ACCUPLACER Advanced Algebra & Functions Quiz: Linear Function Word Problems
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Linear Function Word ProblemsQuestion 1 of 20

A large water tank is being filled at a constant rate. After 5 minutes, the tank contains 300 liters of water. After 12 minutes, it contains 475 liters. Assuming the tank was not empty when filling began, how much water was in the tank initially?

25 liters
125 liters
175 liters
200 liters
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ACCUPLACER Advanced Algebra & Functions Quiz

ACCUPLACER Advanced Algebra & Functions Quiz: Linear Function Word Problems

Practice Linear Function Word Problems in ACCUPLACER Advanced Algebra & Functions with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Linear Function Word Problems, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Advanced Algebra & Functions.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A large water tank is being filled at a constant rate. After 5 minutes, the tank contains 300 liters of water. After 12 minutes, it contains 475 liters. Assuming the tank was not empty when filling began, how much water was in the tank initially?

  1. 25 liters
  2. 125 liters
  3. 175 liters (correct answer)
  4. 200 liters
Explanation: First, find the rate of filling (the slope). The points are (5, 300) and (12, 475). The slope m=475−30012−5=1757=25m = \frac{475 - 300}{12 - 5} = \frac{175}{7} = 25 liters per minute. Now use the point-slope form with the point (5, 300): L(t)−300=25(t−5)L(t) - 300 = 25(t - 5). The initial amount is the value when t=0t=0. L(0)−300=25(0−5)  ⟹  L(0)−300=−125  ⟹  L(0)=175L(0) - 300 = 25(0 - 5) \implies L(0) - 300 = -125 \implies L(0) = 175. The initial amount was 175 liters.

Question 2

A salesperson earns a weekly base salary of $400 plus an 8% commission on all sales revenue over $1,000. This week, the salesperson's total earnings were $820. What was the total sales revenue for the week?

  1. $4,250
  2. $5,250
  3. $6,250 (correct answer)
  4. $11,250
Explanation: Let SS be the total sales revenue. The commission is earned on the amount S−1000S - 1000. The earnings function is E(S)=400+0.08(S−1000)E(S) = 400 + 0.08(S - 1000). We are given E(S)=820E(S) = 820. So, 820=400+0.08(S−1000)820 = 400 + 0.08(S - 1000). Subtract 400: 420=0.08(S−1000)420 = 0.08(S - 1000). Divide by 0.08: 5250=S−10005250 = S - 1000. Add 1000: S=6250S = 6250. The total sales revenue was $6,250.

Question 3

A catering company offers two pricing plans. Plan A has a flat fee of $500 plus $25 per guest. Plan B has a flat fee of $350 plus $30 per guest. For how many guests is the total cost of both plans exactly the same?

  1. 25
  2. 30 (correct answer)
  3. 50
  4. 170
Explanation: Let gg be the number of guests. The cost for Plan A is CA=500+25gC_A = 500 + 25g and the cost for Plan B is CB=350+30gC_B = 350 + 30g. To find when the costs are the same, set CA=CBC_A = C_B: 500+25g=350+30g500 + 25g = 350 + 30g. Subtract 350350 from both sides: 150+25g=30g150 + 25g = 30g. Subtract 25g25g from both sides: 150=5g150 = 5g. Divide by 5: g=30g = 30. The cost is the same for 30 guests.

Question 4

A company has fixed monthly costs of $22,000. It costs the company $6.50 to produce each unit of its product, which it sells for $17.50 per unit. How many units must the company sell in a month to break even (i.e., for total revenue to equal total cost)?

  1. 1,257
  2. 2,000 (correct answer)
  3. 3,385
  4. 5,143
Explanation: Let xx be the number of units. The total cost function is C(x)=22000+6.50xC(x) = 22000 + 6.50x. The total revenue function is R(x)=17.50xR(x) = 17.50x. To break even, C(x)=R(x)C(x) = R(x). So, 22000+6.50x=17.50x22000 + 6.50x = 17.50x. Subtract 6.50x6.50x from both sides: 22000=11x22000 = 11x. Divide by 11: x=2000x = 2000. The company must sell 2,000 units to break even.

Question 5

A town's population has been decreasing at a constant rate. In the year 2010, the population was 45,000. In 2020, the population was 39,000. If this linear trend continues, in which year is the population projected to first fall below 30,000?

  1. 2034
  2. 2035 (correct answer)
  3. 2036
  4. 2040
Explanation: Let t=0t=0 represent the year 2010. The two data points are (0, 45000) and (10, 39000). The rate of change (slope) is m=39000−4500010−0=−600010=−600m = \frac{39000 - 45000}{10 - 0} = \frac{-6000}{10} = -600 people per year. The model is P(t)=45000−600tP(t) = 45000 - 600t. We want to find when P(t)<30000P(t) < 30000. Setting P(t)=30000P(t) = 30000: 45000−600t=30000  ⟹  600t=15000  ⟹  t=2545000 - 600t = 30000 \implies 600t = 15000 \implies t = 25. At t=25t = 25 (year 2035), the population equals exactly 30,000. Therefore, the population first falls below 30,000 in the year 2035.

Question 6

Two trees are planted at the same time. Tree A is initially 2 feet tall and grows at a rate of 1.5 feet per year. Tree B is initially 4 feet tall and grows at 1.25 feet per year. After how many full years will Tree A first be taller than Tree B?

  1. 7
  2. 8
  3. 9 (correct answer)
  4. 10
Explanation: Let tt be the number of years. The height of Tree A is HA(t)=2+1.5tH_A(t) = 2 + 1.5t. The height of Tree B is HB(t)=4+1.25tH_B(t) = 4 + 1.25t. We want to find when HA(t)>HB(t)H_A(t) > H_B(t). So, 2+1.5t>4+1.25t2 + 1.5t > 4 + 1.25t. Subtract 1.25t1.25t from both sides: 2+0.25t>42 + 0.25t > 4. Subtract 2 from both sides: 0.25t>20.25t > 2. Divide by 0.25: t>8t > 8. Tree A will be taller than Tree B after more than 8 years. The first full year after 8 years is 9 years.

Question 7

A car rental company charges a flat daily fee plus a fee per mile driven. A 150-mile trip costs a total of $95 for one day. A 250-mile trip costs a total of $125 for one day. How much would a 50-mile trip cost for one day?

  1. $50
  2. $65 (correct answer)
  3. $75
  4. $80
Explanation: Let the cost be C(m)=r⋅m+fC(m) = r \cdot m + f, where mm is miles, rr is the rate per mile, and ff is the flat fee. We have two points: (150, 95) and (250, 125). First, find the rate rr (the slope): r=125−95250−150=30100=0.30r = \frac{125 - 95}{250 - 150} = \frac{30}{100} = 0.30. The cost per mile is $0.30. Now find the flat fee ff using one point: 95=0.30(150)+f  ⟹  95=45+f  ⟹  f=5095 = 0.30(150) + f \implies 95 = 45 + f \implies f = 50. The cost function is C(m)=0.30m+50C(m) = 0.30m + 50. For a 50-mile trip, the cost is C(50)=0.30(50)+50=15+50=65C(50) = 0.30(50) + 50 = 15 + 50 = 65.

Question 8

The value of a phone depreciates linearly over time. A phone purchased for $800 has a resale value of $260 after 3 years. A different phone, purchased for $1,050, has a resale value of $0 after 4 years. What is the difference between the annual depreciation rates of the two phones?

  1. $82.50 (correct answer)
  2. $180.00
  3. $262.50
  4. $442.50
Explanation: The annual depreciation rate is the slope of the value function. For the first phone, the points are (0, 800) and (3, 260). The rate is m1=260−8003−0=−5403=−180m_1 = \frac{260 - 800}{3 - 0} = \frac{-540}{3} = -180. The depreciation is $180/year. For the second phone, the points are (0, 1050) and (4, 0). The rate is m2=0−10504−0=−262.5m_2 = \frac{0 - 1050}{4 - 0} = -262.5. The depreciation is $262.50/year. The difference in the rates is 262.50−180=82.50262.50 - 180 = 82.50.

Question 9

A student's meal plan starts with a balance of $1,500. Each lunch the student buys costs $8.50, and each dinner costs $12.50. If the student bought 40 lunches, how many dinners can the student buy to have a remaining balance of exactly $260?

  1. 20
  2. 72 (correct answer)
  3. 99
  4. 106
Explanation: First, calculate the total amount spent. The starting balance was $1,500 and the desired remaining balance is $260, so the total amount spent is 1500−260=12401500 - 260 = 1240. Next, calculate the amount spent on lunches: 40×8.50=34040 \times 8.50 = 340. The remaining amount available for dinners is 1240−340=9001240 - 340 = 900. Finally, divide this amount by the cost of one dinner to find the number of dinners: 900/12.50=72900 / 12.50 = 72. The student can buy 72 dinners.

Question 10

On a newly discovered planet, temperature is measured in degrees Zorp (Z). Scientists found a linear relationship between Z and Celsius (C). They observed that 10°C is equivalent to 50 Z, and 30°C is equivalent to 90 Z. Based on this relationship, what is the freezing point of water (0°C) in degrees Zorp?

  1. 10 Z
  2. 20 Z
  3. 30 Z (correct answer)
  4. 40 Z
Explanation: We have two points on a line: (10, 50) and (30, 90). The slope is m=90−5030−10=4020=2m = \frac{90 - 50}{30 - 10} = \frac{40}{20} = 2. The linear equation is Z=2C+bZ = 2C + b. We can find the y-intercept, bb, (which corresponds to 0°C) by substituting one of the points: 50=2(10)+b  ⟹  50=20+b  ⟹  b=3050 = 2(10) + b \implies 50 = 20 + b \implies b = 30. Therefore, 0°C is equivalent to 30 Z.

Question 11

Two cyclists are in a race. Cyclist A starts at the starting line and travels at a constant speed of 18 miles per hour. Cyclist B starts with a 5-mile head start and travels at a constant speed of 15 miles per hour along the same route. How long after the start of the race will it take for Cyclist A to catch up to Cyclist B?

  1. 20 minutes
  2. 1 hour and 20 minutes
  3. 1 hour and 40 minutes (correct answer)
  4. 3 hours
Explanation: Let tt be the time in hours. The distance traveled by Cyclist A is DA=18tD_A = 18t. The distance traveled by Cyclist B is DB=15t+5D_B = 15t + 5 (initial distance plus distance traveled). Cyclist A catches up when their distances are equal: 18t=15t+518t = 15t + 5. Subtracting 15t15t from both sides gives 3t=53t = 5, so t=53t = \frac{5}{3} hours. To convert this to hours and minutes: 53 hours=123 hours\frac{5}{3} \text{ hours} = 1 \frac{2}{3} \text{ hours}. Since 23\frac{2}{3} of an hour is 23×60=40\frac{2}{3} \times 60 = 40 minutes, the time is 1 hour and 40 minutes.

Question 12

To host a banquet, a hotel charges a one-time reservation fee plus a cost per person. A banquet for 50 people costs a total of $2,900. A banquet for 80 people costs a total of $4,400. What is the hotel's one-time reservation fee?

  1. $50
  2. $400 (correct answer)
  3. $500
  4. $1,500
Explanation: This problem describes a linear relationship where the reservation fee is the y-intercept. Let the points be (number of people, total cost). So we have (50, 2900) and (80, 4400). First, find the slope (cost per person): m=4400−290080−50=150030=50m = \frac{4400 - 2900}{80 - 50} = \frac{1500}{30} = 50. Now use the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) to find the equation of the line: y−2900=50(x−50)y - 2900 = 50(x - 50). The reservation fee is the cost when x=0x=0. y−2900=50(0−50)  ⟹  y−2900=−2500  ⟹  y=400y - 2900 = 50(0 - 50) \implies y - 2900 = -2500 \implies y = 400. The fee is $400.

Question 13

The pressure in a certain liquid increases by 10.2 kilopascals (kPa) for every 1 meter increase in depth. At a depth of 2 meters, the total pressure is measured to be 121.9 kPa. What is the pressure at the surface (a depth of 0 meters)?

  1. 101.5 kPa (correct answer)
  2. 20.4 kPa
  3. 111.7 kPa
  4. 142.3 kPa
Explanation: Let P(d)P(d) be the pressure at depth dd. The rate of increase is the slope, so m=10.2m=10.2. The function is P(d)=10.2d+bP(d) = 10.2d + b, where bb is the pressure at the surface (the y-intercept). We are given the point (2, 121.9). Substitute this into the equation: 121.9=10.2(2)+b  ⟹  121.9=20.4+b121.9 = 10.2(2) + b \implies 121.9 = 20.4 + b. Solving for bb gives b=121.9−20.4=101.5b = 121.9 - 20.4 = 101.5. The pressure at the surface is 101.5 kPa.

Question 14

An employee earns $18 per hour for the first 40 hours worked in a week and $27 per hour for any hours worked over 40. Which function E(h)E(h) represents the employee's total weekly earnings for working hh hours, where h>40h > 40?

  1. E(h)=18h+27(h−40)E(h) = 18h + 27(h-40)
  2. E(h)=27hE(h) = 27h
  3. E(h)=18(40)+27hE(h) = 18(40) + 27h
  4. E(h)=18(40)+27(h−40)E(h) = 18(40) + 27(h-40) (correct answer)
Explanation: For h>40h > 40, the employee's earnings consist of two parts. First, the pay for the initial 40 hours, which is 18×40=72018 \times 40 = 720. Second, the pay for overtime hours. The number of overtime hours is h−40h - 40, and the rate for these hours is $27. So, the overtime pay is 27(h−40)27(h-40). The total earnings are the sum of these two parts: E(h)=18(40)+27(h−40)E(h) = 18(40) + 27(h-40).

Question 15

The total cost to manufacture a product is a linear function of the number of units produced. The cost to produce 100 units is $5,000, and the cost to produce 400 units is $14,000. What is the cost per unit to produce this product?

  1. $20
  2. $30 (correct answer)
  3. $35
  4. $50
Explanation: The cost per unit is the rate of change of the linear function, which is its slope. Using the two given points, (100, 5000) and (400, 14000), we can calculate the slope: m=change in costchange in units=14000−5000400−100=9000300=30m = \frac{\text{change in cost}}{\text{change in units}} = \frac{14000 - 5000}{400 - 100} = \frac{9000}{300} = 30. The cost per unit is $30.

Question 16

The value, VV, in dollars, of a piece of industrial equipment after tt years is modeled by the function V(t)=45,000−2,500tV(t) = 45,000 - 2,500t. Which of the following is the best interpretation of the number 2,500 in this context?

  1. The initial value of the equipment is $2,500.
  2. The value of the equipment increases by $2,500 each year.
  3. The value of the equipment decreases by $2,500 each year. (correct answer)
  4. The equipment will have no value after 2,500 years.
Explanation: In the linear model y=b+mxy = b + mx, mm represents the rate of change. In V(t)=45,000−2,500tV(t) = 45,000 - 2,500t, the slope is -2,500. This means that for each one-year increase in tt, the value VV changes by -2,500. Therefore, the value of the equipment decreases by $2,500 each year.

Question 17

The percentage of charge remaining in a device battery decreases linearly with hours of use. After 2 hours of use, the battery is at 70% charge. After 5 hours of use, it is at 25% charge. Let C(h)C(h) be the linear function that models the charge CC after hh hours. What is the practical meaning of the y-intercept of the graph of this function?

  1. The rate, in percent per hour, at which the battery loses charge.
  2. The total number of hours the battery can be used until it is empty.
  3. The initial charge of the battery before any use. (correct answer)
  4. The average charge of the battery over the first 5 hours.
Explanation: The y-intercept of a function represents the value of the dependent variable when the independent variable is zero. In this context, the independent variable is hh, hours of use, and the dependent variable is CC, the charge percentage. When h=0h=0, this corresponds to the moment before any use has occurred. Therefore, the y-intercept represents the initial charge of the battery.

Question 18

An online streaming service offers two monthly plans. Plan X costs $25 per month plus $1.50 per movie rental. Plan Y costs $10 per month plus $2.25 per movie rental. At what number of movie rentals per month does Plan X become the less expensive option?

  1. 15 rentals or more
  2. 16 rentals or more
  3. 20 rentals or more
  4. 21 rentals or more (correct answer)
Explanation: Let mm be the number of movie rentals. The cost of Plan X is CX=25+1.50mC_X = 25 + 1.50m. The cost of Plan Y is CY=10+2.25mC_Y = 10 + 2.25m. We want to find when Plan X is less expensive, so CX<CYC_X < C_Y. 25+1.50m<10+2.25m25 + 1.50m < 10 + 2.25m. Subtract (1.50m) from both sides: 25<10+0.75m25 < 10 + 0.75m. Subtract 10 from both sides: 15<0.75m15 < 0.75m. Divide by 0.75: m>150.75  ⟹  m>20m > \frac{15}{0.75} \implies m > 20. Plan X becomes less expensive when the number of rentals is greater than 20. The smallest integer number of rentals for which this is true is 21.

Question 19

A candle is initially 25 centimeters tall. After burning for 4 hours, its height is 19 centimeters. The candle burns at a constant rate. What will be the height of the candle after it has been burning for a total of 6 hours?

  1. 9 cm
  2. 13 cm
  3. 15 cm
  4. 16 cm (correct answer)
Explanation: The initial height (at t=0t=0) is 25 cm. After 4 hours, the height is 19 cm. The amount burned is 25−19=625 - 19 = 6 cm in 4 hours. The rate of burning is 6 cm/4 hours=1.56 \text{ cm} / 4 \text{ hours} = 1.5 cm per hour. The linear model for the height HH after tt hours is H(t)=25−1.5tH(t) = 25 - 1.5t. To find the height after 6 hours, calculate H(6)H(6): H(6)=25−1.5(6)=25−9=16H(6) = 25 - 1.5(6) = 25 - 9 = 16. The height will be 16 cm.

Question 20

A submarine is at a depth of 450 meters below sea level and ascends toward the surface at a constant rate of 75 meters per minute. The function D(t)=450−75tD(t) = 450 - 75t gives the submarine's depth in meters after tt minutes. What does the t-intercept of the graph of this function represent?

  1. The initial depth of the submarine.
  2. The speed at which the submarine ascends.
  3. The depth of the submarine after one minute.
  4. The time it takes for the submarine to reach the sea surface. (correct answer)
Explanation: The t-intercept is the value of tt for which the function's value is zero. In this context, it's the time tt when the depth D(t)D(t) is 0. A depth of 0 meters represents the sea surface. Therefore, the t-intercept represents the time it takes for the submarine to reach the surface.