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Question 1 of 20 · Counting, combinations and permutations
No calculator
Dana is packing for a trip. She brings 5 shirts, 3 pairs of pants, and 4 pairs of shoes. An outfit is one shirt, one pair of pants, and one pair of shoes.
How many different outfits can she make?
Hint
Build an outfit in three steps. For each shirt, how many pants-and-shoes pairs could go with it?
Step-by-step solution
Building an outfit has 3 stages: shirt, pants, shoes.
The fundamental counting principle says to multiply the number of choices at each stage.
5 \times 3 \times 4 = 60
Dana can make 60 different outfits.
Check: Each shirt goes with 3 \times 4 = 12 pants-and-shoes pairs. Five shirts give 5 \times 12 = 60 ✓
Tip: When you pick one item from each of several separate lists, multiply the list sizes. Adding (5 + 3 + 4 = 12) counts the pieces of clothing in the suitcase, not the outfits.
Question 2 of 20 · Evaluate linear and quadratic functions
A function is given by q(x) = 2x^2 + 3. The output of q is 53. What are all the possible inputs?
Hint
Set 2x^2 + 3 = 53. Undo the steps in reverse order: subtract, divide, then take the square root.
Step-by-step solution
Set the function equal to the output: 2x^2 + 3 = 53.
Subtract 3 from both sides: 2x^2 = 50.
Divide both sides by 2: x^2 = 25.
Take the square root, keeping both signs: x = 5 or x = -5.
Tip: Undo operations in the reverse order they were done to x: square, times 2, plus 3 → minus 3, divide by 2, square root. And x^2 = 25 always has two answers.
Question 3 of 20 · Pyramids and cones
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A candle maker pours wax into a cone-shaped mold. The mold has a radius of 5 cm and a height of 12 cm.
How many cubic centimeters of wax fill the mold? Round to the nearest whole number.
cm³
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Use V = \tfrac{1}{3}\pi r^2 h. Square the radius first, and press the \pi key only at the end.
Step-by-step solution
Formula (on the formula sheet): V = \tfrac{1}{3}\pi r^2 h.
Square the radius: r^2 = 5^2 = 25.
Multiply by the height: 25 \times 12 = 300.
Take one-third: \tfrac{1}{3}(300) = 100, so V = 100\pi.
Use the \pi key: 100\pi \approx 314.16, which rounds to 314.
Answer: about 314 cm³
Check: A cylinder with the same radius and height holds \pi(25)(12) = 300\pi \approx 942.5\text{ cm}^3. One-third of 942.5 is about 314 ✓.
Tip: Keep the exact form (100\pi) until the last step. With \pi \approx 3.14 you get 314 as well, so either way of using \pi gives the same rounded answer here.
Question 4 of 20 · Evaluate linear and quadratic functions
No calculatorFill in
Let f(x) = \dfrac{2}{3}x + 5. What is f(-9)?
f(-9) =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Put -9 in parentheses. Two-thirds of a number: divide by 3, then multiply by 2.
Step-by-step solution
Substitute -9 for x in parentheses: f(-9) = \dfrac{2}{3}(-9) + 5.
Multiply the fraction by -9: divide by the denominator first, -9 \div 3 = -3, then multiply by the numerator, -3 \times 2 = -6.
Add 5: -6 + 5 = -1.
Answer: f(-9) = -1
Check:f(0) = 5 and the slope is \tfrac{2}{3}, so moving 9 units left lowers the output by \tfrac{2}{3} \times 9 = 6: 5 - 6 = -1 ✓
Tip: When the input is a multiple of the denominator, divide first; the numbers stay small. A positive times a negative is negative, so \tfrac{2}{3}(-9) must be negative before you add 5.
Question 5 of 20 · Line from slope and a point
A phone battery loses charge at a steady rate of 6% per hour while streaming video. After 2.5 hours of streaming, the battery shows 71%.
Which equation gives the battery level B, in percent, after t hours of streaming?
Hint
A loss means a negative slope. Work backwards from 71% to find the level at t = 0.
Step-by-step solution
The battery loses 6% per hour, so m = -6: B = -6t + b.
The point is (t, B) = (2.5, 71). Substitute:
71 = -6(2.5) + b
Multiply: -6 \times 2.5 = -15, so 71 = -15 + b.
Add 15: b = 86. The battery started at 86%.
Answer: B = -6t + 86
Check:-6(2.5) + 86 = -15 + 86 = 71 ✓. It makes sense that the start (86%) is higher than 71%, because the battery is draining.
Tip: For a decreasing quantity, the starting value is bigger than any later value. Use that common-sense test to eliminate answers fast.
Question 6 of 20 · When is an expression undefined?
No calculator
The expression \dfrac{x + 3}{x^2 + 2x - 15} has no value for certain inputs. Which choice lists every such input?
Hint
Factor the denominator into two binomials, then set each factor equal to 0.
Step-by-step solution
Set the denominator equal to 0: x^2 + 2x - 15 = 0.
Factor: find two numbers that multiply to -15 and add to +2. They are 5 and -3. So x^2 + 2x - 15 = (x + 5)(x - 3).
Set each factor to 0: x + 5 = 0 \Rightarrow x = -5; x - 3 = 0 \Rightarrow x = 3.
Tip: The numerator x + 3 does not matter — at x = -3 the expression is \tfrac{0}{-18} = 0, which is defined. After factoring, each factor (x + a) gives x = -a: the sign flips.
Question 7 of 20 · Rational expression operations
No calculator
For which values of x is \dfrac{x + 1}{x^2 - 9} undefined?
Hint
A fraction is undefined when its denominator equals 0. Solve x^2 - 9 = 0.
Step-by-step solution
A rational expression is undefined when the denominator is 0.
Set the denominator equal to 0: x^2 - 9 = 0.
Factor: (x - 3)(x + 3) = 0.
So x - 3 = 0 or x + 3 = 0, giving x = 3 or x = -3.
The numerator (x + 1) does not affect where the expression is undefined.
Answer: x = 3 and x = -3
Check: At x = 3: denominator 9 - 9 = 0 ✓. At x = -3: 9 - 9 = 0 ✓. At x = -1: denominator 1 - 9 = -8, so the expression is \dfrac{0}{-8} = 0, which is defined.
Tip:x^2 = 9 has two solutions, 3 and -3. Do not forget the negative root.
Question 8 of 20 · Solve one-variable linear equations
No calculator
Which equation has the same solution as the equation below?
3(x + 4) = 2x + 19
Hint
Simplify the original equation one step at a time and look for one of those steps among the choices.
Step-by-step solution
Solve the original equation and watch the steps.
Distribute: 3x + 12 = 2x + 19.
Subtract 2x from both sides: x + 12 = 19. ← this is choice A.
Subtract 12: x = 7.
Now check the other choices:
B: 5x + 12 = 19 gives 5x = 7, x = 1.4.
C: x = 31 is not 7.
D: 3x + 4 = 2x + 19 gives x = 15.
The answer is A: x + 12 = 19.
Check:x = 7 in the original: 3(11) = 33 and 2(7) + 19 = 33 ✓; in A: 7 + 12 = 19 ✓
Tip: Two equations are equivalent when they have the same solution. Every correct step of a solution produces an equivalent equation.
Question 9 of 20 · Rules of exponents
No calculator
What is the value of 27^{\frac{2}{3}}?
Hint
In a^{m/n}, the bottom number n is a root and the top number m is a power. Take the root first to keep numbers small.
Step-by-step solution
Rule: a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m. Here m = 2 (power) and n = 3 (cube root).
Check (other order):27^2 = 729 and \sqrt[3]{729} = 9 because 9 \times 9 \times 9 = 729. Same answer. ✓
Tip: "Denominator = root, numerator = power." Doing the root first keeps the numbers small enough for the no-calculator section.
Question 10 of 20 · Mean, median, mode, range, weighted average
A swimmer timed four laps at practice. Her times were 1.8, 2.1, 1.9, and 2.2 minutes. What was her mean lap time?
Hint
Add the four times, then divide by the number of laps.
Step-by-step solution
Add the four lap times:
1.8 + 2.1 + 1.9 + 2.2 = 8.0
Count the values: there are 4 laps.
Divide the sum by the count (mean formula, on the GED formula sheet):
\text{mean} = \frac{8.0}{4} = 2.0
Her mean lap time was 2.0 minutes.
Check: The mean must be between the fastest time (1.8) and the slowest (2.2). It is: 1.8 < 2.0 < 2.2 ✓ Also, 2.0 \times 4 = 8.0, the same total.
Tip: Decimals do not change the method: add, then divide by how many values there are. Line up the decimal points when adding by hand, and count the values carefully before you divide.
Question 11 of 20 · Write rational expressions from context
Fill in
A school plans a field trip to a science museum. Renting the bus costs $900, and museum tickets cost $45 per student.
Write an expression for the average cost per student when s students go. Then use it to find the average cost per student when 60 students go.
$ per student
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The total cost is the bus plus the tickets for s students. Divide the total by s.
Check: The bus cost shared by 60 students is \dfrac{900}{60} = 15 dollars each, plus the $45 ticket: 15 + 45 = 60 ✓.
Tip: Average cost can be split into two parts: the fixed cost ÷ number of people, plus the per-person cost. This is a fast way to check your answer.
Question 12 of 20 · Compare proportional relationships
Marco's pay is given by p = 17.40h, where p is pay in dollars and h is hours worked. Ellie is paid a constant hourly wage that is $2.50 more per hour than Marco's. Part of Ellie's pay record is shown.
Hours
6
15
Pay ($)
119.40
?
What number belongs in the missing cell?
Hint
Find Ellie's hourly wage first, then multiply by 15. Check it against the 6-hour column.
Step-by-step solution
Marco's unit rate: $17.40 per hour.
Ellie's unit rate: 17.40 + 2.50 = 19.90 dollars per hour.
Check with the table: 6 \times 19.90 = 119.40 ✓.
Missing pay: 15 \times 19.90 = 298.50.
Answer: $298.50
Check: Ellie's ratio must be the same in both columns: \tfrac{298.50}{15} = 19.90 and \tfrac{119.40}{6} = 19.90 ✓.
Tip: A condition like "$2.50 more per hour" changes the rate, not the total. Add it to the unit rate, then multiply.
Question 13 of 20 · Probability of simple and compound events
At a school fair, each player rolls two fair number cubes (1–6) once and wins a prize if the sum is 7. The organizers expect 180 players.
About how many prizes should they expect to give away?
Hint
How many of the 36 possible (first, second) outcomes give a sum of 7? Use that probability for 180 players.
Step-by-step solution
Two cubes have 6 \times 6 = 36 equally likely outcomes.
Outcomes with sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes.
P(\text{sum } 7) = \frac{6}{36} = \frac{1}{6}
Expected winners: \frac{1}{6} \times 180 = 30
They should expect about 30 prizes.
Check: Sum 7 is the most common sum; every first roll has exactly one partner that makes 7, so 1 in 6 ✓ 180 \div 6 = 30 ✓
Tip: Sums of two dice (2 through 12) are NOT equally likely. List the ordered pairs, or remember that 7 has the most ways (6 of 36).
Question 14 of 20 · Dot plots, histograms and box plots
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The dot plot shows how many times each of 16 adults exercised last week. Each dot stands for one adult.
Two more adults join the group. They both exercised the same number of times last week. After they are added, the mean for all 18 adults is exactly 2.5 times. How many times did each new adult exercise?
times
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find the total for the 16 adults from the dot plot, then the total the 18 adults need to have a mean of 2.5.
Step-by-step solution
Total for the original 16 adults (value × dots):
0(2) + 1(3) + 2(4) + 3(3) + 4(2) + 5(1) + 6(1) = 39
For 18 adults to have a mean of 2.5, the new total must be
2.5 \times 18 = 45
The two new adults together add 45 - 39 = 6 times.
They exercised the same number of times, so each one exercised 6 \div 2 = 3 times.
Each new adult exercised 3 times.
Check: New total 39 + 3 + 3 = 45; 45 \div 18 = 2.5 ✓.
Tip: For any "what value gives this mean" problem: (new mean × new count) − old total = what the new values must add up to.
Question 15 of 20 · Graph linear equations
No calculator
At which point does the graph of y = \tfrac{1}{2}x + 4 cross the x-axis?
Hint
Every point on the x-axis has the same y-value. Substitute that value and solve for x.
Step-by-step solution
Points on the x-axis have y = 0. Substitute:
0 = \tfrac{1}{2}x + 4
Subtract 4 from both sides:
-4 = \tfrac{1}{2}x
Multiply both sides by 2 (dividing by \tfrac{1}{2} is the same as multiplying by 2):
x = -8
So the line crosses the x-axis at (-8,\ 0).
The x-intercept is (-8,\ 0).
Check:\tfrac{1}{2}(-8) + 4 = -4 + 4 = 0 ✓
Tip: To undo "times \tfrac{1}{2}", multiply by 2 — do not multiply by \tfrac{1}{2} again. The point (0,\ 4) is the y-intercept, a different point on the same line.
Question 16 of 20 · Composite 2-D figures
No calculatorDrag and drop
The L-shaped figure is split by a dashed line into rectangles A and B.
Drag the correct area into each box. Not every value is used.
Area of A (ft²)
Area of B (ft²)
Total area (ft²)
Tap a tile, then tap a box to place it.
Hint
Rectangle A is as wide as the top edge and as tall as the left side. How tall is rectangle B — look at the right side.
Step-by-step solution
Rectangle A: width 7 ft (the top edge), height 9 ft (the left side): 7 \times 9 = 63 ft².
Rectangle B: width 12 - 7 = 5 ft, height 5 ft (the right side): 5 \times 5 = 25 ft².
Total:63 + 25 = 88 ft².
Unused: 108 = 12 \times 9 (the full rectangle, ignoring the cut-out corner) and 45 = 5 \times 9 (rectangle B with the full 9 ft height).
Tip: When you split an L-shape, each rectangle gets its own height. Rectangle B only reaches up to the step, so its height is 5 ft, not 9 ft.
Question 17 of 20 · Write linear expressions from context
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An internet company charges a one-time installation fee of $49.99 and $34.50 per month for service. The total cost, in dollars, for m months can be written as am + b.
Enter the values of a and b.
a =
b =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
a is multiplied by the number of months. b is paid only once.
Step-by-step solution
The monthly charge is paid every month. For m months it is 34.50m, so a = 34.50.
The installation fee is paid once. It is the constant: b = 49.99.
Total cost: 34.50m + 49.99.
The answers are a = 34.50 and b = 49.99.
Check: for one year (m = 12): 34.50(12) + 49.99 = 414 + 49.99 = 463.99 dollars. Counting directly: 12 months at $34.50 is $414, plus $49.99 installation, so $463.99 ✓
Tip: in am + b, the order of the terms is swapped compared with how the problem was written. Match by meaning (per month vs. one time), not by position.
Question 18 of 20 · Evaluate linear and quadratic functions
No calculator
The graph of the function y = f(x) is shown.
What is the value of f(-1) + f(2)?
Hint
f(-1) is the y-value of the point on the curve where x = -1. Do the same for x = 2, then add the two outputs.
Step-by-step solution
f(-1): go to x = -1 on the horizontal axis; the curve passes through (-1, 0). So f(-1) = 0.
f(2): go to x = 2; the curve passes through (2, -3). So f(2) = -3.
Tip: Add the outputs (y-values), not the inputs. And read carefully: the lowest point (1, -4) is at x = 1, not x = 2.
Question 19 of 20 · Solve linear inequalities
No calculator
Keisha solved the inequality 2(5 - x) > 4. Her work is shown.
Step
Work
1
10 - 2x > 4
2
-2x > -6
3
x > 3
4
Solution: all numbers greater than 3
In which step did Keisha make her first error?
Hint
Check each step on its own. What must happen when you divide both sides by a negative number?
Step-by-step solution
Check each step:
Step 1: Distribute the 2 to both terms: 2 \cdot 5 - 2 \cdot x = 10 - 2x. So 10 - 2x > 4 ✓.
Step 2: Subtract 10 from both sides: -2x > 4 - 10 = -6 ✓.
Step 3: Divide both sides by -2. Because -2 is negative, the sign must flip:
x < \frac{-6}{-2} \quad\Rightarrow\quad x < 3
Keisha wrote x > 3 — she forgot to flip. ✗
Step 4 only restates Step 3, so it is wrong because of Step 3.
Answer: Step 3
Check: Test x = 4 (in Keisha's answer): 2(5 - 4) = 2, and 2 > 4 is false. Test x = 0 (in the correct answer x < 3): 2(5) = 10 > 4 ✓. So the correct solution is x < 3.
Tip: In "find the error" items, test one number from the student's final answer in the original inequality. If it fails, trace back to the first step where a rule was broken.
Question 20 of 20 · Bar graphs and circle graphs
Drop-down
The double bar graph shows the quarterly sales of two stores, in thousands of dollars.
Choose the correct answer for each blank.
The greatest difference between the two stores' sales occurred in .
Store B had higher sales than Store A in .
Choose an answer from each menu.
Hint
For each quarter, compare the two bars side by side. The difference is the gap between their tops, not the height of the taller bar.
Step-by-step solution
Read every bar (in thousands of dollars):
Quarter
Store A
Store B
Difference
Q1
40
30
40 - 30 = 10
Q2
55
60
60 - 55 = 5
Q3
70
45
70 - 45 = 25
Q4
35
65
65 - 35 = 30
The largest difference is 30 thousand dollars, in Q4.
Store B's bar is taller than Store A's in Q2 (60 > 55) and Q4 (65 > 35). So the answer is Q2 and Q4.
Check: In Q1 and Q3 the Store A (blue) bar is taller, so those quarters do not belong in the second blank ✓
Tip: In a double bar graph, the legend tells you which color is which. Build a quick table of the values — it turns a picture problem into an arithmetic problem.