Troy School District · Mathematics · Grade 7
Form A · 120 minutes · 100 points

Grade 7 Mathematics — Testing-Out Examination

A student who passes this examination has demonstrated mastery of the Common Core Grade 7 standards (7.RP, 7.NS, 7.EE, 7.G, 7.SP) and is eligible to advance directly to Grade 8 Mathematics.

Student Name
Proctor
Score

Instructions

Part I · No Calculator · Problems 1–4 · 35 pts · ~45 min
1. Operations with rational numbers 10 points
  1. Compute (show your work, give exact answers in simplest form):

    (i) \( -7 + 12 - (-5) \)     (ii) \( -8 \cdot \frac{3}{4} \)     (iii) \( \frac{-15}{-3} - \frac{8}{4} \)

  2. Compute exactly:

    (i) \( \dfrac{2}{3} - \dfrac{5}{6} \)     (ii) \( \dfrac{3}{4} \cdot \dfrac{8}{15} \)     (iii) \( \dfrac{7}{8} \div \dfrac{3}{4} \)

  3. Compute (without converting to decimal): \( \dfrac{2}{5} + \left(-\dfrac{3}{4}\right) \cdot \dfrac{2}{3} \). Use the order of operations (PEMDAS) and show every step.
  4. The temperature in Anchorage was \(-12°\) F at midnight. By 6 a.m. it had dropped 7 degrees; by noon it had risen 18 degrees from midnight. Find the noon temperature. Show your arithmetic with signed numbers.
2. Ratios and proportional relationships 10 points
  1. Determine whether the relationship in the table is proportional. If so, identify the constant of proportionality \(k\) and write the relationship as \(y = kx\). If not, justify in one sentence.
    \(x\)25812
    \(y\)512.52030
  2. Solve the proportion: \( \dfrac{4}{x} = \dfrac{14}{21} \). Show the cross-multiplication step.
  3. A jacket on sale for $51 has been marked down 15% from its original price. Find the original price. Show your equation.
  4. A recipe calls for 3 cups of flour and \( 1\dfrac{1}{2} \) cups of sugar. Suppose a baker uses 7 cups of flour. How much sugar should they use, in proportion? Express as a mixed number.
3. Expressions, equations, and inequalities 10 points
  1. Simplify by combining like terms: \( 5x + 3 - 2x + 7 - 4x \).
  2. Expand and simplify: \( 3(2x - 5) - 4(x + 1) \). Show the distribution.
  3. Solve for \(x\): \( 2(x - 4) + 5 = 3x - 11 \). Show every step.
  4. Solve the inequality and graph the solution on a number line: \( -3x + 7 < 22 \). Remember to flip the inequality when dividing by a negative number, and explain why this is necessary in one sentence.
  5. A student earns $9 per hour at a part-time job and an additional $25 per week as a stipend. The student wants their weekly pay to be at least $115. Set up an inequality in terms of \(h\), the number of hours worked, and solve for \(h\). State the smallest whole number of hours that satisfies the constraint.
4. Geometry — scale, angles, and triangle conditions 5 points
  1. On a map drawn to a scale of 1 in : 12 mi, two cities are 4.5 inches apart. Find the actual distance.
  2. In a triangle, two angles measure 47° and 68°. Find the third angle. Justify using the triangle-angle-sum theorem.
  3. Two angles are supplementary. One measures \( (3x + 10)° \) and the other measures \( (x - 6)° \). Find \(x\), then find each angle. Set up an equation explicitly.
Part II · Calculator Permitted · Problems 5–10 · 65 pts · ~75 min
5. Multistep percent application 10 points

A laptop is listed at $850. The store offers a 20% discount, then applies a 6% sales tax to the discounted price.

  1. Find the discounted price (after the 20% off, before tax).
  2. Find the final price the customer pays (after sales tax).
  3. What single percent of the original $850 is the final price? (i.e., \( \text{final price} / 850 \cdot 100\% \)). Round to two decimals.
  4. Suppose tax were applied before the discount instead of after. Compute the final price under this ordering. Is it more, less, or the same? Justify in one sentence using a property of multiplication.
6. Circles — area and circumference 10 points
  1. Find the circumference and area of a circle of radius 7 cm. Express each exactly (in terms of \(\pi\)) and decimally (using \(\pi \approx 3.14\)) to the nearest 0.01.
  2. A circular pizza of diameter 14 inches is cut into 8 equal slices. Find the area of one slice. Round to the nearest 0.01 in\(^2\).
  3. A circular running track has radius 30 m. A runner runs 5 full laps. How many meters did they cover? Round to the nearest meter.
  4. The area of a circular flower bed is 200 ft\(^2\). Find its radius. Round to the nearest 0.01 ft. Show the equation you set up.
7. Volume & surface area of prisms and pyramids 10 points
  1. A rectangular prism has dimensions 8 cm by 5 cm by 12 cm. Find its volume and total surface area.
  2. A triangular prism has a right-triangle base with legs 3 cm and 4 cm; its height (depth) is 10 cm. Find the volume.
  3. A pyramid has a square base of side 6 cm and height 9 cm. Find its volume.
  4. A swimming pool is shaped like a rectangular prism: 25 m long, 12 m wide, and 2 m deep. Compute the volume in cubic meters, then convert to liters (1 m\(^3\) = 1000 L).
8. Probability — simple and compound 12 points
  1. A bag contains 4 red, 5 blue, and 3 green marbles. One marble is drawn at random. Find: \(P(\text{red})\); \(P(\text{not blue})\); \(P(\text{red or green})\). Express each as a fraction in simplest form.
  2. A fair six-sided die is rolled twice. Are the two rolls independent? Find the probability of rolling two 6s in a row. Express as a fraction and as a decimal to four places.
  3. From the bag in (a), two marbles are drawn without replacement. Find the probability that both are red. Show the dependence: the second probability uses the new (smaller) total. Express as a fraction.
  4. A student conducts 200 trials of an experiment in which they flip a thumbtack and record whether it lands point-up or point-down. Out of 200 trials, the tack landed point-up 132 times. Use this to estimate the probability \(P(\text{point-up})\) — this is the relative frequency. Then predict the number of point-up landings in 1000 future flips.
  5. Explain in one sentence the difference between a theoretical probability (such as 1/2 for heads on a fair coin) and an experimental (relative-frequency) probability. Why might these differ for the thumbtack in (d)?
9. Comparing populations & sampling 8 points

A researcher is comparing the daily exercise (in minutes) of two random samples of high-school students at two different schools.

\(n\)MeanMedianIQR
School A4052.35022
School B4038.74018
  1. Which school's students appear, on average, to exercise more? Use the medians (or means) to justify in one sentence.
  2. State the difference of the medians and the difference of the means between the two schools.
  3. Each school's IQR is about 20 minutes, which is roughly twice the difference of the medians. In one or two sentences, explain informally whether the gap between schools is large compared to the spread within each school.
  4. Suppose only 5 students from School A had been sampled instead of 40. Would you trust the inference about average exercise as much? Explain in one or two sentences.
10. Multi-step modeling — paint coverage 15 points

A homeowner is repainting the four exterior walls of a rectangular garage. The garage is 24 ft long, 18 ft wide, and 10 ft tall. There is one rectangular door (8 ft × 7 ft) and two square windows (4 ft × 4 ft) that will not be painted. Paint costs $42 per gallon; one gallon covers 320 ft\(^2\).

  1. Compute the total surface area of the four exterior walls (do not subtract the door or windows yet). Show the four wall areas as a sum.
  2. Subtract the area of the door and the two windows. State the net paintable surface area.
  3. How many gallons of paint are needed? You cannot purchase a fraction of a gallon — round up to the next whole gallon and explain why rounding up is necessary.
  4. Find the total cost of the paint.
  5. If the homeowner applies two coats, how many gallons are needed? How much will that cost?
  6. The hardware store offers a 12% discount on purchases of 4 or more gallons. Recompute the cost in (e) with the discount applied. Round all dollar amounts to the nearest cent.