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.
Tip: Ask "how many groups of ___ fit into ___?" — that is always division. Since each portion is less than 1 cup, the answer must be more than 5\tfrac{1}{3}, which rules out choices smaller than that right away.
Question 2 of 5 · Rules of exponents
No calculator
Which value is equal to the expression below?
\frac{3^{4} \cdot 3^{-1}}{3^{5}}
Hint
Same base: when you multiply, add the exponents; when you divide, subtract the bottom exponent from the top one.
Step-by-step solution
Numerator (multiply, same base → add exponents):3^{4} \cdot 3^{-1} = 3^{4 + (-1)} = 3^{3}.
Divide (same base → subtract exponents, top minus bottom):\dfrac{3^{3}}{3^{5}} = 3^{3 - 5} = 3^{-2}.
Negative exponent means reciprocal:3^{-2} = \dfrac{1}{3^{2}} = \dfrac{1}{9}.
The value is \tfrac{1}{9}.
Check: Write it out with plain numbers: 3^4 = 81 and 3^{-1} = \tfrac{1}{3}, so the top is 81 \times \tfrac{1}{3} = 27. The bottom is 3^5 = 243. Then \tfrac{27}{243} = \tfrac{1}{9} ✓.
Tip: A negative exponent never makes the number negative — it flips it to the other side of the fraction bar. And when dividing powers, always subtract top minus bottom; if the bottom exponent is bigger, the result goes in the denominator.
Question 3 of 5 · Squares and square roots
No calculatorFill in
What is the value of the expression below?
5 + 2\sqrt{36} - (7 - 10)^{2}
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Work inside the parentheses and the square root first, then the exponent, then multiply, then add and subtract from left to right.
Root and exponent:\sqrt{36} = 6 and (-3)^{2} = (-3)(-3) = 9. Now we have 5 + 2 \cdot 6 - 9.
Multiply:2 \cdot 6 = 12, giving 5 + 12 - 9.
Add and subtract left to right:5 + 12 = 17, then 17 - 9 = 8.
The value is 8.
Check: Two common slips give other numbers. Adding first, (5 + 2) \cdot 6 - 9 = 33, breaks the rule that multiplication comes before addition. Writing (-3)^2 as -9 gives 5 + 12 + 9 = 26. Neither follows the order of operations, so 8 stands ✓.
Tip: A square root bar and parentheses are both grouping symbols — simplify what is inside them first. A negative number in parentheses that is squared always becomes positive.
Question 4 of 5 · Cubes and cube roots
No calculator
What is the value of the expression below?
\sqrt[3]{-64} - \sqrt{16}
Hint
Ask yourself: which number, used as a factor three times, gives -64? Can a negative number times itself three times stay negative?
Step-by-step solution
Cube root of a negative number: we need a number n with n \cdot n \cdot n = -64. Since (-4)(-4)(-4) = 16 \cdot (-4) = -64, we get \sqrt[3]{-64} = -4.
Square root:\sqrt{16} = 4 because 4 \cdot 4 = 16.
Subtract:-4 - 4 = -8.
The value is -8.
Check:(-4)^3 = -64 ✓ and 4^2 = 16 ✓. Starting at -4 on a number line and moving 4 more to the left lands on -8 ✓.
Tip: Square roots of negative numbers are not real numbers, but cube roots of negative numbers are fine: an odd number of negative factors gives a negative product. Memorize the cubes 1, 8, 27, 64, 125 for the no-calculator part.
Question 5 of 5 · Absolute value and distance on a number line
No calculatorDrop-down
On a beach vacation, Theo is snorkeling 18 feet below the surface of the water (an elevation of -18 feet). A seagull is flying 25 feet above the surface (an elevation of +25 feet).
Select the correct values to complete the sentences.
The vertical distance between Theo and the seagull is feet.
The value of \lvert -9 \rvert - \lvert 4 - 11 \rvert is .
Choose an answer from each menu.
Hint
Distance on a number line is the absolute value of the difference. For absolute value, simplify inside the bars first, then make the result non-negative.
Step-by-step solution
Distance between Theo and the seagull
Distance between two numbers is \lvert a - b \rvert: \lvert 25 - (-18) \rvert = \lvert 25 + 18 \rvert = \lvert 43 \rvert = 43.
Think of it in two pieces: 18 feet up to the surface, then 25 more feet up to the bird: 18 + 25 = 43 feet.
The distance is 43 feet.
The expression
\lvert -9 \rvert = 9.
Inside the second bars first: 4 - 11 = -7, so \lvert 4 - 11 \rvert = \lvert -7 \rvert = 7.
Subtract: 9 - 7 = 2.
The value is 2.
Check: Order does not matter for distance: \lvert -18 - 25 \rvert = \lvert -43 \rvert = 43 ✓. For the expression, the absolute values are 9 and 7, both positive, and 9 - 7 = 2 ✓.
Tip: When two points are on opposite sides of zero, add their distances from zero. A distance is never negative. Inside absolute-value bars, simplify completely before removing the bars.
Check: One-step version: 480 \times 0.85 \times 1.07 = 436.56 ✓. The answer is less than $480 (the discount is bigger than the tax) but more than the $408 sale price ✓.
Tip: You cannot just combine the percents (-15\% + 7\% = -8\%), because the tax is figured on a different, smaller amount than the discount. Use multipliers: "15% off" means \times 0.85, "add 7%" means \times 1.07.
Question 2 of 41 · Real-world linear equations
Fill in
A community recreation center charges a one-time sign-up fee of $35 plus $18 per month. Jamal has paid a total of $269 so far.
For how many months has Jamal been a member?
months
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Write an equation: fee + (cost per month) × (number of months) = total paid. Undo the fee first.
Step-by-step solution
Let m = number of months. Total paid = sign-up fee + monthly cost: 35 + 18m = 269.
Subtract 35 from both sides: 18m = 269 - 35 = 234.
Divide both sides by 18: m = \dfrac{234}{18} = 13.
Jamal has been a member for 13 months.
Check:35 + 18 \times 13 = 35 + 234 = 269 ✓.
Tip: A one-time fee is paid only once, so it is added once — it is not multiplied by the number of months. Undo operations in reverse order: subtract the fee first, then divide by the monthly rate. (Dividing $269 by 18 right away gives about 14.9, which is not a whole number of months — a sign the fee was not removed.)
Question 3 of 41 · Circles: area and circumference
Ruth is putting plastic edging around the outside of the circular flower bed shown below.
About how many feet of edging does she need? Round to the nearest tenth of a foot.
Hint
Edging goes around the circle, so you need the circumference. Is 12 ft the radius or the diameter?
Step-by-step solution
Edging goes around the edge, so find the circumference.
The 12-ft line goes all the way across through the center, so it is the diameter: d = 12 ft. (The radius is r = 6 ft.)
Formula (on the formula sheet): C = \pi d (same as 2\pi r).
Compute: C = \pi \times 12 \approx 3.14159 \times 12 \approx 37.70 ft.
She needs about 37.7 feet of edging.
Check: Using 2\pi r: 2 \times \pi \times 6 \approx 37.7 ✓. Estimate: circumference is a little more than 3 times the diameter, and 3 \times 12 = 36 ✓.
Tip: "Around" = circumference (feet); "covers" = area (square feet). Always check whether a circle's measurement is a radius (center to edge) or a diameter (edge to edge).
Question 4 of 41 · Evaluate linear expressions
A ride-share company figures the fare, in dollars, with the expression
2.50 + 1.75m + 0.30t
where m is the number of miles and t is the number of minutes of the ride. What is the fare for an 8-mile ride that takes 20 minutes?
Hint
Replace m with the number of miles and t with the number of minutes, then follow the order of operations.
Step-by-step solution
Substitute m = 8 and t = 20: 2.50 + 1.75(8) + 0.30(20).
Check: Break it down in words: $2.50 to start, $1.75 for each of 8 miles ($14), and 30¢ for each of 20 minutes ($6). 2.50 + 14 + 6 = 22.50 ✓.
Tip: Match each variable to its meaning before substituting — here miles go with $1.75 and minutes go with $0.30. A term with no variable (the $2.50 base fare) is always included exactly once.
Question 5 of 41 · Bar graphs and circle graphs
The bar graph shows how many minutes per day Kayla spends on different types of phone apps.
How many more minutes per day does Kayla spend on social apps than on games and music combined?
Hint
First combine the two smaller categories. Then compare that total with the social bar.
Step-by-step solution
Read the bars: Social = 95, Games = 45, Music = 30 minutes.
Games and music combined: 45 + 30 = 75 minutes.
Difference: 95 - 75 = 20 minutes.
Kayla spends 20 more minutes per day on social apps.
Check: Add back: 75 + 20 = 95 ✓. Each bar's value is printed above it, and the bar heights match the scale (the Social bar is just under the 100 line).
Tip: "How many more" means subtract. When a question names two categories "combined," add them first, then compare. Underline the exact categories in the question so you do not compare the wrong bars.
Question 6 of 41 · Slope from a graph, equation or table
On a grid map of a state park, the trailhead is at the point (-3, 4) and a waterfall is at the point (5, -2).
What is the slope of the straight line through these two points?
Hint
Slope = change in y ÷ change in x. Subtract in the same order on top and bottom, and watch the negative signs.
Step-by-step solution
Slope formula (on the formula sheet): m = \dfrac{y_2 - y_1}{x_2 - x_1}.
Let (x_1, y_1) = (-3, 4) and (x_2, y_2) = (5, -2).
Change in y: -2 - 4 = -6.
Change in x: 5 - (-3) = 5 + 3 = 8.
Slope: m = \dfrac{-6}{8} = -\dfrac{3}{4}.
The slope is -\tfrac{3}{4}.
Check: Use the points in the other order: \dfrac{4 - (-2)}{-3 - 5} = \dfrac{6}{-8} = -\dfrac{3}{4} ✓. Also, moving right from the trailhead to the waterfall, the y-value goes down, so the slope must be negative ✓.
Tip: Subtracting a negative is adding: 5 - (-3) = 8. Quick sense check: a line that falls from left to right has a negative slope.
Question 7 of 41 · Cylinders
Fill in
A rain barrel for watering a garden is shaped like a cylinder. It is 34 inches tall, and the circular top has a diameter of 22 inches.
What is the volume of the rain barrel, in cubic inches? Round your answer to the nearest whole cubic inch.
cubic inches
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The formula sheet gives V = \pi r^{2} h. You need the radius, not the diameter.
Step-by-step solution
Formula (on the formula sheet): V = \pi r^{2} h.
Radius is half the diameter: r = 22 \div 2 = 11 in. Height: h = 34 in.
Check: Estimate with \pi \approx 3: 3 \times 121 \times 34 = 12{,}342, a little less than our answer, as expected because \pi is a bit more than 3 ✓. (Using \pi \approx 3.14 gives 12{,}918, also accepted.)
Tip: Using the diameter in place of the radius makes the answer 4 times too big (\pi \times 22^2 \times 34 \approx 51{,}698). Always halve the diameter first. On the TI-30XS, use the \pi key for the most accurate result.
Question 8 of 41 · Solve linear inequalities
Which inequality is equivalent to
-4x + 7 \le 3x - 14\,?
Hint
Get all the x-terms on one side and the numbers on the other. What must happen to the inequality sign if you divide by a negative number?
Step-by-step solution
Subtract 3x from both sides: -4x - 3x + 7 \le -14, so -7x + 7 \le -14.
Subtract 7 from both sides: -7x \le -21.
Divide both sides by -7. Dividing by a negative flips the sign: x \ge \dfrac{-21}{-7}, so x \ge 3.
The answer is x \ge 3.
Check: Test x = 4 (it should work): left -16 + 7 = -9, right 12 - 14 = -2, and -9 \le -2 ✓. Test x = 0 (it should fail): 7 \le -14 is false ✓. Test the boundary x = 3: -5 \le -5 ✓ (equal is allowed).
Tip: To avoid flipping, you can move the x-terms to the side where they stay positive: add 4x to both sides and add 14 to both sides to get 21 \le 7x, so 3 \le x — same answer, no sign flip.
Question 9 of 41 · Probability of simple and compound events
Lena's phone playlist has 12 rock songs, 8 pop songs, and 5 country songs. She puts it on shuffle. The phone plays songs in random order and does not repeat a song.
What is the probability that the first two songs played are both country songs?
Hint
Find the probability of a country song first. After one country song has been played, how many country songs and how many total songs are left?
Step-by-step solution
Total songs: 12 + 8 + 5 = 25.
First song country: P = \dfrac{5}{25}.
The song is not repeated, so 4 country songs remain out of 24 songs: P(\text{2nd country} \mid \text{1st country}) = \dfrac{4}{24}.
Both events must happen, so multiply: \dfrac{5}{25} \times \dfrac{4}{24} = \dfrac{20}{600} = \dfrac{1}{30}.
The probability is \tfrac{1}{30}.
Check: Count pairs: there are 5 \times 4 = 20 ordered ways to pick two different country songs and 25 \times 24 = 600 ordered ways to pick any two different songs. \tfrac{20}{600} = \tfrac{1}{30} ✓. As a decimal that is about 0.033, or about 3%.
Tip: "Both/and" means multiply. When items are not replaced (no repeats), reduce both the favorable count and the total by 1 for the second pick.
Question 10 of 41 · Line from two points
A campground charges a one-time reservation fee plus the same price for each night. The total cost of a 2-night stay is $90, and the total cost of a 5-night stay is $195.
Which equation gives the total cost y, in dollars, for a stay of x nights?
Hint
The two stays give two points, (2, 90) and (5, 195). Find the slope (price per night) first, then the y-intercept (the fee).
Step-by-step solution
The two stays give the points (2, 90) and (5, 195).
Slope = cost per night: m = \dfrac{195 - 90}{5 - 2} = \dfrac{105}{3} = 35.
Use y = mx + b with the point (2, 90): 90 = 35(2) + b, so 90 = 70 + b and b = 20.
Equation: y = 35x + 20.
The answer is y = 35x + 20 (a $20 fee plus $35 per night).
Tip: Always test both points in your final equation. An equation like y = 45x fits the first stay but not the second — a sign that the relationship has a fixed fee and is not proportional.
Question 11 of 41 · Mean, median, mode, range, weighted average
Fill in
During a 5-day hiking vacation, Ana hiked an average (mean) of 6.5 miles per day for the first 4 days.
How many miles must she hike on day 5 so that her mean for all 5 days is exactly 7 miles per day?
miles
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Change each average into a total: total = mean × number of days.
Step-by-step solution
Total for the first 4 days: 6.5 \times 4 = 26 miles.
Total needed for 5 days at a mean of 7: 7 \times 5 = 35 miles.
Tip: You cannot average averages directly. Turn every mean into a total (mean × count), do the adding or subtracting with totals, and only divide at the end. Shortcut: each of the first 4 days is 0.5 mile below the target of 7, a total shortfall of 4 \times 0.5 = 2 miles, so day 5 must be 7 + 2 = 9.
Question 12 of 41 · Solve quadratic equations
Select all
Select all the solutions of the equation
3x^{2} - 5x - 12 = 0
Select all that apply.
Hint
Try factoring into (3x + \_)(x + \_), or use the quadratic formula from the formula sheet with a = 3, b = -5, c = -12.
Step-by-step solution
Factor 3x^{2} - 5x - 12. We need (3x + p)(x + q) with pq = -12 and 3q + p = -5. Try p = 4, q = -3: 3(-3) + 4 = -5 ✓.
So 3x^{2} - 5x - 12 = (3x + 4)(x - 3).
Zero-product rule: 3x + 4 = 0 or x - 3 = 0.
Solve: 3x = -4 \Rightarrow x = -\tfrac{4}{3}, and x = 3.
The solutions are x = 3 and x = -\tfrac{4}{3}.
Check: Use the quadratic formula: x = \dfrac{5 \pm \sqrt{(-5)^2 - 4(3)(-12)}}{2(3)} = \dfrac{5 \pm \sqrt{25 + 144}}{6} = \dfrac{5 \pm 13}{6}, giving \tfrac{18}{6} = 3 and \tfrac{-8}{6} = -\tfrac{4}{3} ✓.
Tip: When the leading coefficient is not 1, a factor like (3x + 4) gives x = -\tfrac{4}{3}, not x = -4. Divide by the coefficient of x. If factoring stalls, go straight to the quadratic formula.
Question 13 of 41 · Composite 2-D figures
A backyard lawn has the shape shown below: a rectangle with a half circle attached to one of its 12-foot sides.
What is the area of the lawn, to the nearest tenth of a square foot?
Hint
Split the shape into a rectangle and a half circle. The 12-ft side is the half circle's diameter.
Step-by-step solution
Rectangle:A = lw = 20 \times 12 = 240 sq ft.
Half circle: its diameter is the 12-ft side, so the radius is r = 6 ft. A full circle would be \pi r^{2} = \pi \times 36 \approx 113.10 sq ft, so the half circle is \tfrac{1}{2} \times 113.10 \approx 56.55 sq ft.
Total:240 + 56.55 = 296.55, which rounds to 296.5 sq ft. (Exactly: 240 + 18\pi \approx 296.549.)
The area is about 296.5 square feet.
Check: The half circle should be smaller than a 12 ft × 6 ft rectangle (72 sq ft) but bigger than half of it (36 sq ft). 56.5 fits ✓. So the total should be between 276 and 312 ✓.
Tip: For composite figures, sketch the pieces, label each one's dimensions, and find each area separately. Watch for "half circle" — divide the circle's area by 2 — and remember the straight side of a half circle is a diameter.
Question 14 of 41 · Evaluate linear and quadratic functions
Omar launches a model rocket from a small platform. The rocket's height, in feet, t seconds after launch is given by
h(t) = -16t^{2} + 48t + 5
What is h(2.5), the height of the rocket after 2.5 seconds?
Hint
Replace every t with 2.5. Square 2.5 first, then multiply by -16.
Check: On the TI-30XS, enter (−) 16 × 2.5 x² + 48 × 2.5 + 5 and press enter: 25 ✓. The answer is positive and less than the rocket's highest point (h(1.5) = 41 ft), which makes sense because the rocket is on its way down ✓.
Tip: In -16t^{2} only t is squared, then the result is multiplied by -16, so the term is always negative (or zero). Put negative inputs in parentheses on the calculator.
Question 15 of 41 · Unit rates
On a road trip, the Nguyen family drove 378 miles in 6 hours 45 minutes, not counting stops.
What was their average speed, in miles per hour?
Hint
Change 45 minutes into a fraction of an hour before dividing. Is 45 minutes the same as 0.45 hour?
Step-by-step solution
Average speed = distance ÷ time.
Convert the time to hours: 45 \text{ min} = \dfrac{45}{60} = 0.75 hour, so the time is 6.75 hours.
Divide: \dfrac{378}{6.75} = 56.
Their average speed was 56 miles per hour.
Check:56 \times 6.75 = 378 ✓. Also 56 \times 6 = 336 miles in the first 6 hours and 56 \times 0.75 = 42 miles in the last 45 minutes; 336 + 42 = 378 ✓.
Tip: Minutes are not decimal parts of an hour: 45 minutes is \tfrac{3}{4} hour = 0.75, not 0.45. Always divide minutes by 60. A rate "per hour" means distance divided by hours, never multiplied.
Question 16 of 41 · Polynomial operations
A rectangular raised garden bed is (x + 5) feet long and (2x - 3) feet wide.
Which expression represents the area of the bed, in square feet?
Hint
Area = length × width. Multiply every term in the first binomial by every term in the second (FOIL), then combine like terms.
Check: Let x = 4. The bed is 9 by 5, so the area is 45. The expression gives 2(16) + 7(4) - 15 = 32 + 28 - 15 = 45 ✓.
Tip: Plugging in a small number like x = 4 into the original product and into each answer choice is a fast way to test polynomial answers on the GED. Keep the sign with each term: the -3 makes the outer product negative.
Question 17 of 41 · Scatter plots and two-variable data
A repair shop tested the battery health of phones of different ages. The scatter plot shows the results along with a line of best fit.
Based on the line of best fit, what battery health would you predict for a phone that is 4.5 years old?
Hint
Find the slope of the line from two points on it, such as (0, 100) and (6, 64). Then go 4.5 years along the line.
Step-by-step solution
The line passes through (0, 100) and (6, 64).
Slope: \dfrac{64 - 100}{6 - 0} = \dfrac{-36}{6} = -6. Battery health drops about 6 percentage points per year.
Equation of the line: y = 100 - 6x.
Substitute x = 4.5: y = 100 - 6(4.5) = 100 - 27 = 73.
The predicted battery health is about 73%.
Check: On the graph, halfway between 4 years (100 - 24 = 76) and 5 years (100 - 30 = 70) is 73 ✓. Nearby data points (74 and 77 at 4 years, 70 and 72 at 5 years) agree ✓.
Tip: When a question says "based on the line of best fit," use the line, not a single data point. Writing the line's equation is more precise than eyeballing between grid lines.
Question 18 of 41 · Systems of two linear equations
A group of 9 people bought tickets to a theme park. Adult tickets cost $54 each and child tickets cost $38 each. The group paid $406 in all.
How many child tickets did the group buy?
Hint
Write two equations: one for the number of tickets and one for the total cost. Replace one variable using the first equation.
Step-by-step solution
Let a = adult tickets and c = child tickets.
Number of tickets: a + c = 9
Cost: 54a + 38c = 406
From the first equation, a = 9 - c. Substitute: 54(9 - c) + 38c = 406.
Tip: After solving a system, reread the question to see which variable it asks for. Plugging each answer choice into both conditions is a quick backup method: only c = 5 (with a = 4) makes both true.
Question 19 of 41 · Pyramids and cones
A landscaper dumped a pile of mulch that is shaped like a cone. The pile is 4 feet tall, and its circular base has a diameter of 9 feet.
What is the volume of the mulch pile, to the nearest tenth of a cubic foot?
Hint
Use the cone formula from the formula sheet, V = \tfrac{1}{3}\pi r^{2} h. What is the radius?
Step-by-step solution
Formula (on the formula sheet): V = \dfrac{1}{3}\pi r^{2} h.
Radius = half the diameter: r = 9 \div 2 = 4.5 ft. Height: h = 4 ft.
Check: A cylinder with the same base and height would hold \pi \times 20.25 \times 4 \approx 254.5 cubic feet. A cone is exactly one third of that: 254.5 \div 3 \approx 84.8 ✓.
Tip: Cones and pyramids get the \tfrac{1}{3}; cylinders and prisms do not. Before using any circle formula, decide whether the number given is a radius or a diameter.
Question 20 of 41 · Key features of graphs and tables
Hot spot
Priya flies a camera drone. The graph shows the drone's height above the ground during a 10-minute flight. Points A, B, C and D are marked on the graph.
The drone hovers (stays at the same height) for part of the flight. Click the point that marks the moment the drone stops hovering and starts climbing again.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
A flat (horizontal) piece of the graph means the height is not changing. Where does the flat piece end?
Step-by-step solution
Read the graph in order of time:
From 0 to 2 minutes, the line goes up from 0 m to 40 m (climbing) and reaches point A.
From 2 to 5 minutes, the line is flat at 40 m — the height does not change, so the drone is hovering. This flat piece runs from A to B.
From 5 to 6 minutes, the line goes up again to 60 m at point C (climbing).
From 6 to 10 minutes, the line goes down to 0 m at point D (landing).
The hover ends and the climb begins at the right end of the flat piece: point B, at (5, 40).
The answer is point B.
Check: Just before B the slope is 0 (flat); just after B the slope is \dfrac{60 - 40}{6 - 5} = 20 m per minute, positive ✓.
Tip: On a graph over time, rising = increasing, flat = constant, falling = decreasing. Read from left to right, the way time passes, and match each phrase in the question to a piece of the graph.
Question 21 of 41 · Proportions and unit conversion
Fill in
While driving on vacation in Canada, Grace sees a road sign that says the next city is 250 kilometers away.
Using 1 \text{ mile} \approx 1.61 \text{ kilometers}, how many miles away is the city? Round your answer to the nearest tenth of a mile.
miles
Type your answer in the box. Fractions like 3/4 are fine.
Hint
A mile is longer than a kilometer, so the number of miles should be smaller than 250. Set up a proportion or multiply by a conversion fraction.
Step-by-step solution
Set up the conversion so kilometers cancel: 250 \text{ km} \times \dfrac{1 \text{ mi}}{1.61 \text{ km}}.
Compute: 250 \div 1.61 \approx 155.28 miles.
Round to the nearest tenth: 155.3 miles.
The city is about 155.3 miles away.
Check: Convert back: 155.3 \times 1.61 \approx 250.0 km ✓. And the answer is smaller than 250, as expected because each mile is longer than a kilometer ✓.
Tip: Write the conversion factor as a fraction with the unit you want to get rid of on the bottom. Multiplying 250 \times 1.61 = 402.5 would be the error of putting the units upside down — always ask whether the answer should be bigger or smaller than the starting number.
Question 22 of 41 · Factor polynomials
Drag and drop
Drag the completely factored form of each polynomial into the box below it. Not every tile will be used.
x^{2} + 3x - 28
3x^{2} - 12
x^{2} - 10x + 25
Tap a tile, then tap a box to place it.
Hint
Always take out a common factor first. Then look for two numbers that multiply to the constant term and add to the middle coefficient, or for special patterns (difference of squares, perfect square).
Step-by-step solution
1. x^{2} + 3x - 28
Need two numbers with product -28 and sum +3: 7 and -4 (7 \times -4 = -28, 7 + (-4) = 3).
Factored: (x + 7)(x - 4).
2. 3x^{2} - 12
Common factor 3: 3(x^{2} - 4).
x^{2} - 4 is a difference of squares: x^{2} - 2^{2} = (x + 2)(x - 2).
Completely factored: 3(x + 2)(x - 2).
3. x^{2} - 10x + 25
Perfect square trinomial: 25 = 5^{2} and -10x = 2 \cdot x \cdot (-5).
Tip: "Completely factored" means no factor can be broken down further. (3x + 6)(x - 2) multiplies out correctly, but 3x + 6 still has a common factor of 3, so it is not complete.
Question 23 of 41 · Pythagorean theorem
TV screens are measured along the diagonal. The screen shown below has a 55-inch diagonal and is 48 inches wide.
What is the height h of the screen, to the nearest tenth of an inch?
Hint
The width, height and diagonal form a right triangle. Which side is the hypotenuse?
Step-by-step solution
The width, height and diagonal form a right triangle. The diagonal (55 in) is across from the right angle, so it is the hypotenusec.
Pythagorean theorem (on the formula sheet): a^{2} + b^{2} = c^{2}, so 48^{2} + h^{2} = 55^{2}.
Squares: 2304 + h^{2} = 3025.
Subtract: h^{2} = 3025 - 2304 = 721.
Square root: h = \sqrt{721} \approx 26.85, which rounds to 26.9 in.
The screen is about 26.9 inches tall.
Check:48^2 + 26.85^2 \approx 2304 + 720.9 = 3024.9 \approx 55^2 ✓. The height must be shorter than the diagonal, and it is ✓.
Tip: When you know the hypotenuse and want a leg, subtract the squares: \text{leg} = \sqrt{c^2 - a^2}. If your "leg" comes out longer than the hypotenuse, you added when you should have subtracted.
Question 24 of 41 · Real-world inequalities
Carlos has $420 saved for a vacation. He will save $65 more each week. He needs at least $1,250 for the trip.
What is the fewest number of weeks Carlos must save to have enough money?
Hint
Write an inequality: starting amount + 65 × weeks ≥ 1,250. Weeks must be a whole number — should you round up or down?
Step-by-step solution
Let w = number of weeks. Inequality: 420 + 65w \ge 1250.
Subtract 420: 65w \ge 830.
Divide by 65: w \ge \dfrac{830}{65} \approx 12.77.
Weeks must be a whole number, and 12 weeks is not quite enough, so round up: w = 13.
Carlos must save for at least 13 weeks.
Check: After 12 weeks: 420 + 65(12) = 1200, which is less than $1,250 ✗. After 13 weeks: 420 + 65(13) = 1265, which is at least $1,250 ✓.
Tip: In "at least / enough" problems the answer rounds up, even if the decimal is small, because the smaller whole number falls short. Always test the whole number just below your answer.
Question 25 of 41 · Counting, combinations and permutations
Fill in
A national park has 9 hiking trails. On her vacation, Lucia has time to hike 3 different trails. The order in which she hikes them does not matter.
How many different groups of 3 trails can Lucia choose?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Order does not matter, so this is a combination: {}_{n}C_{r} with n = 9 and r = 3.
Step-by-step solution
Order does not matter, so use a combination: {}_{9}C_{3} = \dfrac{9!}{3!\,(9-3)!}.
Check: If order mattered, there would be 9 \times 8 \times 7 = 504 ordered lists. Each group of 3 trails can be listed in 3 \times 2 \times 1 = 6 orders, so 504 \div 6 = 84 groups ✓. On the TI-30XS, 9 nCr 3 also gives 84 (the nCr function is in the prb menu).
Tip: Ask "does order matter?" Picking a group (team, set of trails, toppings) is a combination; arranging in order (1st/2nd/3rd, a PIN code, a schedule) is a permutation. Permutations are always bigger.
Question 26 of 41 · Unit rate as slope
The graph shows how much cell-phone data Darius used during one billing cycle. He used data at a steady rate.
What is the slope of the line, and what does it mean?
Hint
Slope = change in y ÷ change in x. What are the units of y? Of x?
Step-by-step solution
Pick two points on the line: (0, 0) and (30, 12).
Slope = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{12 - 0}{30 - 0} = \dfrac{12}{30} = 0.4.
Units: y is in gigabytes (GB) and x is in days, so the slope is in GB per day.
The slope is 0.4 GB per day: Darius uses 0.4 gigabyte of data each day.
Check: Use other marked points: from (10, 4) to (20, 8) the slope is \dfrac{8 - 4}{20 - 10} = \dfrac{4}{10} = 0.4 ✓. Also 0.4 \times 30 = 12 GB for the whole month ✓.
Tip: The units of a slope are always "y-units per x-unit." For a line through the origin, the slope is the unit rate, and you can find it from any single point: y \div x.
Question 27 of 41 · Identify functions
Drop-down
The table shows a relation between x and y.
x
1
2
1
4
y
4
7
9
10
Select the correct words to complete the sentence.
The relation a function, because .
Choose an answer from each menu.
Hint
In a function, each input (x-value) is paired with exactly one output (y-value). Look for repeated x-values.
Step-by-step solution
A relation is a function when every input (x) has exactly one output (y). Outputs may repeat; inputs may not be paired with different outputs.
Look at the inputs: the x-value 1 appears twice — once with y = 4 and once with y = 9.
One input with two different outputs breaks the rule.
The relation is not a function, because the input 1 is paired with two different outputs (4 and 9).
Check: Plot the points (1, 4) and (1, 9): they lie on the same vertical line x = 1, so the graph fails the vertical line test ✓.
Tip: Scan the input row for repeats. A repeated input with the same output is fine; a repeated input with a different output means "not a function." Whether the outputs increase evenly only matters for being linear, not for being a function.
Question 28 of 41 · Dot plots, histograms and box plots
A music teacher asked 15 guitar students how many hours they practiced last week. The dot plot shows the results.
What is the median number of practice hours?
Hint
With 15 values in order, the median is the 8th value. Count the dots from the left.
Step-by-step solution
There are 15 dots, so the median is the \tfrac{15 + 1}{2} = 8th value when the data are in order.
Count dots from the left (smallest values first):
1 hour: 1 dot (running total 1)
2 hours: 2 dots (running total 3)
3 hours: 4 dots (running total 7)
4 hours: 3 dots (running total 10)
The 8th value falls in the 4-hour column (values 8, 9 and 10 are all 4).
The median is 4 hours.
Check: 7 values are below the median position and 7 are above (positions 9–15) ✓. For comparison, the mode is 3 (tallest column), the range is 7 - 1 = 6, and the mean is \tfrac{58}{15} \approx 3.9 — all different measures.
Tip: On a dot plot, keep a running count from the left until you reach the middle position. The tallest column shows the mode, not the median.
Question 29 of 41 · Write quadratic equations from context
Elena is framing an 8-inch by 10-inch photo for her scrapbook display. She puts a mat of the same width, x inches, on all four sides of the photo. The total area of the photo and mat together must be 120 square inches.
Which equation can be used to find x?
Hint
Draw it. Adding a border of width x on both the left and right adds 2x to the width; the same is true for the height.
Step-by-step solution
The mat adds x on the left andx on the right, so the new width is 8 + 2x. Likewise the new length is 10 + 2x.
Area of photo plus mat: (8 + 2x)(10 + 2x) = 120.
Expand with FOIL: 80 + 16x + 20x + 4x^{2} = 120, so 4x^{2} + 36x + 80 = 120.
Set equal to zero by subtracting 120: 4x^{2} + 36x - 40 = 0.
The equation is 4x^{2} + 36x - 40 = 0.
Check: Divide by 4: x^{2} + 9x - 10 = 0, which factors to (x + 10)(x - 1) = 0. The positive solution is x = 1 inch. Then the framed size is 10 \times 12 = 120 square inches ✓.
Tip: A uniform border adds twice its width to each dimension. To solve a quadratic, it must be set equal to zero — subtract the area first, then factor or use the quadratic formula.
Question 30 of 41 · Scale factors and scale drawings
On a hiking map, 1 inch represents 2.5 miles. The trail from the parking lot to the lake measures 3\tfrac{3}{4} inches on the map.
What is the actual length of the trail?
Hint
Each inch on the map stands for 2.5 miles. How many miles do 3\tfrac{3}{4} inches stand for?
Step-by-step solution
Write the scale as a rate: 2.5 miles per inch.
Convert the map length to a decimal: 3\tfrac{3}{4} = 3.75 inches.
Multiply: 3.75 \times 2.5 = 9.375 miles.
The trail is 9.375 miles long.
Check: Proportion: \dfrac{1 \text{ in}}{2.5 \text{ mi}} = \dfrac{3.75 \text{ in}}{x}, so x = 3.75 \times 2.5 = 9.375 ✓. Estimate: about 4 inches × 2.5 = 10 miles, a bit more than our answer, as expected ✓.
Tip: Map length → real length: multiply by the scale. Real length → map length: divide. A real distance should be much larger than the map distance when 1 inch stands for miles.
Question 31 of 41 · Parallel and perpendicular lines
Line k has the equation 2x - 3y = 12.
Which equation describes the line that is perpendicular to line k and passes through the point (4, -1)?
Hint
First solve 2x - 3y = 12 for y to find the slope of line k. Perpendicular slopes are negative reciprocals.
Step-by-step solution
Find the slope of line k. Solve for y: -3y = -2x + 12, so y = \dfrac{2}{3}x - 4. The slope is \dfrac{2}{3}.
Tip: Never read the slope straight off standard form Ax + By = C — solve for y first (the slope is -\tfrac{A}{B}). For perpendicular lines, flip the fraction and change the sign.
Question 32 of 41 · Mean, median, mode, range, weighted average
On a weekend trip, Dev drove 120 miles to a lake at an average speed of 60 miles per hour. Because of heavy traffic, he drove the same 120 miles home at an average speed of only 40 miles per hour.
What was Dev's average speed for the whole round trip?
Hint
Average speed is total distance ÷ total time. Find how long each half of the trip took.
Step-by-step solution
Time going: \dfrac{120 \text{ mi}}{60 \text{ mph}} = 2 hours.
Time returning: \dfrac{120 \text{ mi}}{40 \text{ mph}} = 3 hours.
Total distance: 120 + 120 = 240 miles. Total time: 2 + 3 = 5 hours.
Average speed: \dfrac{240}{5} = 48 mph.
Dev's average speed was 48 miles per hour.
Check: As a weighted average of the speeds, weighted by time: \dfrac{60 \times 2 + 40 \times 3}{2 + 3} = \dfrac{120 + 120}{5} = 48 ✓. The answer is closer to 40 than to 60 because Dev spent more time driving at 40 mph ✓.
Tip: You cannot simply average two rates (\tfrac{60 + 40}{2} = 50) unless the same time is spent at each rate. Go back to totals: total distance over total time.
Question 33 of 41 · Solve one-variable linear equations
Fill in
Solve for x.
\frac{3(x - 4)}{5} = x - 6
x =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Clear the fraction first by multiplying both sides by 5. Then distribute and collect the x-terms on one side.
Step-by-step solution
Multiply both sides by 5 to clear the fraction: 3(x - 4) = 5(x - 6).
Distribute: 3x - 12 = 5x - 30.
Subtract 3x from both sides: -12 = 2x - 30.
Add 30 to both sides: 18 = 2x.
Divide by 2: x = 9.
The solution is x = 9.
Check: Left side: \dfrac{3(9 - 4)}{5} = \dfrac{3 \cdot 5}{5} = 3. Right side: 9 - 6 = 3 ✓.
Tip: When you multiply both sides by 5, multiply the whole right side, 5(x - 6) = 5x - 30, not just the x. Moving the x-terms toward the side with the larger coefficient (5x) keeps the coefficient positive and avoids sign slips.
Question 34 of 41 · Compare proportional relationships
Two friends are training for a charity bike ride.
Rider A's distance d (in miles) after t hours is d = 14.5t.
Rider B's distances are shown in the table.
Time (hours)
2
3
5
Distance (miles)
27
40.5
67.5
Which statement is true?
Hint
Find Rider B's unit rate (miles per hour) from the table, then compare it with the 14.5 in Rider A's equation.
Step-by-step solution
Rider A's unit rate is the coefficient in d = 14.5t: 14.5 mph.
Rider B's unit rate from the table: \dfrac{27}{2} = 13.5, \dfrac{40.5}{3} = 13.5, \dfrac{67.5}{5} = 13.5. Every ratio is the same, so B rides at 13.5 mph.
Compare: 14.5 - 13.5 = 1.
Rider A is faster, by 1 mile per hour.
Check: In 2 hours, A rides 14.5 \times 2 = 29 miles and B rides 27 miles. The gap is 2 miles in 2 hours, which is 1 mile per hour ✓.
Tip: To compare proportional relationships in different forms, turn each into a unit rate (the k in y = kx, or y \div x from a table). Do not compare a distance from a table with a rate from an equation.
Question 35 of 41 · Write linear expressions from context
At a garden center, seed packets cost $2.75 each and bags of potting soil cost $8 each. Mia buys n seed packets and 3 bags of potting soil.
Which expression represents the total cost, in dollars?
Hint
Write the cost of the seeds and the cost of the soil separately, then add them.
Step-by-step solution
Seeds: n packets at $2.75 each cost 2.75n dollars.
Soil: 3 bags at $8 each cost 3 \times 8 = 24 dollars.
Total: 2.75n + 24.
The expression is 2.75n + 24.
Check: Try n = 4 packets: 4 \times 2.75 = 11 for seeds plus $24 for soil = $35. The expression gives 2.75(4) + 24 = 11 + 24 = 35 ✓.
Tip: The price that goes with the unknown quantity is the coefficient of the variable. Fixed quantities (3 bags) become a number that you can multiply out. Testing a small value for the variable is a fast way to check any expression.
Question 36 of 41 · Spheres
A beach ball is a sphere with a diameter of 16 inches.
How much air does the beach ball hold, to the nearest tenth of a cubic inch?
Hint
Use the sphere volume formula from the formula sheet, V = \tfrac{4}{3}\pi r^{3}. Find the radius first.
Step-by-step solution
Formula (on the formula sheet): V = \dfrac{4}{3}\pi r^{3}.
The beach ball holds about 2,144.7 cubic inches of air.
Check: A cube that just fits around the ball is 16 \times 16 \times 16 = 4096 cubic inches. A sphere fills a little more than half of that box (\tfrac{\pi}{6} \approx 0.52), and 4096 \times 0.52 \approx 2130 — close to our answer ✓.
Tip: "Holds" or "fills" means volume (r^3, cubic units). "Covers the outside" means surface area (4\pi r^2, square units). Using the diameter in r^3 makes the answer 8 times too large.
Question 37 of 41 · Write inequalities from context
Drop-down
For a flight, Rosa's carry-on bag may weigh at most 22 pounds. The bag and the items already packed weigh 6.5 pounds. She wants to add souvenirs that weigh 0.75 pound each.
Let n be the number of souvenirs. Select the correct choices to complete the statements.
An inequality for the situation is 6.5 + 0.75n22.
The greatest number of souvenirs Rosa can pack is .
Choose an answer from each menu.
Hint
"At most" means the weight can be equal to 22 or less. After solving, remember n must be a whole number that keeps the bag within the limit.
Step-by-step solution
The inequality
Total weight = 6.5 + 0.75n.
"At most 22" means the weight is less than or equal to 22: 6.5 + 0.75n \le 22.
Solve it
Subtract 6.5: 0.75n \le 15.5.
Divide by 0.75: n \le 20.67 (to two decimal places).
n must be a whole number no greater than 20.67, so the greatest value is n = 20.
The inequality uses \le, and the greatest number of souvenirs is 20.
Tip: Translate carefully: "at most / no more than" → \le; "at least / no less than" → \ge. For a maximum count, round down — rounding up would break the limit.
Question 38 of 41 · Points in the coordinate plane
Hot spot
Jordan's geocaching app shows a hidden treasure box on a coordinate grid. The treasure is at the point (-3, 2).
Click the location of the treasure, (-3, 2), 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 from the origin; the second tells you how far to move up or down.
Step-by-step solution
Start at the origin (0, 0), where the axes cross.
The x-coordinate is -3: move 3 units left along the x-axis.
The y-coordinate is 2: from there, move 2 units up.
You land in the upper-left part of the grid (Quadrant II).
The treasure is at the point 3 left and 2 up from the origin.
Check: Points in Quadrant II have a negative x and a positive y, and (-3, 2) does ✓. The point (2, -3) is a different point — order matters in an ordered pair.
Tip: Remember "run before you jump": move along the x-axis (left/right) first, then along the y-direction (up/down). A negative x goes left; a negative y goes down.
Question 39 of 41 · Compare functions in different forms
Ella and Ben are each saving money for a summer trip.
Ella's savings, in dollars, after w weeks are given by E(w) = 150 + 40w.
Ben's savings are shown in the table.
Weeks, w
0
2
4
6
Savings ($)
60
150
240
330
Both keep saving at the same steady rates. After how many weeks will Ella and Ben have the same amount saved?
Hint
Find Ben's weekly rate from the table — notice the weeks go up by 2 each step. Then write an equation for Ben and set it equal to Ella's.
Step-by-step solution
Ben's starting amount (at w = 0) is $60.
Ben's rate: each step in the table is 2 weeks and adds $90, so the rate is \dfrac{90}{2} = 45 dollars per week. Ben: B(w) = 60 + 45w.
Set the savings equal: 150 + 40w = 60 + 45w.
Subtract 40w: 150 = 60 + 5w. Subtract 60: 90 = 5w. Divide by 5: w = 18.
They will have the same amount after 18 weeks.
Check: Ella: 150 + 40(18) = 870. Ben: 60 + 45(18) = 870 ✓. Ben starts $90 behind but gains $5 per week, and 90 \div 5 = 18 ✓.
Tip: When a table skips values (0, 2, 4, …), divide the change in output by the change in input to get the true rate. "When are they equal?" means set the two expressions equal and solve.
Question 40 of 41 · Write linear equations from context
A bike rental shop charges a flat fee of $12 plus $8 for each hour. Kenji paid $52 to rent a bike for h hours.
Which equation can be used to find h?
Hint
Which amount is charged once, and which amount is charged for each hour?
Step-by-step solution
The $8 is charged each hour, so the hourly cost is 8h.
The $12 flat fee is charged once.
Total paid: 12 + 8h = 52.
The equation is 12 + 8h = 52.
Check: Solve it: 8h = 40, so h = 5 hours. Then 12 + 8(5) = 12 + 40 = 52 ✓.
Tip: In a "flat fee plus rate" situation, the rate (the word each or per) is multiplied by the variable, and the flat fee is added on its own. Solving the equation you chose and checking it against the story is a quick way to confirm it.
Question 41 of 41 · Line from slope and a point
Drag and drop
A line has a slope of -2 and passes through the point (3, 5).
Drag numbers into the boxes to write the equation of the line in slope-intercept form. Not every tile will be used.
y =x \; +
Tap a tile, then tap a box to place it.
Hint
In y = mx + b, m is the slope. Substitute the point to find b.
Step-by-step solution
Slope-intercept form (on the formula sheet): y = mx + b, where m is the slope and b is the y-intercept.
The slope is given: m = -2.
Substitute the point (3, 5): 5 = -2(3) + b, so 5 = -6 + b.
Add 6 to both sides: b = 11.
Equation: y = -2x + 11.
The equation is y = -2x + 11.
Check: Plug in x = 3: -2(3) + 11 = -6 + 11 = 5 ✓, so the line passes through (3, 5). The slope (coefficient of x) is -2 ✓.
Tip: The y-intercept is not the y-coordinate of just any point — only of a point with x = 0. Always solve for b by substituting the point. When undoing -6, add 6 to both sides.