent applies before the minus sign, so it equals −16.

7.2 Fractions, decimals and percents

These are three ways to write the same quantity. Convert freely:

• Adding fractions: common denominator first. 2/3 + 1/4 = 8/12 + 3/12 = 11/12.
• Multiplying: multiply straight across. 2/3 × 3/5 = 6/15 = 2/5.
• Dividing: multiply by the reciprocal. 3/4 ÷ 1/2 = 3/4 × 2 = 3/2.

7.3 Percent and percent change

Most percent problems are one of three questions: What is 15% of 80? (0.15 × 80 = 12). 12 is what percent of 80? (12 ÷ 80 = 0.15 = 15%). 12 is 15% of what? (12 ÷ 0.15 = 80).

Percent change = (new − old) ÷ old × 100%. A price rising from $40 to $50 is a (10 ÷ 40) = 25% increase. Always divide by the old (starting) value. The Bureau of Labor Statistics explains the idea with its price index: "An index of 110, for example, means there has been a 10-percent increase in price since the reference period"^[3]. It also shows why the starting value matters: two items with "the same change in index points" can have different percent changes, and the one with "the lower starting index value" has the greater percent change^[3].

The multiplier shortcut. An aligned readiness standard in the Test at a Glance states it as an example: "a + 0.05a = 1.05a means that 'increase by 5%' is the same as 'multiply by 1.05'"^[4]. So:

• increase by 20% → multiply by 1.20
• decrease by 20% → multiply by 0.80
• a 20% increase followed by a 20% decrease → 1.20 × 0.80 = 0.96, a 4% overall decrease, not zero.

Settings the test uses: sales tax, discounts and markups, tips, commissions, simple interest (I = Prt: principal × rate × time), and percent error.

7.4 Ratios, rates and proportions

A ratio compares quantities (3 cups of flour to 2 cups of water, 3:2). A rate is a ratio with different units ($12 per hour, 55 miles per hour). A unit rate has 1 in the denominator.

A proportion says two ratios are equal. Cross-multiply to solve: 3/2 = x/10 → 2x = 30 → x = 15.

Distance = rate × time is not on the formula sheet^[1]. A car at 60 miles per hour for 2.5 hours goes 150 miles; to travel 210 miles at 60 mph takes 210 ÷ 60 = 3.5 hours.

Best-buy questions: compare unit prices. 24 ounces for $4.80 is $0.20 per ounce; 32 ounces for $6.08 is $0.19 per ounce, so the larger package is the better buy.

7.5 Exponents and radicals

The Test at a Glance's own example is "an equivalent expression to the cube root of 27x⁵y⁶"^[2] — a descriptor in this category. Work it: ∛27 = 3; ∛y⁶ = y²; ∛x⁵ = ∛(x³ · x²) = x∛(x²). So the cube root is 3xy²∛(x²).

Simplifying square roots: pull out perfect-square factors. √72 = √(36 × 2) = 6√2. √50 + √8 = 5√2 + 2√2 = 7√2.

7.6 Scientific notation

A number in scientific notation is written as a number from 1 up to (but not including) 10, times a power of 10: 4,500,000 = 4.5 × 10⁶; 0.00072 = 7.2 × 10⁻⁴. The Test at a Glance names adding, subtracting, multiplying and dividing numbers in this form^[2].

• Multiply: multiply the front numbers, add the exponents. (3 × 10⁴)(2 × 10⁵) = 6 × 10⁹.
• Divide: divide the front numbers, subtract the exponents. (8 × 10⁹) ÷ (2 × 10³) = 4 × 10⁶.
• Adjust: (5 × 10³)(4 × 10²) = 20 × 10⁵ = 2 × 10⁶.
• Add or subtract: first rewrite with the same power of 10. 3 × 10⁵ + 4 × 10⁴ = 30 × 10⁴ + 4 × 10⁴ = 34 × 10⁴ = 3.4 × 10⁵.

The metric prefixes are powers of ten; NIST lists "kilo" as 1000 and "milli" as one-thousandth^[5]. A kilometer is 10³ meters; a milligram is 10⁻³ grams.

7.7 Units and appropriate accuracy

Reasoning with units is its own descriptor: "Reason quantitatively and use units to solve problems," and "Choose a level of accuracy appropriate to limitations on measurement"^[2].

Unit conversions. Multiply by a conversion fraction equal to 1, arranged so the unwanted unit cancels. 3.5 miles × (5,280 feet / 1 mile) = 18,480 feet. The formula sheet lists "1 mile= 5,280 feet"^[6], "1 gallon= 4 quarts"^[6], "1 pound= 16 ounces"^[6] and "1 inch = 2.54 centimeters"^[7], among others. Some conversions on it are approximate, marked with ≈ — for example 1 kilogram ≈ 2.2 pounds^[6].

Appropriate accuracy. An answer should not claim more precision than the measurements behind it. If a board is measured to the nearest inch, reporting its area to a thousandth of a square inch is false precision. If a question asks how many buses are needed for 130 people with 48 seats each, 130 ÷ 48 = 2.7, and the answer is 3 buses — you round up because you cannot leave people behind. If it asks how many full boxes of 12 can be packed from 130 items, the answer is 10 — you round down.

7.8 Averages

The mean (average) is the sum of the values divided by how many there are^[8]. Two useful twists:

• Missing value: four scores average 82. What fifth score makes the average 85? Total needed = 5 × 85 = 425; current total = 4 × 82 = 328; needed score = 97.
• Weighted average: 10 workers earn $15 an hour and 30 earn $19. Mean = (10 × 15 + 30 × 19) ÷ 40 = (150 + 570) ÷ 40 = $18. Not $17 — the larger group pulls the average toward $19.

Worked example

A store raises the price of a $60 jacket by 25%, then puts it on sale at 20% off the new price. What is the sale price?

Step 1. 60 × 1.25 = 75. Step 2. 75 × 0.80 = 60.

The sale price is $60 — the same as the original. The tempting answer is $63 (thinking +25% − 20% = +5%), but each percent applies to a different base.

Common traps in Number and Operations

• Dividing by the new value instead of the old one in percent change.
• Adding percents that apply to different bases.
• −2⁴ vs. (−2)⁴.
• Rounding down when the situation requires rounding up (buses, boxes to hold everything).
• Forgetting to convert units before computing.
• Choosing an intermediate result that appears among the options.

Key numbers — Chapter 7

• Number and Operations on Numbers: about 19 percent of Mathematics^[1].
• Mathematics: 55 questions in 90 minutes^[1].
• Not on the formula sheet: d = rt, the Pythagorean theorem, the quadratic formula^[1].
• Increase by p% = multiply by (1 + p/100)^[4].

Key takeaways — Chapter 7

1. Percent change divides by the starting value.
2. Chain percent changes by multiplying the multipliers.
3. Know the exponent rules, including negative and fractional exponents.
4. In scientific notation, add exponents to multiply and subtract to divide; align powers of ten to add.
5. Let the units guide the operation.
6. Round to fit the situation, not the calculator.

Chapter 7 quiz

1. A gym raised its monthly fee from $32 to $40. By what percent did the fee increase?

• A. 8%
• B. 80%
• C. 25%
• D. 20%

2. Which expression is equivalent to (2 × 10⁵)(4 × 10³)?

• A. 6 × 10⁸
• B. 8 × 10²
• C. 8 × 10¹⁵
• D. 8 × 10⁸

3. Which of these numbers is irrational?

• A. 2/9
• B. 0.125
• C. √18
• D. √49

4. A recipe uses 3 cups of rice for every 5 cups of water. How many cups of water are needed for 12 cups of rice?

• A. 20
• B. 14
• C. 7.2
• D. 36

5. What is the value of 8 − 3 × (−2)²?

• A. 44
• B. 20
• C. 100
• D. −4

6. A charter company has 130 passengers and buses that each seat 48. What is the fewest number of buses it needs?

• A. 3
• B. 2
• C. 2.7
• D. 4

7. A jacket priced at $80 is marked down 25%, and then 8% sales tax is added to the sale price. What is the total cost?

• A. $60.00
• B. $64.80
• C. $86.40
• D. $66.40

8. Which expression is equivalent to √75?

• A. 25√3
• B. 3√5
• C. 15√5
• D. 5√3

9. Four test scores average 78. What must a fifth score be for the average of all five to be 80?

• A. 80
• B. 88
• C. 90
• D. 82

10. A trail is 2.5 miles long. How many feet long is it?

• A. 15,840 feet
• B. 13,200 feet
• C. 2,112 feet
• D. 6,336 feet

11. What is the value of 16^(3/2)?

• A. 64
• B. 24
• C. 48
• D. 4,096

12. A car travels 195 miles in 3 hours at a constant speed. At that speed, how long will it take to travel 325 miles?

• A. 5.5 hours
• B. 6 hours
• C. 5 hours
• D. 4 hours

Chapter 7 quiz — answers and explanations

1. C. Percent change is the change divided by the starting value: (40 − 32) ÷ 32 = 8 ÷ 32 = 0.25, or 25%. The tempting 20% comes from dividing by the new fee, 8 ÷ 40. The BLS explains percent change from a starting value — an index moving from 100 to 110 is "a 10-percent increase"^[3].

2. D. Multiply the front numbers (2 × 4 = 8) and add the exponents (5 + 3 = 8). Multiplying the exponents gives the tempting 10¹⁵, which is the rule for a power raised to a power, not for a product. Operations in scientific notation are a named descriptor^{[2, 5]}.

3. C. 18 is not a perfect square, so √18 = 3√2 has a decimal that never ends or repeats. √49 is exactly 7, 0.125 is 1/8, and 2/9 is a fraction of integers, so all three are rational. The Test at a Glance names identifying rational and irrational numbers under this category^[2].

4. A. Set up the proportion 3/5 = 12/x; cross-multiplying gives 3x = 60, so x = 20. The tempting 7.2 comes from flipping the ratio (12 × 3/5). Proportional relationships are listed among the multistep settings in this category^[2].

5. D. Exponent first: (−2)² = 4. Then multiply: 3 × 4 = 12. Then subtract: 8 − 12 = −4. The tempting 20 comes from working left to right (8 − 3 = 5, then 5 × 4). Order of operations governs multistep numerical work in this category^[2].

6. A. 130 ÷ 48 ≈ 2.7, and a fraction of a bus is impossible; 2 buses seat only 96, so 3 are needed. The tempting 2 comes from rounding to the nearest whole number instead of rounding up to fit the situation. Choosing accuracy appropriate to the situation is a named descriptor^[2].

7. B. Sale price: 80 × 0.75 = 60. With tax: 60 × 1.08 = 64.80. The tempting $66.40 comes from adding the tax to the original price's 8% ($6.40) instead of taxing the sale price. The Test at a Glance's standard states that increasing by a percent is the same as multiplying by one plus that percent^[4].

8. D. 75 = 25 × 3, and √25 = 5, so √75 = 5√3. The tempting 3√5 swaps the factors: (3√5)² = 45, not 75. Rewriting expressions with radicals is a named descriptor in this category^[2].

9. B. The five scores must total 5 × 80 = 400; the four so far total 4 × 78 = 312; so the fifth must be 400 − 312 = 88. The tempting 82 raises 80 by the 2-point gap, ignoring that the new score must lift all four earlier scores. The mean is the sum of the values divided by their number^[8].

10. B. 2.5 × 5,280 = 13,200 feet, using the formula sheet's "1 mile= 5,280 feet". The tempting 15,840 uses 3 miles instead of 2.5. Reasoning with units is a named descriptor^{[2, 6]}.

11. A. A fractional exponent means root first, then power: √16 = 4, and 4³ = 64. The tempting 24 multiplies 16 by 3/2. Rewriting rational exponents is a named descriptor^[2].

12. C. Speed = 195 ÷ 3 = 65 miles per hour; time = 325 ÷ 65 = 5 hours. Distance = rate × time is one of the formulas the Test at a Glance says is not on the formula sheet, so it must be remembered^[1].

Sources cited in this excerpt

1. HiSET Test at a Glance — Mathematics. PSI Services LLC, 2025 (copyright year; retrieved 2026-09-24). https://hiset.org/wp-content/uploads/2025/05/taag.pdf
2. HiSET Test at a Glance — Mathematics Content Category I. PSI Services LLC, 2025 (copyright year; retrieved 2026-09-24). https://hiset.org/wp-content/uploads/2025/05/taag.pdf
3. Consumer Price Index: Questions and Answers (Internet Archive copy captured 2026-01-13). U.S. Bureau of Labor Statistics, captured 2026-01-13. http://web.archive.org/web/20260113062334id_/https://www.bls.gov/cpi/questions-and-answers.htm
4. HiSET Test at a Glance — Mathematics, aligned College and Career Readiness Standard 7.EE.2. PSI Services LLC, 2025 (copyright year; retrieved 2026-09-24). https://hiset.org/wp-content/uploads/2025/05/taag.pdf
5. Metric (SI) Prefixes. National Institute of Standards and Technology, Office of Weights and Measures, retrieved 2026-09-24. https://www.nist.gov/pml/owm/metric-si-prefixes
6. HiSET Mathematics Formula Sheet. PSI Services LLC, retrieved 2026-09-24. https://hiset.org/wp-content/uploads/2025/05/hiset-mathematics-formula-sheet.pdf
7. HiSET Mathematics Formula Sheet, as printed in Free Half-Length Practice Test FPT9 — Mathematics. PSI Services LLC, 2025 (released). https://hiset.org/wp-content/uploads/2025/11/hiset_fpt9_en_mathematics.pdf
8. NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.1 Measures of Location. National Institute of Standards and Technology, retrieved 2026-09-24. https://www.itl.nist.gov/div898/handbook/eda/section3/eda351.htm