This is an estimate based on your percent correct on our practice tests. The official GED score conversion is not public, so your real score may differ. Scores of 145+ pass; 165+ is College Ready.
Simplify (divide top and bottom by 6): \dfrac{18}{12} = \dfrac{3}{2} = 1\tfrac{1}{2}.
Dana should use 1\tfrac{1}{2} cups of flour.
Check:1\tfrac{1}{2} is less than 2\tfrac{1}{4}, which makes sense for a smaller batch. Also 1\tfrac{1}{2} \div 2\tfrac{1}{4} = \tfrac{3}{2} \times \tfrac{4}{9} = \tfrac{12}{18} = \tfrac{2}{3} ✓.
Tip: Never multiply only the whole-number part of a mixed number. Convert to an improper fraction, multiply straight across, then simplify. You can cancel before multiplying: \tfrac{9}{4} \times \tfrac{2}{3} = \tfrac{3}{2} \times \tfrac{1}{1}.
Question 2 of 5 · Add, subtract, multiply and divide rational numbers
No calculatorFill in
What is the value of the expression below?
20 - 2(5 - 8)^2 \div 3
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Follow the order of operations: parentheses, exponents, then multiply/divide left to right, then add/subtract.
Step-by-step solution
Parentheses:5 - 8 = -3, so the expression is 20 - 2(-3)^2 \div 3.
Exponent:(-3)^2 = (-3)(-3) = 9. Now we have 20 - 2 \cdot 9 \div 3.
Multiply and divide, left to right:2 \cdot 9 = 18, then 18 \div 3 = 6.
Subtract:20 - 6 = 14.
The value is 14.
Check: A common slip is subtracting first: (20 - 2) \cdot 9 \div 3 = 54. That ignores the rule that multiplication comes before subtraction. Another slip is -3^2 = -9; but here the parentheses mean the whole -3 is squared, giving +9.
Tip: Rewrite the expression after every single step. On the no-calculator part, one clean line per step prevents sign and order mistakes.
Question 3 of 5 · Rules of exponents
No calculator
Which expression is equivalent to \dfrac{a^5 \cdot a^3}{a^2}, where a \ne 0?
Hint
When you multiply powers with the same base, add the exponents. When you divide, subtract them.
Tip: Same base: multiply → add exponents; divide → subtract exponents; power of a power → multiply exponents. If you are unsure, test with a small number like a = 2 — it takes seconds even without a calculator.
Question 4 of 5 · Absolute value and distance on a number line
No calculator
The thermometer in Luis's freezer reads -8^\circ\text{F}. The thermometer in his kitchen reads 68^\circ\text{F}.
How many degrees apart are the two temperatures?
Hint
The distance between two numbers on a number line is the absolute value of their difference.
Step-by-step solution
The distance between two numbers a and b is \lvert a - b \rvert.
Subtract: 68 - (-8) = 68 + 8 = 76.
Take the absolute value: \lvert 76 \rvert = 76.
The temperatures are 76^\circF apart.
Check: Count on a number line in two pieces: from -8 up to 0 is 8 degrees, and from 0 up to 68 is 68 degrees. 8 + 68 = 76 ✓.
Tip: When one value is below zero and one is above, the distance is the sum of their distances from zero. Subtracting a negative is the same as adding. A distance is never negative.
Question 5 of 5 · Squares and square roots
No calculatorHot spot
Four points are marked on the number line below.
Click the point that best represents \sqrt{30}.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
Find the two perfect squares that 30 falls between. Is 30 closer to the smaller one or the larger one?
Step-by-step solution
List perfect squares near 30: 5^2 = 25 and 6^2 = 36.
Since 25 < 30 < 36, we know 5 < \sqrt{30} < 6. Only point C (at 5.5) lies between 5 and 6.
Fine-tune: 30 is 5 more than 25 and 6 less than 36, so \sqrt{30} is a little below the middle of 5 and 6. Check: 5.5^2 = 30.25, very close to 30. (Exactly, \sqrt{30} \approx 5.48.)
The best point is C.
Check: Point A (about 3.1) is near the cube root of 30 (3.1^3 \approx 29.8), not the square root. Point B: 4.5^2 = 20.25, too small. Point D: 6.5^2 = 42.25, too big.
Tip: Memorize the perfect squares from 1^2 to 15^2 (1, 4, 9, \ldots, 225). They let you place any square root between two whole numbers without a calculator.
Question 1 of 41 · Unit rates
At the grocery store, a 24-ounce jar of peanut butter costs $4.32 and a 40-ounce jar costs $6.80.
What is the price per ounce of the jar that is the better buy?
Hint
Unit price = total price ÷ number of ounces. The better buy has the lower unit price.
Step-by-step solution
24-oz jar:\dfrac{\$4.32}{24\text{ oz}} = \$0.18 per ounce.
40-oz jar:\dfrac{\$6.80}{40\text{ oz}} = \$0.17 per ounce.
Compare: \$0.17 < \$0.18, so the 40-ounce jar is the better buy.
Tip: For "price per unit," always put money on top and the units on the bottom. If you divide the other way (40 \div 6.80), you get ounces per dollar — a real number, but it answers a different question.
Question 2 of 41 · Linear expressions: combine, factor, expand
Which expression is equivalent to 3(2x - 4) - (x - 5)?
Hint
Distribute the 3 into the first parentheses and the -1 into the second. Watch the signs.
Step-by-step solution
Distribute the 3: 3(2x - 4) = 6x - 12.
Distribute the minus sign (think -1): -(x - 5) = -x + 5.
Tip: A minus sign in front of parentheses changes the sign of every term inside. Plugging in a small number like x = 2 is a fast way to test any answer choice.
Question 3 of 41 · Pythagorean theorem
Marcus leans a 17-foot ladder against the wall of his house to clean the gutters. The foot of the ladder is 8 feet from the wall, as shown.
How high up the wall, h, does the ladder reach?
Hint
The wall, the floor and the ladder form a right triangle. Which side is the hypotenuse?
Step-by-step solution
The ladder is across from the right angle, so it is the hypotenuse: c = 17. The legs are a = 8 and b = h.
Pythagorean theorem (on the formula sheet): a^2 + b^2 = c^2.
Check:8^2 + 15^2 = 64 + 225 = 289 = 17^2 ✓. Also, a leg must be shorter than the hypotenuse, and 15 < 17 ✓.
Tip: When you are finding a leg, subtract the squares: b = \sqrt{c^2 - a^2}. Only add the squares when you are finding the hypotenuse. 8-15-17 is a common right-triangle triple worth recognizing.
Question 4 of 41 · Write linear equations from context
A home cleaning service charges $40 per visit plus a one-time supply fee of $25. So far, Jen has paid the service a total of $345.
Which equation can be used to find v, the number of visits Jen has paid for?
Hint
Which amount changes with the number of visits, and which amount is charged only once?
Step-by-step solution
The rate is $40 for each visit, so visits cost 40v dollars.
The supply fee of $25 is charged once, so it is added one time: 40v + 25.
This total equals what Jen paid: 40v + 25 = 345.
The equation is 40v + 25 = 345.
Check: Solve it: 40v = 320, so v = 8. Eight visits cost 8 \times 40 = 320, plus the $25 fee is $345 ✓.
Tip: In "$A per item plus a one-time $B" problems, the per-item amount is the coefficient of the variable and the one-time amount is the constant. Ask yourself which number gets multiplied by the variable.
Question 5 of 41 · Bar graphs and circle graphs
The bar graph shows the Nguyen family's grocery spending for six months.
How much more did the family spend in the month with the highest spending than in the month with the lowest spending?
Hint
First find the tallest bar and the shortest bar. Look closely — two of the shorter bars are close in height.
Step-by-step solution
Highest month: April, with $510.
Lowest month: compare the short bars — January $420, February $385, June $395. The lowest is February, $385.
Difference:510 - 385 = 125.
The family spent $125 more in April than in February.
Check: Add back: 385 + 125 = 510 ✓. And $385 is less than every other bar value ($395, $420, $450, $470, $510) ✓.
Tip: "How much more" means subtract. On bar graphs, list the values you need before computing — bars that look alike (here February and June) are a favorite GED trap.
Question 6 of 41 · Points in the coordinate plane
Hot spot
Rosa is planning her backyard on a coordinate grid. Each unit is 1 yard, and the back door of the house is at the origin. She wants to put a birdbath at the point (-3,\ 4).
Click the location of the birdbath on the coordinate plane.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
The first number tells you how far to move left or right. The second tells you how far to move up or down.
Step-by-step solution
An ordered pair is written (x,\ y). Here x = -3 and y = 4.
Start at the origin (0,\ 0).
x = -3: move 3 units left (negative means left).
y = 4: from there, move 4 units up (positive means up).
The birdbath is in the upper-left part of the grid (Quadrant II), at (-3,\ 4).
Check: The point is directly above -3 on the x-axis and directly to the left of 4 on the y-axis ✓.
Tip: Remember "run before you jump": move along the x-axis first, then up or down. Mixing up the order gives (4,\ -3), which is a completely different spot.
Check: One line on the calculator: 850 \times 0.8 \times 1.07 = 727.6 ✓.
Tip: You cannot combine "20% off" and "7% tax" into "13% off," because the two percents are taken of different amounts. Use multipliers: p\% off → multiply by (1 - p/100); p\% tax → multiply by (1 + p/100).
Question 8 of 41 · Solve quadratic equations
Select all
Select all the solutions of the equation
x^2 - 5x - 14 = 0
Select all that apply.
Hint
Look for two numbers that multiply to -14 and add to -5.
Step-by-step solution
Find two numbers with product -14 and sum -5: -7 and 2 (since -7 \times 2 = -14 and -7 + 2 = -5).
Tip: The numbers inside the factors have the opposite signs of the solutions: (x - 7) gives x = +7. If factoring is hard, the quadratic formula on the formula sheet always works.
Question 9 of 41 · Order fractions and decimals, including on a number line
Drag and drop
Kevin is sorting drill bits by size. The bit sizes, in inches, are shown on the tiles.
Drag each size into the table so the bits are in order from smallest to largest.
Smallest
2nd
3rd
Largest
Tap a tile, then tap a box to place it.
Hint
Change every fraction to a decimal so you can compare them the same way.
Step-by-step solution
Convert the fractions to decimals:
\tfrac{3}{8} = 3 \div 8 = 0.375
\tfrac{5}{16} = 5 \div 16 = 0.3125
Write all four with the same number of decimal places: 0.3000,\ 0.3125,\ 0.3500,\ 0.3750.
Order from smallest to largest: 0.3 < 0.3125 < 0.35 < 0.375.
The order is 0.3 in, \tfrac{5}{16} in, 0.35 in, \tfrac{3}{8} in.
Check: As sixteenths: 0.3 = 4.8, \tfrac{5}{16} = 5, 0.35 = 5.6, \tfrac{3}{8} = 6 sixteenths — still increasing ✓.
Tip: Do not judge fractions by the size of the denominator: \tfrac{5}{16} has a bigger denominator than \tfrac{3}{8} but is the smaller number. Converting everything to decimals (or to a common denominator) removes the guesswork.
Question 10 of 41 · Compare functions in different forms
Two lawn-care companies charge by the hour. Company A's prices are shown in the table. Company B charges according to the equation C = 12h + 42, where C is the cost in dollars for h hours.
Hours, h
1
3
5
7
Company A cost, C ($)
45
75
105
135
For how many hours of work would the two companies charge the same amount?
Hint
Write an equation for Company A from the table (find its rate per hour and its starting fee), then set the two costs equal.
Step-by-step solution
Company A's rate: each 2 extra hours add $30, so the rate is \dfrac{75 - 45}{3 - 1} = \dfrac{30}{2} = \$15 per hour.
Company A's starting fee: at 1 hour the cost is $45, so 45 - 15 = \$30. Company A: C = 15h + 30.
Set the costs equal:15h + 30 = 12h + 42.
Subtract 12h: 3h + 30 = 42. Subtract 30: 3h = 12. Divide by 3: h = 4.
The companies charge the same for 4 hours.
Check: Company A: 15(4) + 30 = \$90. Company B: 12(4) + 42 = \$90 ✓.
Tip: To compare a table with an equation, turn the table into an equation first. The rate is the slope (\tfrac{\text{change in } C}{\text{change in } h}) — be careful when the table's inputs jump by 2, not by 1.
Question 11 of 41 · Real-world linear equations
Fill in
A cooking guide says to roast a turkey for 15 minutes per pound and then let it rest for 20 minutes before carving. Alicia's turkey needs a total of 3 hours 50 minutes for roasting and resting.
How many pounds does her turkey weigh?
pounds
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Change the total time to minutes first, then write an equation with w for the weight in pounds.
Step-by-step solution
Convert to minutes:3\text{ h }50\text{ min} = 3 \times 60 + 50 = 230 minutes.
Equation: 15 minutes per pound plus 20 minutes of rest: 15w + 20 = 230.
Subtract 20 from both sides: 15w = 210.
Divide by 15: w = \dfrac{210}{15} = 14.
The turkey weighs 14 pounds.
Check:15 \times 14 = 210 minutes of roasting, plus 20 minutes of rest = 230 minutes = 3 h 50 min ✓.
Tip: Make units match before you build the equation. Writing 3 h 50 min as "3.50 hours" is a trap — 50 minutes is \tfrac{50}{60} \approx 0.83 hour, not 0.50 hour.
Question 12 of 41 · Rectangular prisms
A fish tank in the Parkers' living room is shaped like a rectangular prism, as shown.
What is the volume of the tank, in cubic inches?
Hint
Volume of a rectangular prism = length × width × height.
Step-by-step solution
Volume of a rectangular prism (on the formula sheet): V = lwh.
Tip: Volume (how much water fits inside) uses all three dimensions multiplied. Surface area (how much glass) adds the areas of the six faces. Read the question to see which one is asked.
Question 13 of 41 · Write inequalities from context
Tasha can spend at most $150 on her son's birthday party. The cake costs $45, and each guest's food and party favor cost $7.50.
Which inequality shows the number of guests, g, that Tasha can invite?
Hint
"At most" means the total can equal $150 or be less than $150. Which symbol says that?
Step-by-step solution
Cost per guest: $7.50 for each guest gives 7.50g.
One-time cost: the cake, $45, is added once: total cost = 45 + 7.50g.
"At most $150": the total must be less than or equal to 150, so use \le.
The inequality is 45 + 7.50g \le 150.
Check: Solving gives 7.50g \le 105, so g \le 14. With 14 guests: 45 + 7.50(14) = 45 + 105 = \$150, which is allowed because "at most" includes $150 exactly ✓.
Tip: Key phrases: "at most / no more than" → \le; "at least / no less than" → \ge; "more than" → >; "less than" → <.
Question 14 of 41 · Mean, median, mode, range, weighted average
Drop-down
The Johnsons recorded their grocery bill (in dollars) for 7 weeks:
112,\ 98,\ 140,\ 120,\ 98,\ 105,\ 132
Select the correct values to complete the sentence.
The median weekly bill is $ and the mean weekly bill is $.
Choose an answer from each menu.
Hint
For the median, put the numbers in order first. For the mean, add them all and divide by how many there are.
Step-by-step solution
Median
Order the bills: 98,\ 98,\ 105,\ 112,\ 120,\ 132,\ 140.
With 7 values, the median is the 4th one: $112.
Mean
Add: 112 + 98 + 140 + 120 + 98 + 105 + 132 = 805.
Divide by the number of weeks: 805 \div 7 = 115. The mean is $115.
Check: Mean: 7 \times 115 = 805 ✓. Median: three values (98, 98, 105) are below 112 and three (120, 132, 140) are above ✓.
Tip: The median of the list as written (120 here) is a trap — always sort first. Also keep the four "centers" straight: mean = average, median = middle, mode = most frequent (98), and the midrange (98 + 140) \div 2 = 119 is not a GED measure of center.
Question 15 of 41 · Slope from a graph, equation or table
Water is running into a bathtub that already had some water in it. The table shows the amount of water in the tub.
Time (minutes)
0
2
4
6
Water (gallons)
5
13
21
29
What is the rate of change of the water in the tub, in gallons per minute?
Hint
Rate of change = change in gallons ÷ change in minutes. Use any two columns.
Step-by-step solution
Slope (rate of change) = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{y_2 - y_1}{x_2 - x_1} (on the formula sheet).
Use the first two columns (0,\ 5) and (2,\ 13): \dfrac{13 - 5}{2 - 0} = \dfrac{8}{2} = 4.
Confirm with other columns (2,\ 13) and (6,\ 29): \dfrac{29 - 13}{6 - 2} = \dfrac{16}{4} = 4 ✓.
The water rises at 4 gallons per minute.
Check: Start at 5 gallons and add 4 per minute: after 6 minutes, 5 + 4 \times 6 = 29 gallons, matching the table ✓.
Tip: In a table, the change between neighboring columns is not the slope unless the x-values go up by exactly 1. Here they go up by 2, so divide the change of 8 by 2.
Question 16 of 41 · Composite 2-D figures
Fill in
Ben is putting new vinyl flooring in the L-shaped room shown below. The flooring is sold in boxes, and each box covers 20 square feet. He can only buy whole boxes.
What is the least number of boxes Ben must buy to cover the whole floor?
boxes
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Split the L into two rectangles, add their areas, then divide by 20. Can Ben buy part of a box?
Step-by-step solution
Split the room with a horizontal cut at the notch:
Top rectangle: 14\text{ ft} \times 8\text{ ft} = 112\text{ ft}^2.
Seven boxes cover only 7 \times 20 = 140\text{ ft}^2, which is not enough. Ben must round up.
Ben must buy 8 boxes.
Check: Another way — big rectangle minus the missing corner: 14 \times 12 - 5 \times 4 = 168 - 20 = 148\text{ ft}^2 ✓. And 8 \times 20 = 160 \ge 148 ✓.
Tip: When you buy supplies (boxes, cans of paint, bags of soil), always round up, even if the decimal is small. Rounding to the nearest whole number (7) would leave part of the floor bare.
Question 17 of 41 · Write linear expressions from context
A pizza shop charges $12.50 for each large pizza plus a $4 delivery fee for the whole order.
Which expression gives the total cost, in dollars, of having p large pizzas delivered?
Hint
Which amount is multiplied by the number of pizzas, and which is paid only once?
Step-by-step solution
Each pizza costs $12.50, so p pizzas cost 12.50 \times p = 12.50p dollars.
The delivery fee of $4 is paid once for the whole order, so add it one time.
Total cost: 12.50p + 4.
The expression is 12.50p + 4.
Check: Try p = 2: two pizzas cost $25, plus $4 delivery = $29. The expression gives 12.50(2) + 4 = 29 ✓.
Tip: Test any expression with a small, easy number (like 1 or 2 items) and compare it with the cost you work out in your head. A wrong expression usually fails this test right away.
Question 18 of 41 · Parallel and perpendicular lines
Drag and drop
Look at the line y = 2x - 3.
Drag one equation into each box: a line that is parallel to y = 2x - 3 and a line that is perpendicular to y = 2x - 3. Not every equation will be used.
Parallel
Perpendicular
Tap a tile, then tap a box to place it.
Hint
Parallel lines have the same slope. Perpendicular slopes are opposite reciprocals — flip the fraction AND change the sign.
Step-by-step solution
The given line y = 2x - 3 has slope m = 2.
Parallel: same slope, m = 2. Only y = 2x + 5 has slope 2 (a different y-intercept makes it a separate line).
Perpendicular: the opposite reciprocal of 2 = \tfrac{2}{1} is -\tfrac{1}{2}. The equation with slope -\tfrac{1}{2} is y = -\tfrac{1}{2}x + 1.
Not used: y = -2x + 3 (slope -2: sign changed but not flipped) and y = \tfrac{1}{2}x - 3 (flipped but sign not changed).
Parallel: y = 2x + 5. Perpendicular: y = -\tfrac{1}{2}x + 1.
Check: Perpendicular slopes multiply to -1: 2 \times \left(-\tfrac{1}{2}\right) = -1 ✓. The unused choices give 2 \times (-2) = -4 and 2 \times \tfrac{1}{2} = 1 ✗.
Tip: For perpendicular lines you must do two things to the slope: take the reciprocal and change the sign. The test "slopes multiply to -1" catches both kinds of half-done answers.
Question 19 of 41 · Probability of simple and compound events
A drawer holds 12 loose socks: 6 black, 4 white and 2 gray. Without looking, Tom pulls out one sock and then, without putting it back, pulls out a second sock.
What is the probability that both socks are black?
Hint
Find the probability of black on the first pull. After one black sock is gone, how many socks — and how many black socks — are left?
Step-by-step solution
First sock: 6 of the 12 socks are black: P(\text{black}_1) = \dfrac{6}{12}.
Second sock: one black sock is gone, so 5 black remain out of 11 socks: P(\text{black}_2) = \dfrac{5}{11}.
Both (multiply):\dfrac{6}{12} \times \dfrac{5}{11} = \dfrac{30}{132}.
Simplify (divide by 6): \dfrac{30}{132} = \dfrac{5}{22}.
The probability is \dfrac{5}{22} (about 0.227, or 22.7%).
Check: Count pairs: there are \tfrac{6 \times 5}{2} = 15 black pairs out of \tfrac{12 \times 11}{2} = 66 possible pairs, and \tfrac{15}{66} = \tfrac{5}{22} ✓.
Tip: "Without replacement" means both the top and the bottom of the second fraction go down by 1. "And" in probability means multiply.
Question 20 of 41 · Solve one-variable linear equations
What value of x makes the equation 4x - 7 = 2x + 9 true?
Hint
Get all the x-terms on one side and all the plain numbers on the other side.
Step-by-step solution
Subtract 2x from both sides: 4x - 2x - 7 = 9 \;\Rightarrow\; 2x - 7 = 9.
Add 7 to both sides: 2x = 16.
Divide both sides by 2: x = 8.
The solution is x = 8.
Check: Left side: 4(8) - 7 = 32 - 7 = 25. Right side: 2(8) + 9 = 16 + 9 = 25. Both sides equal 25 ✓.
Tip: Whatever you do to one side, do to the other. When a term "moves" across the equals sign, its sign flips (-7 becomes +7). Always finish by dividing by the coefficient of x, and plug your answer back in.
Question 21 of 41 · Circles: area and circumference
A round rug has a diameter of 6 feet. What is the area of the rug, in square feet? Use \pi \approx 3.14.
Hint
The area formula uses the radius, which is half the diameter.
Step-by-step solution
Radius:r = \dfrac{d}{2} = \dfrac{6}{2} = 3 ft.
Area of a circle (on the formula sheet): A = \pi r^2.
Substitute: A = 3.14 \times 3^2 = 3.14 \times 9.
Multiply: A = 28.26.
The rug's area is 28.26 square feet.
Check: A square 6 ft on each side has area 36 ft², and the rug fits inside it, so the area should be a bit less than 36 ✓ (\pi r^2 \approx 0.785 \times 36 = 28.26).
Tip: Always ask "radius or diameter?" before using a circle formula. Area uses r^2 (square units); circumference uses 2\pi r or \pi d (plain units).
Question 22 of 41 · Key features of graphs and tables
Drop-down
Ana holds a garden hose at ground level and sprays water into the air. The graph shows the height of the stream of water at each horizontal distance from the nozzle.
Select the correct values to complete the sentences.
The maximum height of the water is feet. The water hits the ground feet from the nozzle.
Choose an answer from each menu.
Hint
The maximum is the highest point of the curve — read its height on the vertical axis. Where the curve comes back to height 0 is where the water lands.
Step-by-step solution
Maximum height: the highest point (vertex) of the curve is at (3,\ 4.5). The height is the y-value: 4.5 feet. (The point is halfway between the grid lines for 4 and 5.)
Where it lands: the water starts at (0,\ 0) and returns to height 0 at (6,\ 0). That x-intercept tells us it lands 6 feet from the nozzle.
Answers: 4.5 feet; 6 feet.
Check: A parabola is symmetric about its vertex. The vertex is at x = 3, and the starting point x = 0 is 3 units to its left, so the landing point is 3 units to its right: x = 6 ✓.
Tip: For key features, decide which coordinate the question wants. A maximum value is a y-value; where something happens (lands, peaks) is an x-value. The vertex (3,\ 4.5) answers "how high" with 4.5 and "where" with 3.
Question 23 of 41 · Real-world inequalities
Fill in
To move to a new apartment, Derek rents a truck for $29.95 plus $0.79 per mile. He wants to spend no more than $100 on the truck.
What is the greatest whole number of miles Derek can drive and stay within his budget?
miles
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Write an inequality: fixed cost + cost per mile × miles ≤ 100. Then think carefully about how to round.
Step-by-step solution
Inequality:29.95 + 0.79m \le 100.
Subtract 29.95: 0.79m \le 70.05.
Divide by 0.79 (a positive number, so the sign stays the same): m \le \dfrac{70.05}{0.79} \approx 88.67.
Derek can drive at most 88.67 miles, so the greatest whole number of miles is 88. Rounding up to 89 would go over budget.
Tip: With a "no more than" limit, round down to stay under it, even if the decimal is .67. Always test your whole number and the next one up in the original inequality.
Question 24 of 41 · Multi-step real-world problems
The Garcia family takes home $3,840 each month. Rent is 35% of their take-home pay. Groceries cost $620 a month and utilities cost $185 a month. They split whatever money is left equally between savings and spending money.
How much do they put into savings each month?
Hint
Find the rent first. Subtract all three expenses from the take-home pay, then split what is left in half.
Step-by-step solution
Rent:35\% of \$3{,}840 = 0.35 \times 3{,}840 = \$1{,}344.
Tip: In multi-step budget problems, write a short label next to every number ("rent," "left," "savings"). It keeps you from stopping one step too early — a very common GED trap — or skipping an expense.
Question 25 of 41 · Factor polynomials
Drop-down
The area of a rectangular rug, in square feet, is given by the expression x^2 + 2x - 24.
Select the expression that makes the statement true.
The length and width of the rug can be written as the factors .
Choose an answer from each menu.
Hint
Look for two numbers that multiply to -24 and add to +2.
Step-by-step solution
For x^2 + bx + c, find two numbers that multiply to c = -24 and add to b = 2.
List factor pairs of -24 and their sums: (-1, 24) \to 23; (-2, 12) \to 10; (-3, 8) \to 5; (-4, 6) \to 2 ✓.
The numbers are -4 and 6, so x^2 + 2x - 24 = (x + 6)(x - 4).
The factors are (x + 6)(x - 4).
Check: Expand with FOIL: x \cdot x + 6x - 4x - 24 = x^2 + 2x - 24 ✓.
Tip: When the constant is negative, the two numbers have opposite signs, and the larger one (in size) takes the sign of the middle term. Here the middle term +2x is positive, so the 6 is positive and the 4 is negative.
Question 26 of 41 · Scatter plots and two-variable data
The scatter plot shows a family's monthly heating cost and the average outdoor temperature for 9 months.
Which statement best describes the relationship shown?
Hint
Read the points from left to right. As temperature goes up, do the costs tend to go up or down?
Step-by-step solution
Read the points from left to right (colder to warmer months): at 20^\circF the cost is about $210; at 40^\circF about $140; at 60^\circF about $62.
As the temperature increases, the heating cost decreases. The points fall from upper left to lower right.
That pattern is a negative association (a negative correlation). The points lie close to a line, so the association is fairly strong.
Answer: As the outdoor temperature increases, the heating cost tends to decrease.
Check: Pick any two points — e.g. (30,\ \$175) and (50,\ \$100). Higher temperature, lower cost ✓. The drops between neighboring points are not all equal ($20, $15, $25, …), so no "exact" rule fits.
Tip: Upper-left to lower-right = negative association; lower-left to upper-right = positive. Scatter plots show tendencies, so be suspicious of answer choices with words like "exactly" or "always."
Question 27 of 41 · Line from slope and a point
A line has a slope of -3 and passes through the point (2,\ 5).
Which is the equation of the line?
Hint
Use y = mx + b. Substitute the slope and the point, then solve for b.
Step-by-step solution
Slope-intercept form: y = mx + b, with m = -3: y = -3x + b.
Substitute the point (2,\ 5): 5 = -3(2) + b \;\Rightarrow\; 5 = -6 + b.
Add 6 to both sides: b = 11.
Equation: y = -3x + 11.
(Point-slope form, on the formula sheet, gives the same thing: y - 5 = -3(x - 2) \Rightarrow y = -3x + 6 + 5 = -3x + 11.)
The equation is y = -3x + 11.
Check: Put x = 2 into the answer: y = -3(2) + 11 = 5 ✓, so the line passes through (2,\ 5), and its slope is -3 ✓.
Tip: The y-coordinate of a point is the y-intercept only if the point's x-coordinate is 0. Otherwise, substitute and solve for b.
Question 28 of 41 · Proportions and unit conversion
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A recipe uses 3 cups of uncooked rice to make 8 servings. Carla is cooking for a family reunion and needs 20 servings.
At this rate, how many cups of uncooked rice does she need?
cups
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Set up a proportion: cups over servings equals cups over servings.
Carla needs 7.5 cups (7\tfrac{1}{2} cups) of rice.
Check: Unit rate: 3 \div 8 = 0.375 cup per serving, and 0.375 \times 20 = 7.5 ✓. Also, 20 servings is 2.5 times 8 servings, and 3 \times 2.5 = 7.5 ✓.
Tip: Keep the same units in the same position (cups on top, servings on the bottom on both sides). Flipping one ratio, \tfrac{3}{8} = \tfrac{20}{x}, gives 53.3 cups — far too much, which a quick estimate would catch.
Question 29 of 41 · Write quadratic equations from context
Will has a square patio. He plans to make it longer by adding 3 feet to one side, while the other side stays the same. The new rectangular patio will have an area of 108 square feet.
What is the side length, in feet, of the original square patio?
Hint
Let s be the original side. Write the new length and width in terms of s, then set length × width equal to 108.
Step-by-step solution
Define: original side = s. New patio: width s, length s + 3.
Equation:s(s + 3) = 108, so s^2 + 3s - 108 = 0.
Factor: find two numbers that multiply to -108 and add to 3: 12 and -9. So (s + 12)(s - 9) = 0.
Solve:s = -12 or s = 9. A length cannot be negative, so s = 9.
Tip: In real-world quadratics, solve completely, then throw out any answer that makes no sense (negative lengths, times or counts). The quadratic formula on the formula sheet works too: s = \dfrac{-3 \pm \sqrt{9 + 432}}{2} = \dfrac{-3 \pm 21}{2}.
Question 30 of 41 · Counting, combinations and permutations
For a kitchen remodel, the Lees can choose from 4 countertop materials, 3 cabinet colors and 5 styles of cabinet handles. They will pick one of each.
How many different combinations of countertop, cabinet color and handle style are possible?
Hint
For a choice made in several independent steps, multiply the number of options at each step.
Step-by-step solution
The Lees make three independent choices: countertop (4 options), cabinet color (3 options), handle style (5 options).
Fundamental counting principle: multiply the number of options for each choice.
4 \times 3 \times 5 = 12 \times 5 = 60.
There are 60 possible combinations.
Check: For one countertop there are 3 \times 5 = 15 cabinet-and-handle pairs. With 4 countertops: 4 \times 15 = 60 ✓.
Tip: "Pick one from each group" → multiply. Adding the options (12) counts single items, not complete combinations. Combination formulas such as {}_nC_r are for choosing several items from one group.
Question 31 of 41 · Systems of two linear equations
At a farmers market, 3 pounds of apples and 2 pounds of pears cost $9.70. Also, 1 pound of apples and 4 pounds of pears cost $10.40.
What is the price of one pound of pears?
Hint
Let a = price per pound of apples and p = price per pound of pears. Write two equations, then solve the second one for a and substitute.
Step-by-step solution
Equations:3a + 2p = 9.70 and a + 4p = 10.40.
Solve the second for a:a = 10.40 - 4p.
Substitute into the first:3(10.40 - 4p) + 2p = 9.70 \;\Rightarrow\; 31.20 - 12p + 2p = 9.70.
Tip: Substitution is easiest when a variable has coefficient 1 (here a). Read the question again at the end — after finding both prices, make sure you choose the one asked for.
Question 32 of 41 · Identify functions
Each set of ordered pairs shows (number of guests, cups of punch). Which set of ordered pairs represents a function?
Hint
In a function, each input (x-value) is paired with exactly one output. Look for repeated x-values.
Step-by-step solution
A relation is a function if every input (x-value) has exactly one output (y-value). So no x-value may appear twice with different y-values.
Check each set:
\{(1, 4), (2, 6), (1, 8)\}: input 1 has outputs 4 and 8 → not a function.
\{(3, 2), (3, 5), (4, 7)\}: input 3 has outputs 2 and 5 → not a function.
\{(2, 5), (4, 5), (6, 5)\}: inputs 2, 4, 6 are all different → function.
\{(0, 1), (1, 2), (0, 3)\}: input 0 has outputs 1 and 3 → not a function.
The function is \{(2, 5), (4, 5), (6, 5)\}.
Check: Repeated outputs are allowed: different guest counts may need the same amount of punch. Plot it and use the vertical line test — no vertical line hits two points ✓.
Tip: Only scan the first numbers. If any first number repeats with a different second number, it is not a function.
Question 33 of 41 · Cylinders
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A water pitcher is shaped like a cylinder, as shown.
What is the volume of the pitcher, to the nearest cubic inch? Use \pi \approx 3.14.
cubic inches
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The volume formula for a cylinder uses the radius. The figure gives the diameter.
Step-by-step solution
Radius: the diameter is 8 in, so r = 8 \div 2 = 4 in.
Volume of a cylinder (on the formula sheet): V = \pi r^2 h.
The volume is about 502 cubic inches. (With the calculator's \pi key you get 502.65 \approx 503, which is also accepted.)
Check: Estimate with \pi \approx 3: 3 \times 16 \times 10 = 480, close to 502 ✓.
Tip: Using the diameter as the radius makes the answer 4 times too big (3.14 \times 64 \times 10 = 2{,}009.6). Halve the diameter before you do anything else.
Question 34 of 41 · Solve linear inequalities
Which inequality is the solution of -3x + 5 \ge 20?
Hint
Solve it like an equation. What happens to the inequality sign when you divide by a negative number?
Step-by-step solution
Subtract 5 from both sides: -3x \ge 15.
Divide both sides by -3. Dividing by a negative number reverses the inequality sign: x \le \dfrac{15}{-3}.
Simplify: x \le -5.
The solution is x \le -5.
Check: Try x = -6 (should work): -3(-6) + 5 = 18 + 5 = 23 \ge 20 ✓. Try x = 0 (should fail): 0 + 5 = 5, which is not \ge 20 ✓. The boundary x = -5 gives exactly 20 ✓.
Tip: Multiplying or dividing both sides by a negative number flips the sign: \ge becomes \le. Testing one number from your answer and one outside it catches a forgotten flip.
Question 35 of 41 · Unit rate as slope
Jordan is saving for a new refrigerator. The graph shows the total amount he has saved over 8 weeks. He already had some money saved at week 0, and he saves the same amount each week.
How much money does Jordan save each week?
Hint
The amount saved each week is the slope of the line: change in dollars ÷ change in weeks.
Step-by-step solution
Use the two labeled points (0,\ 80) and (8,\ 480).
Slope (on the formula sheet): m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{480 - 80}{8 - 0} = \dfrac{400}{8} = 50.
The slope is the unit rate: dollars per week.
Jordan saves $50 per week.
Check: Start with $80 and add $50 for 8 weeks: 80 + 50 \times 8 = 80 + 400 = 480 ✓.
Tip: A unit rate from a graph is the change in y over the change in x. Dividing the final total by the number of weeks (480 \div 8 = 60) only works when the line starts at the origin — this one starts at $80.
Question 36 of 41 · Mean, median, mode, range, weighted average
For the first 5 months of the year, the Browns' electric bill averaged $96 per month. They want their average for the first 6 months to be exactly $100 per month.
What must their electric bill be in month 6?
Hint
An average times the number of items gives the total. Compare the total you need with the total you already have.
Step-by-step solution
Total so far: mean × count = 96 \times 5 = \$480.
Total needed for a $100 average over 6 months:100 \times 6 = \$600.
Tip: For "what score/value do I need" questions, work with totals, not averages: needed total − current total = missing value. Simply adding the difference in averages ($4) to $100 ignores that the new bill must make up $4 for each of the 5 earlier months.
Question 37 of 41 · Polynomial operations
A tabletop Sam is building is (2x - 5) feet long and (x + 3) feet wide.
Which expression represents the area of the tabletop, in square feet?
Hint
Area = length × width. Multiply every term in the first binomial by every term in the second (FOIL).
Step-by-step solution
Area = (2x - 5)(x + 3).
FOIL:
First: 2x \cdot x = 2x^2
Outer: 2x \cdot 3 = 6x
Inner: -5 \cdot x = -5x
Last: -5 \cdot 3 = -15
Combine: 2x^2 + 6x - 5x - 15 = 2x^2 + x - 15.
The area is 2x^2 + x - 15 square feet.
Check: Let x = 4: length 2(4) - 5 = 3, width 4 + 3 = 7, area = 21. Expression: 2(16) + 4 - 15 = 21 ✓.
Tip: Multiplying binomials always gives four products before you combine. Skipping the outer and inner terms (getting just 2x^2 - 15) is the most common mistake. Keep each term's sign attached to it.
Question 38 of 41 · Line from two points
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A plumber charges a flat fee for each house call plus an hourly rate. A 2-hour job costs $190, and a 5-hour job costs $385. The total cost is a linear function of the number of hours.
What is the plumber's flat fee, in dollars?
$
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Treat the jobs as points (2,\ 190) and (5,\ 385). The slope is the hourly rate; the y-intercept is the flat fee.
Tip: A common trap is dividing one cost by its hours (190 \div 2 = 95) to get the rate. That mixes the flat fee into the rate. Always use the difference between two jobs to find the rate, then work back to the fee.
Question 39 of 41 · Evaluate linear and quadratic functions
If f(x) = 3x^2 - 2x + 1, what is the value of f(-2)?
Hint
Replace every x with (-2) in parentheses. Square before you multiply by 3.
Step-by-step solution
Substitute x = -2 using parentheses: f(-2) = 3(-2)^2 - 2(-2) + 1.
Check: On the calculator, type 3((−)2)x²−2((−)2)+1 — it shows 17 ✓.
Tip: Always put a substituted negative number in parentheses. (-2)^2 = 4, but -2^2 = -4. Also remember the order of operations: square the -2 first, then multiply by 3.
Question 40 of 41 · Real-world linear equations
Maria has $240 to spend on new curtains. She has already spent $36 on curtain rods. Each curtain panel costs $25.50.
If she spends exactly the rest of her money on panels, how many panels can she buy?
Hint
Write an equation: rods + (price per panel × number of panels) = 240.
Step-by-step solution
Equation:36 + 25.50p = 240.
Subtract 36 from both sides: 25.50p = 204.
Divide by 25.50: p = \dfrac{204}{25.50} = 8.
Maria can buy 8 panels.
Check:8 \times 25.50 = 204, and 204 + 36 = 240 ✓.
Tip: Subtract the one-time cost before dividing by the per-item price. Dividing the whole budget first (240 \div 25.50 \approx 9.4) forgets that some money is already spent.
Question 41 of 41 · Solve quadratic equations
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What is the positive solution of the equation below?
2x^2 - 3x - 20 = 0
x =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Use the quadratic formula from the formula sheet with a = 2, b = -3, c = -20. Be careful with the signs.
Step-by-step solution
Identify a = 2, b = -3, c = -20.
Quadratic formula (on the formula sheet): x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}.