GED Mathematical Reasoning Preparation: Lessons 2026

Lesson 1: The GED Math test, the calculator and a plan

The GED® test is a US high school equivalency test from GED Testing Service. It has 4 subjects: Mathematical Reasoning, Reasoning Through Language Arts, Science and Social Studies. You can take them one at a time, at your own pace. To earn the credential you need at least 145 on each subject.

This course covers the US GED Math test, with US units and dollars. It is not a Canadian credential: in Canada, the CAEC replaced the GED® in 2024.

How the Math test is set up

GED Testing Service's guide for educators splits the test into about 45% quantitative problem solving (numbers, measurement, data) and about 55% algebraic problem solving (expressions, equations, graphs and functions).

Scores

Only correct answers count, and there is no penalty for guessing, so answer every question. At a test center, if you need to retake a subject you get 2 more tries at any time; after 3 attempts you wait 60 days. Online, you get 1 retake before a 60-day wait.

Testing online

You must be in the United States and score “green” on the GED Ready® practice test for that subject within the last 60 days. You cannot use a physical calculator or scratch paper; you get an onscreen calculator, scratch pad and whiteboard.

The calculator and the formula sheet

The TI-30XS follows the order of operations, so 6 + 4 × 3 gives 18, not 30. Practice entering fractions, percents, powers, roots and scientific notation. The answer toggle key switches an answer between forms: fraction and decimal, exact root and decimal, and exact π and decimal.

The formula sheet lists area, perimeter and circumference; surface area and volume; mean and median; slope and the equations of a line; the quadratic formula; the Pythagorean theorem; simple interest; distance (d = rt); and total cost. You must know when to use each one. It uses π ≈ 3.14. Learn percents, the mode and range, probability and US unit conversions (12 inches = 1 foot) yourself.

Part 1 without a calculator

The GED's own sample asks you to put fractions and decimals in order without a calculator. First write them all in the same form: for 0.7, 2/3 and 3/5, use 0.7, about 0.667 and 0.6.

A plan for every word problem

1. Find exactly what the question asks.
2. Pick out the numbers you need. Some are extra.
3. Estimate with rounded numbers.
4. Solve step by step.
5. Check the units and ask: is the answer reasonable?

Example: $19.89 + $30.15 + $49.95 is about $20 + $30 + $50 = $100. The exact total is $99.99. If you cut a 12-foot board into 4 pieces and get 48 feet, something is wrong: pieces must be shorter than the board (12 ÷ 4 = 3 feet).

Topic quiz 1: The GED Math test, the calculator and a plan

Lesson 2: Number sense, operations, ratios and percents

Ordering numbers and absolute value

To compare fractions and decimals, change them all to decimals: 3/8 = 0.375 and 2/5 = 0.4. Absolute value is the distance from 0, so |−9| = 9. The distance between two numbers on a number line is the absolute value of their difference: from −7 to 5 is |5 − (−7)| = 12.

Factors and multiples

The greatest common factor (GCF) is the largest number that divides both numbers: for 24 and 36 it is 12. The least common multiple (LCM) is the smallest number both divide into: for 6 and 8 it is 24.

Fractions

• Multiply: change mixed numbers to improper fractions, then multiply tops and bottoms. 2 1/2 × 1 3/5 = 5/2 × 8/5 = 40/10 = 4.
• Divide: multiply by the reciprocal (flip the second fraction). 3/4 ÷ 3/8 = 3/4 × 8/3 = 2.
• Add or subtract: use a common denominator first.
• You can never divide by 0. An expression like 8 ÷ 0 is undefined (it has no value), but 0 ÷ 8 = 0.

Order of operations, powers and roots

Work in this order: parentheses, exponents, multiplication and division (left to right), then addition and subtraction (left to right). Example: 18 − 3 × (2 + 4)² ÷ 9 = 18 − 3 × 36 ÷ 9 = 18 − 12 = 6.

x² means x × x and x³ means x × x × x. The square root undoes squaring (√81 = 9) and the cube root undoes cubing (∛64 = 4). Take each root separately: √81 + ∛64 = 9 + 4 = 13. An exponent of 1/2 means a square root, so 16^(1/2) = √16 = 4.

Scientific notation writes big and small numbers with powers of 10. 4.2 × 10⁵: move the decimal point 5 places right to get 420,000.

Rates, ratios and proportions

A unit rate is an amount for one unit. To find the better buy, divide price by size: $5.40 ÷ 20 ounces = $0.27 per ounce beats $3.48 ÷ 12 ounces = $0.29 per ounce.

A proportion says two ratios are equal. If 3 cups of flour make 24 cookies, then 3/24 = x/40, and x = 3 × 40 ÷ 24 = 5 cups for 40 cookies. A map scale works the same way: if 1 inch = 15 miles, then 3.5 inches = 3.5 × 15 = 52.5 miles.

Percents

• Percent of a number: change the percent to a decimal and multiply. 20% of $3,200 = 0.20 × $3,200 = $640.
• Discount then tax: $80 at 25% off is $80 × 0.75 = $60. 6% tax: $60 × 0.06 = $3.60, so you pay $63.60.
• Percent change = change ÷ original amount. Rent from $1,200 to $1,290: $90 ÷ $1,200 = 0.075 = 7.5%.
• Simple interest (on the formula sheet): I = Prt. $2,500 at 4% for 3 years: 2,500 × 0.04 × 3 = $300.

Overtime is often paid at 1.5 times the regular rate for hours over 40 a week. At $18 an hour, 46 hours pays 40 × $18 = $720 plus 6 × $27 = $162, or $882.

Topic quiz 2: Number sense, operations, ratios and percents

Lesson 3: Geometry and measurement

The formula sheet gives you these formulas. Pick the right one and use the right units: feet for length, square feet for area, cubic feet for volume.

Perimeter, area and circles

A 14 ft by 9 ft yard has perimeter 2(14) + 2(9) = 46 ft. A circle with radius 5 in has area 3.14 × 5² = 78.5 square inches and circumference 2 × 3.14 × 5 = 31.4 inches. The diameter is twice the radius.

If you know the area and one side, divide: a rectangle with area 84 square inches and width 7 inches is 84 ÷ 7 = 12 inches long.

Composite shapes

Split a complex shape into simple ones. Add areas that are joined together, and subtract areas that are cut out. A 20 ft by 12 ft patio (240 square feet) with a 5 ft by 5 ft planter (25 square feet) has 215 square feet of paving.

The Pythagorean theorem

In a right triangle, a² + b² = c², where c is the hypotenuse, the longest side across from the right angle. Legs of 9 and 12 give c² = 81 + 144 = 225, so c = 15. To find a leg, subtract: a 13 ft ladder with its base 5 ft from a wall reaches h² = 169 − 25 = 144, so h = 12 ft.

Volume and surface area

Surface area is the total area of all the faces, like wrapping paper. For a box, SA = 2(lw + lh + wh). A 2 ft × 3 ft × 4 ft box has volume 24 cubic feet and surface area 2(6 + 8 + 12) = 52 square feet.

Pyramids and cones hold 1/3 of a prism or cylinder with the same base and height. A cone with r = 3 in and h = 8 in has V = 1/3 × 3.14 × 9 × 8 = 75.36 cubic inches.

Topic quiz 3: GED geometry and measurement

Lesson 4: Data, statistics and probability

Measures of center and spread

With an even number of values, the median is the mean of the two middle values: for 7, 9, 11, 12, 15, 20 it is (11 + 12) ÷ 2 = 11.5.

Missing value: if 5 tests have a mean of 84, they add to 84 × 5 = 420. Subtract the known scores to find the missing one. Weighted average: multiply each value by its weight, add, and divide by the total weight. 3 hours at $16 and 5 hours at $20 average ($48 + $100) ÷ 8 = $18.50 an hour.

Graphs and tables

• Bar graphs and tables: find changes by subtracting. From 128 to 150 is +22.
• Circle graphs show parts of a whole. Find a part by multiplying the total by its percent.
• Scatter plots show pairs of data. Points that rise from left to right show a positive association; points that fall show a negative association; no pattern means no association.
• Box plots show five numbers: minimum, first quartile (Q1), median, third quartile (Q3) and maximum. The interquartile range is Q3 − Q1, the spread of the middle half.

Probability

Probability = number of favorable outcomes ÷ number of possible outcomes. It is between 0 (impossible) and 1 (certain). With 4 red, 6 blue and 5 green marbles, P(blue) = 6/15 = 2/5.

• Independent events (one does not affect the other): multiply. P(heads and a 6) = 1/2 × 1/6 = 1/12.
• Mutually exclusive events (cannot both happen): add. P(1 or 6 on one roll) = 1/6 + 1/6 = 1/3.
• Without replacement: after each pick, there is one fewer item. Two red pens from 4 red and 6 black: 4/10 × 3/9 = 2/15.

Counting

Counting principle: multiply the number of choices for each step. 4 shirts × 3 pants × 2 pairs of shoes = 24 outfits.

A permutation counts arrangements when order matters: choosing a president, vice president and secretary from 5 people gives 5 × 4 × 3 = 60. A combination counts groups when order does not matter: choosing 2 of 6 toppings gives (6 × 5) ÷ (2 × 1) = 15.

Topic quiz 4: Data, statistics and probability

Lesson 5: Expressions, equations and inequalities

Expressions

To evaluate, replace the variable and follow the order of operations. Use parentheses for negatives: if x = −2, then 3x² − 2x + 5 = 3(4) + 4 + 5 = 21.

To write an expression, look for a one-time amount and a rate. A $65 visit plus $90 an hour for h hours is 65 + 90h. To simplify, distribute and combine like terms. Watch the signs: 4(x + 3) − 2(x − 5) = 4x + 12 − 2x + 10 = 2x + 22.

Linear equations

Do the same thing to both sides until the variable is alone. 5x − 7 = 3x + 9 → 2x − 7 = 9 → 2x = 16 → x = 8. Check by putting 8 back in: 33 = 33.

• Consecutive numbers: call them n, n + 1 and n + 2. If they add to 72, then 3n + 3 = 72 and n = 23.
• When do two plans cost the same? Set the costs equal. $30 + $0.10m = $45 gives m = 150 minutes.
• Formulas: solve for a letter the same way. From d = rt, divide by r to get t = d/r.

Systems of two equations

Elimination: add or subtract the equations to remove a variable. x + y = 12 and x − y = 4 add to 2x = 16, so x = 8 and y = 4. Substitution: solve one equation for a variable and put it into the other. 20 tickets (a + c = 20) worth $190 at $12 and $7: 12a + 7(20 − a) = 190, so 5a = 50 and a = 10 adult tickets.

Inequalities

Solve like an equation, with one extra rule: when you multiply or divide by a negative number, flip the sign. −2x + 6 ≤ 14 → −2x ≤ 8 → x ≥ −4. On a number line, use an open dot for < or > and a closed dot for ≤ or ≥.

Word problems often use “at most” (≤) or “at least” (≥). With $60, a $15 ticket and $4.50 snacks: 15 + 4.50n ≤ 60, so n ≤ 10 snacks.

Polynomials and factoring

To multiply two binomials, multiply each term by each term: (2x + 3)(x − 4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12.

To factor x² + bx + c, find two numbers that multiply to c and add to b. For x² + 5x − 14, use 7 and −2: (x + 7)(x − 2). A difference of squares factors as a² − b² = (a + b)(a − b), so (x² − 16) ÷ (x + 4) = x − 4.

Quadratic equations

Get 0 on one side and factor. If a product is 0, a factor is 0: x² − 6x + 8 = (x − 2)(x − 4) = 0, so x = 2 or x = 4. If x² = 9, then x = 3 or x = −3.

If it does not factor easily, use the quadratic formula from the formula sheet: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) ÷ 2a. For 2x² + 3x − 2 = 0: x = (−3 ± √25) ÷ 4, so x = 1/2 or x = −2.

Topic quiz 5: Expressions, equations and inequalities

Lesson 6: Graphs, lines and functions

The coordinate plane

A point (x, y) is x steps across and y steps up or down from the origin (0, 0). The axes make 4 quadrants: I (+, +) upper right, II (−, +) upper left, III (−, −) lower left and IV (+, −) lower right.

Slope and intercepts

Slope = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁). Through (2, 3) and (6, 11): 8 ÷ 4 = 2. A line going up from left to right has a positive slope; going down, a negative slope.

In slope-intercept form, y = mx + b, m is the slope and b is the y-intercept (where the line crosses the y-axis). Solve for y to change forms: 2x + y = 6 becomes y = −2x + 6, so b = 6. The x-intercept is where y = 0.

In real life, slope is a unit rate. If pay goes from $18 to $36 to $54 for 1, 2 and 3 hours, the slope is $18 per hour. The y-intercept is the starting amount: a $24,000 car that loses $3,000 a year has value v = −3,000x + 24,000.

Writing equations of lines

• From two points: find the slope, then put one point into y = mx + b to find b. Through (1, 5) and (3, 9): m = 2, 5 = 2 + b, b = 3, so y = 2x + 3.
• From a slope and a point: use point-slope form y − y₁ = m(x − x₁). Slope −4 through (2, −1): y + 1 = −4(x − 2), so y = −4x + 7.

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals: flip the fraction and change the sign. A line perpendicular to y = (2/3)x + 1 has slope −3/2.

Functions

A function gives exactly one output for each input. In a list of points, no x-value may repeat with a different y-value: (2, 1) and (2, 4) together mean it is not a function. Two different x-values can share a y-value.

f(x) is read “f of x.” To find f(3) for f(x) = 2x² − 3x + 1, replace every x with 3: 2(9) − 9 + 1 = 10.

Comparing functions: find each rate of change, whether it is given as an equation, a table or words. Car A goes 210 miles in 3.5 hours, or 60 mph; Car B follows d = 58t, or 58 mph. Car A is faster by 2 mph.

Quadratic functions

The graph of y = ax² + bx + c is a U-shaped parabola. It opens up if a is positive and down if a is negative. Its turning point, the vertex, is at x = −b ÷ (2a). For a ball with height h = −16t² + 64t, the top is at t = −64 ÷ (−32) = 2 seconds, at 64 feet.

The graph crosses the x-axis where y = 0. For y = x² − 4, solve x² − 4 = 0 to get x = 2 and x = −2. The y-intercept is (0, −4), which here is also the lowest point.

Topic quiz 6: Graphs, lines and functions