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GED Math: Number Sense and Operations Concepts

Importance ★★★★★ · 25% (estimate)

Overview

This chapter covers the number skills that sit underneath every other math question: comparing and ordering fractions and decimals, finding factors and multiples, working with exponents and roots, using absolute value, and doing multi-step arithmetic with positive and negative numbers. On the GED test a calculator is provided for most questions, but not all of them, so these skills need to work by hand as well.

To compare numbers written in different forms, change them all to one form first, usually decimals. Factors are numbers that divide evenly into a number, and multiples are what you get when you multiply it by a whole number (1, 2, 3, ...). The greatest common factor helps with cutting things into equal pieces, while the least common multiple tells you when repeating events line up again.

Exponents count repeated multiplication. When the same base is multiplied, the exponents are added, and a power raised to a power multiplies the exponents. For a number that is zero or positive, a square root undoes squaring; a cube root undoes cubing for any number. Absolute value is a distance from zero, so it is never negative, and the distance between two numbers on a number line is the absolute value of their difference.

For long calculations, follow the order of operations: parentheses, exponents, multiplication and division from left to right, and then addition and subtraction from left to right. A fraction with zero on the bottom has no value, so it is called undefined.

Key concepts

  • Order of operations

    Work inside parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

    Memory tipDo not simply work left to right; multiplication and division come before addition and subtraction.

  • GCF and LCM

    The greatest common factor (GCF) is the largest number that divides two numbers evenly. The least common multiple (LCM) is the smallest positive number that both numbers divide into evenly.

    Memory tipEqual pieces with no waste: GCF. Schedules lining up again: LCM.

  • Rules of exponents

    Same base multiplied: add the exponents. Power of a power: multiply the exponents. A negative exponent means one over the positive power, such as 2⁻² = 1/4. A fraction exponent like 2/3 means a cube root, then squaring.

    Memory tipCheck a rule by writing the multiplication out once.

  • Square and cube roots

    The square root of a number is the non-negative number you square to get it (√49 = 7, even though (−7)² is also 49). The cube root is what you cube to get it (∛64 = 4).

    Memory tipA square's side is √area; a cube's edge is ∛volume.

  • Absolute value and distance

    Absolute value is the distance of a number from 0, written |x|. It is never negative. The distance between two numbers a and b on a number line is |a − b|.

    Memory tipFrom −8 to 15 is 8 + 15 = 23 units.

  • Undefined expressions

    Division by zero has no value. An expression such as 12 / (x − 5) is undefined when its bottom equals zero, here at x = 5.

    Memory tipSet the denominator equal to 0 to find where it is undefined.

Often tested

  • Change fractions to decimals before ordering a mixed list.
  • Subtracting a negative number is the same as adding a positive one.
  • To multiply in scientific notation, multiply the front numbers and add the powers of 10. If the front number is 10 or more, move the decimal left and add 1 to the power (20 × 10⁵ = 2 × 10⁶).
  • Read word problems for "of what is left": that amount, not the original, gets multiplied.

Easy to confuse: GCF and LCM

ItemGCFLCM
MeaningLargest number that divides bothSmallest positive number both divide into
Example (12 and 18)636
Typical useCutting into equal, largest piecesWhen repeating schedules meet again

3 sample questions

Verified practice questions from this unit.

Practice questionReviewedQuantitative Problem Solving › Number Sense and Operations★★★★★

Question 1. What is the value of 18 − 3 × (2 + 4) ÷ 9?

  1. ① 2
  2. ② 10
  3. ③ 0
  4. ④ 16
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ④ 16

Key point: Parentheses, exponents, × and ÷, then + and −

  1. Parentheses first: 2 + 4 = 6.
  2. Multiply and divide from left to right: 3 × 6 = 18, then 18 ÷ 9 = 2.
  3. Subtract last: 18 − 2 = 16.

Check: On a calculator that follows the order of operations, 18 − 3 × 6 ÷ 9 also gives 16.

Wrong choices

  • 10: 10 works strictly left to right: (18 − 3) × 6 ÷ 9. Subtraction must wait until multiplication and division are done.
  • 0: 0 subtracts 3 × 6 from 18 first and then divides by 9; the division belongs to the 3 × 6 part only.
  • 2: 2 is only the value of 3 × 6 ÷ 9; it still has to be subtracted from 18.
Practice questionReviewedQuantitative Problem Solving › Number Sense and Operations★★★★★

Question 2. Which list puts these numbers in order from least to greatest? 0.65, 5/8, 0.58, 3/5

  1. ① 3/5, 5/8, 0.58, 0.65
  2. ② 0.65, 5/8, 3/5, 0.58
  3. ③ 0.58, 0.65, 3/5, 5/8
  4. ④ 0.58, 3/5, 5/8, 0.65
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ④ 0.58, 3/5, 5/8, 0.65

Key point: Change every number to a decimal before comparing

  1. Change the fractions to decimals: 3/5 = 0.6 and 5/8 = 0.625.
  2. Now compare four decimals: 0.65, 0.625, 0.58, 0.6.
  3. From least to greatest: 0.58, 0.6, 0.625, 0.65.
  4. Write them in their original form: 0.58, 3/5, 5/8, 0.65.

Check: 0.58 < 0.60 < 0.625 < 0.65, so each number is bigger than the one before it.

Wrong choices

  • 3/5, 5/8, 0.58, 0.65: 3/5 (0.6) is larger than 0.58, so 3/5 cannot come first.
  • 0.58, 0.65, 3/5, 5/8: This lists the decimals first and the fractions after, but 3/5 (0.6) and 5/8 (0.625) are smaller than 0.65.
  • 0.65, 5/8, 3/5, 0.58: This is the right order but backward: it goes from greatest to least.
Practice questionReviewedQuantitative Problem Solving › Number Sense and Operations★★★★★

Question 3. A board is 8 3/4 feet long. A carpenter cuts off a piece 2 5/8 feet long. How long is the board that is left?

  1. ① 6 2/4 feet
  2. ② 11 3/8 feet
  3. ③ 6 1/8 feet
  4. ④ 6 3/8 feet
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ③ 6 1/8 feet

Key point: Use a common denominator before subtracting fractions

  1. Write 3/4 with a denominator of 8: 3/4 = 6/8, so 8 3/4 = 8 6/8.
  2. Subtract the whole numbers: 8 − 2 = 6.
  3. Subtract the fractions: 6/8 − 5/8 = 1/8.
  4. The board left is 6 1/8 feet.

Check: 6 1/8 + 2 5/8 = 8 6/8 = 8 3/4 feet, the original length.

Wrong choices

  • 11 3/8 feet: 11 3/8 feet comes from adding the two lengths instead of subtracting.
  • 6 2/4 feet: 6 2/4 feet comes from subtracting the tops and the bottoms of the fractions separately, which is not how fractions work.
  • 6 3/8 feet: 6 3/8 feet treats 3/4 as a whole 1 before subtracting 5/8; 3/4 must be changed to 6/8, not 1.
Practice 5 questions from this unit →

AI-written · Verified · Written by AI. Only units that passed a separate AI check (re-reading facts and numbers) are published. Tell us if you find a mistake.