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Question 1 of 3 · Order fractions and decimals, including on a number line
No calculatorHot spot
Click the point on the number line that represents \dfrac{5}{4}.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
First figure out how much each small space is worth: how many spaces are between 0 and 1?
Step-by-step solution
Find the size of one space. Between 0 and 1 there are 4 equal spaces, so each space is \tfrac{1}{4}.
Rewrite the fraction as a mixed number: \dfrac{5}{4} = \dfrac{4}{4} + \dfrac{1}{4} = 1\tfrac{1}{4}.
Start at 0 and count 5 spaces to the right: \tfrac14, \tfrac24, \tfrac34, \tfrac44 = 1, \tfrac54.
That lands on the first tick mark after 1. As a decimal, \tfrac{5}{4} = 1.25.
Answer: the first tick mark to the right of 1 (the point 1.25).
Check:1.25 is between 1 and 1.5, and 1.5 is the middle of 1 and 2. The point is a quarter of the way from 1 to 2. ✓
Tip: On any number line, find the value of one space first: (difference between two labels) \div (number of spaces between them). Count spaces, not tick marks.
Question 2 of 3 · Cubes and cube roots
No calculator
What is the value of (-2)^3?
Hint
Write out three factors of -2 and multiply two at a time, watching the sign.
Step-by-step solution
(-2)^3 = (-2) \times (-2) \times (-2).
The first two factors: (-2) \times (-2) = 4 (negative × negative = positive).
Times the third factor: 4 \times (-2) = -8 (positive × negative = negative).
Answer: -8
Check: Three negative factors is an odd number of negatives, so the product must be negative; and 2^3 = 8 ✓.
Tip: A negative number raised to an odd power stays negative; raised to an even power it becomes positive. (-2)^2 = 4 but (-2)^3 = -8.
One-step version: paying 75% of the price, 80 \times 0.75 = 60.
The sale price is $60.
Check:\tfrac{1}{4} of $80 is $20, and \$60 + \$20 = \$80 ✓
Tip: "25% off" means you pay 100\% - 25\% = 75\%. Multiplying by 1 - \text{rate} gives the sale price in one step and keeps you from choosing the discount amount by mistake.
Question 1 of 17 · Pythagorean theorem
No calculator
Find the length of the hypotenuse c.
Hint
Square each leg, add, and then undo the squaring.
Step-by-step solution
The legs are a = 6 m and b = 8 m; c is the hypotenuse.
a^2 + b^2 = c^2 (on the formula sheet).
6^2 + 8^2 = 36 + 64 = 100, so c^2 = 100.
c = \sqrt{100} = 10.
The hypotenuse is 10 m.
Check: 10 is longer than each leg but shorter than 6 + 8 = 14 ✓. Also, 6-8-10 is the 3-4-5 triple doubled ✓.
Tip: Square → add → square root. Stopping before the square root (100) or skipping the squares (14) are the two classic slips.
Question 2 of 17 · Write inequalities from context
No calculatorDrop-down
A roller coaster allows only riders who are taller than 48 inches. Let h be a rider's height in inches. Choose the symbol that completes the inequality.
h48
Choose an answer from each menu.
Hint
Is a rider who is exactly 48 inches tall allowed? Which direction are the allowed heights?
Step-by-step solution
"Taller than 48 inches" means the height must be more than 48.
Is exactly 48 inches allowed? "Taller than 48" means more than 48, so a rider who is exactly 48 inches is not taller — the limit is not included. Use a strict symbol.
Bigger values are allowed → "greater than": >.
Answer: h > 48
Check: A 50-inch rider: 50 > 48 ✓ (may ride). A 48-inch rider: 48 > 48 is false ✓ (not taller than 48). A 40-inch rider: false ✓.
Tip: Ask two questions: Bigger or smaller? (direction) and Is the limit itself allowed? (strict or inclusive).
Question 3 of 17 · Systems of two linear equations
Hot spot
The graph shows the lines y = -x + 3 and y = 2x - 3.
Click the point that represents the solution of the system.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
The solution is the point that lies on both lines at once.
Step-by-step solution
The solution of a system is the point on both lines — where the lines cross.
From the graph, the lines cross at (2, 1): 2 units right of the origin and 1 unit up.
Confirm with algebra. Both equal y, so -x + 3 = 2x - 3.
Add x to both sides: 3 = 3x - 3. Add 3: 6 = 3x, so x = 2.
y = -2 + 3 = 1.
The solution is the point (2, 1).
Check:1 = -(2) + 3 ✓ and 1 = 2(2) - 3 ✓
Tip: Points where a line crosses an axis, like (0, 3) or (0, -3), are on only one line. Only the crossing point of the two lines is on both.
Question 4 of 17 · Counting, combinations and permutations
Fill in
A storage locker uses a 4-letter code. Each letter can be A, B, C, D, or E, and letters may repeat (for example, B B A E is allowed).
How many different codes are possible?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
How many choices are there for each of the 4 positions? Does using a letter once stop you from using it again?
Step-by-step solution
There are 4 positions to fill.
Each position can be any of the 5 letters, and repeats are allowed, so every position has 5 choices.
Tip: When repeats are allowed, the number of choices does not go down: use n^r. If repeats were not allowed it would be 5 \times 4 \times 3 \times 2 = 120.
Question 5 of 17 · Composite 2-D figures
No calculatorFill in
The front wall of a shed is a square with a triangle on top, as shown.
What is the total area of the front wall?
ft²
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find the square's area and the triangle's area separately. The triangle's base is the top of the square.
Step-by-step solution
Square: A = s^2 = 10 \times 10 = 100 ft².
Triangle: its base is the top of the square, b = 10 ft, and its height is the dashed line, h = 4 ft.
A = \tfrac{1}{2}bh = \tfrac{1}{2}(10)(4) = 20 ft².
Add the pieces: 100 + 20 = 120.
The wall's area is 120 ft².
Check: the triangle (20) is half of a 10 by 4 rectangle (40) ✓, and the total is more than the square alone ✓.
Tip: When shapes are joined, find each piece with its own formula and add. Only the triangle gets the \tfrac{1}{2}.
Question 6 of 17 · Mean, median, mode, range, weighted average
Fill in
Luis worked these numbers of hours over 8 weeks:
35, 42, 28, 45, 40, 33, 38, 30
What is the median number of hours?
hours
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Put the numbers in order first. With an even number of values, there are two middle numbers.
Step-by-step solution
Put the values in order from least to greatest:
28,\ 30,\ 33,\ 35,\ 38,\ 40,\ 42,\ 45
There are 8 values (an even number), so there are two middle values — the 4th and 5th: 35 and 38.
The median is their mean:
\frac{35 + 38}{2} = \frac{73}{2} = 36.5
The median is 36.5 hours.
Check: Four values (28, 30, 33, 35) are below 36.5 and four (38, 40, 42, 45) are above ✓
Tip: Never take the middle of the list as written — always sort first. Here the unsorted middle values are 45 and 40, which would give the wrong answer, 42.5.
Question 7 of 17 · Line from two points
A line passes through the points (2, 3) and (4, 7).
Which equation represents the line in point-slope form?
Hint
Find the slope first. Then use y - y_1 = m(x - x_1) with one of the points.
A shirt costs $45. The sales tax rate is 8%. What is the total cost of the shirt, including tax?
Hint
Find 8% of $45 first. Does the question want only the tax, or the price plus the tax?
Step-by-step solution
Change 8% to a decimal: 8\% = 0.08.
Tax: 0.08 \times 45 = \$3.60
Total: 45 + 3.60 = \$48.60
One-step version: 45 \times 1.08 = 48.60.
The total cost is $48.60.
Check: 10% of $45 is $4.50, so 8% should be a bit less — $3.60 fits, and the total is a little under $50 ✓
Tip: "Total cost" means price plus tax. Multiplying by 1 + \text{rate} (here 1.08) does both steps at once and avoids stopping at the tax.
Question 10 of 17 · Solve quadratic equations
Fill in
Solve 2x^2 + 5 = 77.
What is the positive solution?
x =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Get x^2 alone on one side first (undo the addition, then the multiplication).
Step-by-step solution
Subtract 5 from both sides: 2x^2 = 77 - 5 = 72
Divide both sides by 2: x^2 = 36
Take the square root: x = \pm\sqrt{36} = \pm 6
The solutions are 6 and -6. The positive solution is 6.
Check:2(6)^2 + 5 = 2(36) + 5 = 72 + 5 = 77 ✓
Tip: Undo the operations in reverse order (add/subtract first, then multiply/divide) and take the square root last. Also check that you answered the question asked: here only the positive solution, 6, goes in the box.
Question 11 of 17 · Key features of graphs and tables
No calculatorHot spot
Click the minimum point of the graph.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
The minimum is the lowest point on the whole graph — the bottom of the U.
Step-by-step solution
The graph is a U-shaped curve opening upward, so it has a lowest point and no highest point.
Trace the curve: it falls from the upper left, reaches the bottom, then rises.
The bottom of the U is at x = 1, y = -4.
The minimum point is (1,\ -4); the minimum value of the function is -4.
Check: The graph is symmetric about the vertical line x = 1: the x-intercepts -1 and 3 are each 2 units from x = 1, and \frac{-1 + 3}{2} = 1 ✓
Tip: The minimum (or maximum) of a parabola is always on its line of symmetry — halfway between the x-intercepts.
Question 12 of 17 · Unit rate as slope
The graph shows how much Dana earns for the hours she works.
What is Dana's hourly pay rate?
Hint
The unit rate is the amount earned for 1 hour. Use the marked point and divide.
Step-by-step solution
The line starts at (0, 0), so this is a proportional relationship: 0 hours → \$0.
The marked point (4,\ 60) means 4 hours of work earns \$60.
Unit rate = dollars ÷ hours:
\frac{60 \text{ dollars}}{4 \text{ hours}} = 15 \text{ dollars per hour}
This is also the slope of the line: m = \frac{60 - 0}{4 - 0} = 15.
Dana earns $15 per hour.
Check: At 8 hours the line reaches \$120, and 120 \div 8 = 15 ✓
Tip: In a proportional graph, the unit rate = slope = the y-value when x = 1. Here, at 1 hour the line is at \$15.
Question 13 of 17 · Evaluate polynomials
No calculator
What is the value of -x^2 + 4x when x = -2?
Hint
In -x^2, square x first, then take the opposite. Watch the sign of 4x too.
Step-by-step solution
Substitute in parentheses: -(-2)^2 + 4(-2).
Power: (-2)^2 = 4, so -(-2)^2 = -4.
Multiply: 4(-2) = -8.
Add: -4 + (-8) = -12.
The value is -12.
Check:-x^2 is never positive, and 4x is negative when x is negative, so the answer must be negative. Only -12 and -4 are possible, and -4 - 8 = -12 ✓
Tip: the minus sign in -x^2 is applied after squaring. Think of -x^2 as -1 \cdot x^2.
Question 14 of 17 · Right prisms
Fill in
A glass paperweight is a prism with a hexagon-shaped base. The area of the hexagon is 24\text{ cm}^2, and the prism is 7\text{ cm} tall.
What is the volume of the paperweight?
\text{cm}^3
Type your answer in the box. Fractions like 3/4 are fine.
Hint
You don't need a hexagon formula. The base area B is already given; use V = Bh.
Step-by-step solution
For every right prism, V = Bh.
The base area is given: B = 24\text{ cm}^2.
The height of the prism (distance between the two hexagons) is h = 7\text{ cm}.
V = 24 \times 7 = 168.
The volume is 168\text{ cm}^3.
Check:168 \div 7 = 24, the base area we were given ✓.
Tip:V = Bh works for prisms with any base shape: triangle, trapezoid, hexagon, even an irregular shape. If the problem gives you B, skip straight to multiplying.
Question 15 of 17 · Rational expression operations
No calculator
Simplify: \dfrac{x^2 - 25}{x - 5}, for x \ne 5.
Hint
Factor the numerator. It is a difference of two squares.
Step-by-step solution
The numerator is a difference of squares: x^2 - 25 = x^2 - 5^2 = (x - 5)(x + 5).
Rewrite: \dfrac{(x - 5)(x + 5)}{x - 5}.
Cancel the common factor (x - 5), which is allowed because x \ne 5.
Result: x + 5.
Answer: x + 5
Check: Let x = 7: \dfrac{49 - 25}{7 - 5} = \dfrac{24}{2} = 12, and 7 + 5 = 12 ✓.
Tip: Look for a factor of the numerator that matches the denominator exactly. Then the whole factor cancels — not just the numbers inside it.
Question 16 of 17 · Systems of two linear equations
Which ordered pair is the solution to the system of equations?
\begin{cases} y = x + 2 \\ y = 3x - 4 \end{cases}
Hint
Both equations equal y, so set the right sides equal. Or test each choice in both equations.
Step-by-step solution
Both expressions equal y, so they equal each other: x + 2 = 3x - 4.
Subtract x from both sides: 2 = 2x - 4.
Add 4: 6 = 2x. Divide by 2: x = 3.
Find y using the first equation: y = 3 + 2 = 5.
The solution is (3, 5).
Check: In both equations, 5 = 3 + 2 ✓ and 5 = 3(3) - 4 = 9 - 4 ✓
Tip: A wrong choice often works in one of the equations. Always check both — that is what "solution of the system" means.
Question 17 of 17 · Bar graphs and circle graphs
The circle graph shows how the Ramirez family divides its monthly take-home pay of $3,200.
How much money does the family spend on transportation each month?
Hint
Each section of a circle graph is a percent of the whole. What is the whole here?
Step-by-step solution
The Transportation section is labeled 12\%.
The whole circle stands for the monthly pay, \$3{,}200.
Find 12\% of \$3{,}200. Change the percent to a decimal (12\% = 0.12) and multiply:
0.12 \times 3{,}200 = 384
The family spends $384 on transportation.
Check:10\% of \$3{,}200 is \$320 and 2\% is \$64; 320 + 64 = 384 ✓
Tip: Section amount = percent (as a decimal) × whole. Move the decimal point two places left to turn a percent into a decimal: 12\% \to 0.12.