GED Number Operations Practice

Number operations are the foundation under every other part of the GED math test. These questions drill fractions, decimals, signed numbers, exponents, and number sense, mostly without a calculator, the way the first part of the real test asks for them.

Question 1 of 21

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What is 3/4 + 5/6?

Number operations study guide

Number operations are the skills every other GED math question is built on: fractions, decimals, signed numbers, exponents, and the order you do things in. The first part of the test is answered without a calculator, so this guide shows how to do each skill by hand, quickly and accurately.

Order of operations

When an expression has several operations, the order is fixed: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication does not always come before division; whichever appears first, reading left to right, goes first. The same is true for addition and subtraction.

Most wrong answers come from working strictly left to right and ignoring that order. In 18 − 6 ÷ 3, the division happens first, so the answer is 16, not 4.

Worked example

Evaluate 4 + 2(7 − 3)² ÷ 8.

  1. Parentheses: 7 − 3 = 4.
  2. Exponent: 4² = 16.
  3. Multiply and divide from left to right: 2 × 16 = 32, then 32 ÷ 8 = 4.
  4. Add: 4 + 4 = 8.

Fractions and mixed numbers

To add or subtract fractions, they need a common denominator. Rewrite each fraction over the least common denominator, then add or subtract only the numerators. Never add the denominators: 1/2 + 1/3 is 5/6, not 2/5.

To multiply, multiply the numerators and multiply the denominators, simplifying when you can. To divide, keep the first fraction, flip the second, and multiply. Turn mixed numbers into improper fractions before you multiply or divide, since 2 1/4 is 9/4.

Use a size check. Multiplying by a fraction less than 1 makes a number smaller, and dividing by one makes it larger. If you take 2/3 of a recipe and get more than the full amount, you divided when you should have multiplied.

Worked example

A roll holds 12 1/2 yards of ribbon. Each bow uses 5/8 yard. How many bows can be made?

  1. Write the mixed number as an improper fraction: 12 1/2 = 25/2.
  2. Divide by flipping the second fraction: 25/2 × 8/5.
  3. Multiply: 200/10 = 20 bows.

Decimals and percents

To multiply decimals, multiply as if they were whole numbers, then count the total decimal places in both factors and place the point that many places from the right. 0.6 × 0.4 has two decimal places in all, so it is 0.24.

To divide by a decimal, move the decimal point in both numbers the same number of places until the divisor is a whole number. 3.6 ÷ 0.04 becomes 360 ÷ 4 = 90. To change a fraction to a decimal, divide the numerator by the denominator, and multiply a decimal by 100 to write it as a percent.

Negative numbers and absolute value

Adding a negative is the same as subtracting, and subtracting a negative is the same as adding: 5 − (−3) = 8. When you multiply or divide, two numbers with the same sign give a positive result, and two with different signs give a negative result.

Absolute value is distance from zero, so it is never negative: |−9| = 9. Find the value inside the bars first, then take its absolute value. On a number line, the distance between two points is the absolute value of their difference, and a number farther left is always smaller. That is why −0.8 is less than −0.3.

Be careful with exponents and negatives. In (−4)², the negative is squared and the answer is 16. In −4², only the 4 is squared, and the answer is −16.

Worked example

A freezer is at −12°F. It warms 7 degrees, then cools 9 degrees. What is the final temperature?

  1. Warming adds: −12 + 7 = −5.
  2. Cooling subtracts: −5 − 9 = −14.
  3. The final temperature is −14°F.

Exponents, square roots, and scientific notation

An exponent tells how many times the base is used as a factor, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5. Any nonzero number to the zero power is 1. A negative exponent means a reciprocal: 3⁻² = 1/9. When multiplying powers of the same base, add the exponents; when dividing, subtract them.

A square root undoes squaring: √81 = 9 because 9² = 81. Learn the perfect squares from 1 to 225 so you can estimate. √40 lies between 6 and 7 because 36 < 40 < 49.

Scientific notation writes a number as a value from 1 up to 10 times a power of ten. A positive exponent means a large number and a negative exponent means a small one: 5.1 × 10⁴ = 51,000 and 5.1 × 10⁻⁴ = 0.00051. To multiply, multiply the front numbers and add the exponents.

Worked example

Compute (6 × 10⁵) × (4 × 10³) and write the answer in scientific notation.

  1. Multiply the front numbers: 6 × 4 = 24.
  2. Add the exponents: 10⁵ × 10³ = 10⁸.
  3. That gives 24 × 10⁸, and 24 is not between 1 and 10.
  4. Rewrite 24 as 2.4 × 10¹, so the answer is 2.4 × 10⁹.

Factors, multiples, and ordering numbers

A factor divides a number evenly, and a multiple is the result of multiplying it by a whole number. The greatest common factor (GCF) is the largest number that divides two numbers. It answers questions about splitting items into identical groups. The least common multiple (LCM) is the smallest number both divide into. It answers questions about when repeating events line up again.

To order numbers written in different forms, convert them all to decimals first. Then place them on a mental number line, remembering that negatives farther from zero are smaller.

Worked example

One alarm beeps every 8 minutes and another every 20 minutes. They beep together at noon. When do they next beep together?

  1. List multiples of 20: 20, 40, 60.
  2. Check which is also a multiple of 8: 40 = 8 × 5.
  3. The LCM is 40, so they beep together again at 12:40.

Unit conversions and estimation

Many GED word problems give measurements in one unit and ask for an answer in another: feet and yards, ounces and pounds, minutes and hours, or kilometers and miles. Customary conversions are not on the formula sheet, so learn the common ones: 12 inches in a foot, 3 feet in a yard, 16 ounces in a pound, 4 quarts in a gallon, and 60 minutes in an hour. Metric units step by powers of ten, so 1 kilometer is 1,000 meters and 1 liter is 1,000 milliliters.

The safest method is to multiply by a conversion fraction that equals 1. Write it with the unit you want to get rid of on the bottom. To change 90 inches to feet, multiply by 1 foot over 12 inches, and the inches cancel: 90 ÷ 12 = 7.5 feet. Before you move on, do a size check. A larger unit needs a smaller number, so a length in yards should be less than the same length in feet.

Rates need two conversions, one for each unit. A speed in feet per second becomes miles per hour only after you change both the feet and the seconds. Change one unit at a time and write down each step.

Estimation is the other half of this skill. Round each number to one or two nonzero digits, work the easier problem in your head, and use the result two ways. On the no-calculator part, it can point straight to the right choice. On the calculator part, it catches typing errors: if your exact answer is 10 times your estimate, a decimal point slipped. Some questions also need a real-world rounding decision. If 7.2 boxes are needed, you buy 8, because 7 boxes leave part of the job undone.

Worked example

A caterer needs 5 gallons of iced tea. Pitchers hold 48 fluid ounces each. How many pitchers must be filled? (1 gallon = 128 fluid ounces)

  1. Convert gallons to ounces: 5 × 128 = 640 fluid ounces.
  2. Divide by the size of one pitcher: 640 ÷ 48 ≈ 13.3.
  3. Estimate to check: 650 ÷ 50 = 13, so the result is reasonable.
  4. Round up, since 13 pitchers would fall short: the caterer fills 14 pitchers.

Checklist before test day

Practice each item by hand, without a calculator, until it feels routine.

  • Apply the order of operations, doing multiplication and division left to right.
  • Add fractions with unlike denominators and divide by flipping the second fraction.
  • Place the decimal point correctly when multiplying or dividing decimals.
  • Subtract a negative number and square a negative number without a sign error.
  • Estimate a square root and convert between standard and scientific notation.
  • Use the GCF for equal groups and the LCM for events that repeat.

Number Operations practice set 1: 21 questions with answers

Work each question on paper first, then open the answer to check your reasoning.

  1. 1.What is 3/4 + 5/6?

    1. A.1 1/2
    2. B.8/10
    3. C.1 5/6
    4. D.1 7/12
    Show answer

    Answer: D. 1 7/12

    Rewrite with a common denominator of 12: 9/12 + 10/12 = 19/12, which is 1 7/12. Choice 8/10 adds the numerators and the denominators separately, which is never valid.

  1. 2.A board 7 1/2 feet long is cut into pieces that are each 3/4 foot long. How many pieces are there?

    1. A.5 5/8
    2. B.45
    3. C.12
    4. D.10
    Show answer

    Answer: D. 10

    Divide: 7 1/2 ÷ 3/4 = 15/2 × 4/3 = 60/6 = 10. Choice 5 5/8 multiplies by 3/4 instead of dividing.

  1. 3.What is 4.6 × 0.3?

    1. A.0.138
    2. B.4.9
    3. C.13.8
    4. D.1.38
    Show answer

    Answer: D. 1.38

    Multiply 46 × 3 = 138, then count two decimal places because each factor has one: 1.38. Choice 13.8 keeps only one decimal place.

  1. 4.Evaluate: 5 + 3(8 − 2)² ÷ 9

    1. A.17
    2. B.32
    3. C.9
    4. D.12
    Show answer

    Answer: A. 17

    Work inside the parentheses (6), then the exponent (36), then multiply and divide left to right: 3 × 36 = 108 and 108 ÷ 9 = 12. Adding 5 gives 17; adding 5 + 3 first gives 32.

  1. 5.Evaluate: −6 − (−9) + (−4)

    1. A.−1
    2. B.−11
    3. C.−19
    4. D.1
    Show answer

    Answer: A. −1

    Subtracting a negative is the same as adding a positive, so the expression becomes −6 + 9 − 4 = −1. Choice −19 treats “− (−9)” as subtracting 9.

  1. 6.What is the value of |−12| − |5 − 9|?

    1. A.16
    2. B.−8
    3. C.8
    4. D.−16
    Show answer

    Answer: C. 8

    Absolute value is distance from zero, so |−12| = 12 and |5 − 9| = |−4| = 4, giving 12 − 4 = 8. Choice 16 leaves |5 − 9| as −4 and subtracts a negative.

  1. 7.Which number is between √50 and √70?

    1. A.7.0
    2. B.7.9
    3. C.60
    4. D.8.5
    Show answer

    Answer: B. 7.9

    7² = 49 and 8.4² = 70.56, so √50 is just above 7 and √70 is just below 8.4. Only 7.9 falls in that range; 60 is between 50 and 70 but not between their square roots.

  1. 8.Write 0.00042 in scientific notation.

    1. A.4.2 × 10⁻⁵
    2. B.4.2 × 10⁻⁴
    3. C.4.2 × 10⁻³
    4. D.4.2 × 10⁴
    Show answer

    Answer: B. 4.2 × 10⁻⁴

    Move the decimal point 4 places to the right to get 4.2, so the exponent is −4. A positive exponent would describe 42,000, a large number.

  1. 9.Two city buses leave a station at the same time. One returns every 12 minutes and the other every 18 minutes. After how many minutes will both be at the station together again?

    1. A.36
    2. B.6
    3. C.216
    4. D.30
    Show answer

    Answer: A. 36

    The next shared time is the least common multiple of 12 and 18, which is 36. Choice 6 is the greatest common factor, and 216 is a common multiple but not the least one.

  1. 10.A volunteer has 24 bottles of water and 36 granola bars. She wants to pack identical snack bags with nothing left over, using as many bags as possible. How many bags can she make?

    1. A.60
    2. B.72
    3. C.12
    4. D.6
    Show answer

    Answer: C. 12

    The number of bags must divide both 24 and 36, and the largest such number is the greatest common factor, 12, with 2 bottles and 3 bars per bag. Choice 72 is the least common multiple.

  1. 11.On a number line, point P is at −2.5 and point Q is at 4. What is the distance between P and Q?

    1. A.6.5
    2. B.1.5
    3. C.6
    4. D.−6.5
    Show answer

    Answer: A. 6.5

    Distance is the absolute value of the difference: |4 − (−2.5)| = 6.5. Choice 1.5 subtracts 2.5 from 4 as if both points were on the positive side.

  1. 12.Which number is greatest?

    1. A.−0.65
    2. B.−5/8
    3. C.−2/3
    4. D.−0.7
    Show answer

    Answer: B. −5/8

    As decimals, −2/3 ≈ −0.667 and −5/8 = −0.625. Of the four negatives, −0.625 is closest to zero, so it is the greatest. Choice −0.7 is farthest from zero, which makes it the least.

  1. 13.Simplify: √144 + √25

    1. A.17
    2. B.13
    3. C.60
    4. D.169
    Show answer

    Answer: A. 17

    √144 = 12 and √25 = 5, so the sum is 17. Choice 13 adds under one radical first, but √144 + √25 is not the same as √(144 + 25).

  1. 14.A checking account has a balance of $85. The owner writes checks for $120 and $46, then deposits $60. What is the new balance?

    1. A.−$141
    2. B.−$81
    3. C.−$21
    4. D.$21
    Show answer

    Answer: C. −$21

    85 − 120 − 46 = −81, and the $60 deposit brings it to −81 + 60 = −21, so the account is overdrawn by $21. Choice −$81 leaves out the deposit.

  1. 15.Simplify: (2/3 − 1/4) ÷ 5/6

    1. A.25/72
    2. B.5/6
    3. C.2
    4. D.1/2
    Show answer

    Answer: D. 1/2

    Inside the parentheses, 8/12 − 3/12 = 5/12. Dividing by 5/6 means multiplying by 6/5: 5/12 × 6/5 = 1/2. Choice 25/72 multiplies by 5/6 instead of dividing.

  1. 16.What is the value of 10³ × 10⁻⁵?

    1. A.100
    2. B.10⁻¹⁵
    3. C.0.1
    4. D.0.01
    Show answer

    Answer: D. 0.01

    Add the exponents: 3 + (−5) = −2, and 10⁻² = 1/100 = 0.01. Choice 10⁻¹⁵ multiplies the exponents instead of adding them.

  1. 17.What is the value of 5⁰ + 2⁻¹?

    1. A.3
    2. B.−1
    3. C.1.5
    4. D.0.5
    Show answer

    Answer: C. 1.5

    Any nonzero number to the zero power is 1, and a negative exponent means a reciprocal, so 2⁻¹ = 1/2. The sum is 1.5. Choice −1 treats 2⁻¹ as −2.

  1. 18.A runner finished 3/8 of a 24-mile training plan on Monday and 1/4 of it on Tuesday. How many miles of the plan are left?

    1. A.15
    2. B.18
    3. C.9
    4. D.6
    Show answer

    Answer: C. 9

    Monday covered 3/8 × 24 = 9 miles and Tuesday 1/4 × 24 = 6 miles, so 24 − 15 = 9 miles remain. Choice 15 is the distance already run.

  1. 19.Evaluate: −2² + (−3)² − 4 × (−1)

    1. A.−1
    2. B.9
    3. C.1
    4. D.17
    Show answer

    Answer: B. 9

    In −2² the exponent applies before the negative sign, so it equals −4, while (−3)² = 9 and −4 × (−1) = +4. The total is −4 + 9 + 4 = 9. Choice 17 treats −2² as +4.

  1. 20.Point A is at −7 on a number line and point B is at 5. What number is exactly halfway between them?

    1. A.−6
    2. B.1
    3. C.−1
    4. D.6
    Show answer

    Answer: C. −1

    The midpoint is the average of the endpoints: (−7 + 5) ÷ 2 = −1. Choice 6 is half the distance between the points, not the location of the midpoint.

  1. 21.Which list shows these numbers from least to greatest? 2.3 × 10⁻², 2.3 × 10⁻³, 3.2 × 10⁻², 0.03

    1. A.0.03, 2.3 × 10⁻³, 2.3 × 10⁻², 3.2 × 10⁻²
    2. B.2.3 × 10⁻³, 2.3 × 10⁻², 0.03, 3.2 × 10⁻²
    3. C.2.3 × 10⁻², 2.3 × 10⁻³, 0.03, 3.2 × 10⁻²
    4. D.2.3 × 10⁻³, 2.3 × 10⁻², 3.2 × 10⁻², 0.03
    Show answer

    Answer: B. 2.3 × 10⁻³, 2.3 × 10⁻², 0.03, 3.2 × 10⁻²

    In standard form the values are 0.023, 0.0023, 0.032, and 0.03, so the order is 0.0023, 0.023, 0.03, 0.032. Putting 0.03 last misses that 3.2 × 10⁻² = 0.032 is slightly larger.

More practice sets

Number Operations has 3 practice sets. None of these questions appear in the full practice tests.

What GED number operations questions cover

Expect to add, subtract, multiply, and divide fractions and mixed numbers, work with decimals, and evaluate expressions with the correct order of operations. Exponents, square roots, negative numbers, and absolute value appear often, along with scientific notation for very large and very small quantities.

Number sense questions ask you to compare and order values written in different forms, such as a fraction, a decimal, and a negative number in the same list, or to find factors and multiples in a scheduling or packing problem.

Practice these without a calculator

The first part of the Mathematical Reasoning test is answered without a calculator, and its questions lean on exactly these skills. If you reach for a calculator every time you see a fraction, that part will feel slow. Work the drills here by hand until common steps, like finding a common denominator or dividing by a decimal, feel routine.

Hand practice also helps on the calculator part. Knowing that 3.2 ÷ 8 is a little less than one half lets you catch a misplaced decimal point on the TI-30XS before it costs you a question.

Where points are lost

Most errors come from a few habits: adding numerators and denominators straight across, doing operations left to right without regard to order, and dropping a negative sign. Subtracting a negative and squaring a negative cause more wrong answers than any long calculation.

Before choosing an answer, estimate. If 2/3 of a recipe gives you more flour than the full recipe, something went wrong. A quick size check catches most of these mistakes.

Frequently asked questions

Are number operations questions on the no-calculator part of the GED?
Yes. The short first part of the math test is answered without a calculator, and it relies mostly on fractions, order of operations, exponents, and similar number skills.
Do I need to memorize perfect squares for the GED?
It helps. Knowing the squares from 1 through 15 makes square root and estimation questions much faster, especially when no calculator is available.