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1. Solving for a variable from formula:

In optics, the formula 1/f = 1/d0 + 1/di relates the distance of the image from the mirror to the distance of the object (d0) and the focal length (f) of the mirror. Solve this equation for the distance of the image (di).

2. System of Linear Equations (word problem):

A water dispenser manufacturer fills its bottles from a 750 gallon tank of water. The company sells the water bottles in 2-gallon and 5-gallon sizes. One day, 706 gallons were emptied from the tank to fill 182 bottles. If the company makes a $1.25 profit on the 2-gallon bottles and $2.05 profit on the 5-gallon bottles, by what amount - to the nearest dollar - did the profit on the 5-gallon bottles exceed the profit on the 2-gallon bottles on that day?

3. System of Linear Equations (undetermined coefficient):

(1/5)x - (2/3)y = 6
8x + (1-k)y = -13

If the system of equations above has zero solutions, what is the value of the constant k ?

4. Rates (motion):

Stephen rides a bike from town A to a nearby town B. If he travels 27 miles per hour, he will arrive 20 minutes early. If he travels 18 miles per hour, he will be 15 minutes late. At what speed (in miles per hour) will he arrive to town B exactly on time?

5. Rates (relating volumes):

Jane made elderberry juice and stored it into a 1.5 liter pitcher. Mia came over to try the juice and Jane poured it into a glass. The amount of juice that Jane poured into the glass was so that both the glass and the pitcher were now 6/7ths full. What is the volume of the glass in milliliters? (1 liter = 1000 milliliters)

6. Linear Functions (standard form coefficients):

A line in the xy-plane with a standard form equation Ax + By = C passes through points (-6, 7) and (4, -1). If A is an integer greater than 1, what is the smallest possible value of the coefficient B?

7. Multiple Unit Conversions (density):

Students are performing an experiment with hydrogen gas. They have a 900 gallon air-tight empty vessel into which they release 940 grams of hydrogen gas. The gas distributes in the vessel, and they next release an additional 3.6 liters of hydrogen into the vessel. What is the final density of the hydrogen gas in the vessel in kilograms per cubic meter? Round your answer to the nearest tenth of a kilogram. (Density is equal to the mass divided by volume and 100 gallons = 0.3785 cubic meters)

8. Cumulative Percent Discount:

A home appliance store sells refrigerators. If a floor model is designated for sale, its sticker price is marked down 25% with a sign “Floor Model Sale.” If it is not sold in three weeks, it is marked down another 10% off of the reduced price. Every 3 weeks it is not sold, it gets marked down by an additional 10% of its last reduced price. What is the sale price of a refrigerator whose sticker price was $1800 and it was sold 10 weeks after it was designated for sale? Express your answer in the units of dollars.

9. Exponential vs Linear Model (population):

In the year 2010, population of a town was 15,400. At the next census, taken in the year 2020, the population was 19,700. If the population is assumed to have been growing according to the linear model, by what percent would the population in the year 2006 be smaller than the population in 2006 if exponential model of population of growth was assumed? Round your answer to the nearest tenth of a percent.

10. Weighted Average and Percent Change:

Ms. Felps’ class has 23 students. On the test day, 6 were absent, and the remaining students took the test. The test was graded on a scale from 0 to 80 points. The average point score for the students that took the test was 64. At the end of the week, the 6 students that were absent took a make-up test scaled on the same range of points as the original test. Their average was 70 points. Ms. Felps then computed the average of the entire class. By what percent did the original average increase once all the scores were accounted for in the new class average? Round your answer to the nearest tenth of a percent.

11. Probability from Contingency Table Data:

In a study done for a medication that treats back pain, doctors gave patients experiencing back pain either a test drug or a placebo (sugar pill). The patients were split into two equally sized groups, and were not told which group they belonged to. Upon completion of the experiment, the doctors found that of the patients that were given the drug, the drug was effective in treating back pain for 84 patients and was not effective for 29 patients. Additionally, they found that 3 times as many patients found the drug effective as those who were given the sugar pill and whose back pain was cured.

Based on these results, what is the probability, rounded to the nearest percent, that if a test patient’s back pain is not cured, they were given the sugar pill?

12. Statistics - Distribution Parameters (range):

A set of seven integers is ordered from smallest to greatest. The only mode is 12, and the median is 8. What is the smallest possible range of this data set?

13. (Non-linear) System of equations:

y + 2 = |x|

y2 + 4 = x2

If (a, b) is a solution to the above system, what is one possible value of y - x ?

14. Quadratic Equation Solution (Expression):

If 3p/2 is a solution to the equation -2x2 + 12x = 18, what is the value of -p2 + p/2 ?

15. Non-linear Variation (percent):

Electrostatic force is a force experienced between charged matter. If two objects are both charged, the magnitude of the electrostatic force is given by the formula

Fel = k q1 q2 / r2

Where k is a universal constant, q1 and q2 are magnitudes of object charges, and r is the distance between their centers of mass. If the two objects move farther apart, the force between them decreases. If the force between two objects decreases by 80%, and their charges stay the same, by what percent does the distance between them increase? Round your answer to the nearest percent.

16. Function Transformations (absolute value):

The function f(x) = -3|x - 1| + 1 is reflected about the y-axis, then shifted 3 units to the left and 1 unit down. The function that is the result of this transformation is g(x). For what value of x is g(x) = f(0) ?

17. Zeros of Polynomial (synth div/factoring by grouping/sum of solutions):

If x = -1 is a root of the polynomial

p(x) = 3x4 - 4x3 - 4x2 - 4x - 7

What is the sum of all of the roots of this polynomial?

18. Exponential Equation w Substitution:

What is one solution to the equation below?

32 3−𝑥− 10 · 3 3−𝑥 + 32 = 0

19. Area of circular sector

A dartboard is in the shape of a circle with the bull’s eye being a smaller circle in the center of the dartboard. Outside of the bull’s eye, it is divided by radial lines into 20 equal sectors.

If the diameter of the bull’s eye is 2 inches, and the diameter of the dartboard is 18 inches, what is the area of one of the 20 sectors?

20. Special Right Triangles and Complex Numbers Arithmetic

Sarah wants to create a complex number

z = a + b*i

where a and b are real numbers, and i is an imaginary number such that i2 = -1. She has a right triangle with the two acute angles of 300 and 600. The hypotenuse of this triangle is 6. She makes a equal to the shorter leg, and b equal to the longer leg of her triangle. Ryan takes Sarah’s number and squares it. What number does Ryan get?