ACT Math Quiz: Mathematical Modeling

What this quiz covers

This quiz focuses on Mathematical Modeling, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

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.

Question 1

1. The taxi charges a $4.50 starting fee.
2. The taxi charges $4.50 per mile.
3. The taxi charges a $2.25 starting fee.
4. The taxi charges $2.25 per mile. (correct answer)

Explanation: In a linear model $y = mx + b$, the slope $m$ is the amount $y$ changes for each one-unit increase in $x$, while $b$ is the value of $y$ when $x = 0$. In $y = 2.25x + 4.50$ the slope is $2.25$ and it multiplies the miles, so it means the taxi charges $2.25 per mile; the $4.50 is what you owe before traveling any miles. Saying the taxi charges a $4.50 starting fee is a true statement about the model but describes the intercept, not the slope, which is exactly the trap; saying $4.50 per mile attaches the intercept's value to the slope's meaning, and saying $2.25 is a starting fee does the reverse swap. Identify which number is multiplied by the variable and which stands alone: the multiplied one is always the per-unit rate, and the standalone one is always the fixed starting amount.

Question 2

1. $y = -12x + 4$
2. $y = 2x + 20$
3. $y = -4x + 12$
4. $y = -2x + 20$ (correct answer)

Explanation: A quantity changing linearly with time is modeled by $y=mx+b$, where $b$ is the value at $x=0$ and $m$ is the change per unit of time. The temperature is $20$ at $x=0$, so the constant is $20$, and the slope is $\dfrac{-4-20}{12-0}=\dfrac{-24}{12}=-2$ degrees per minute, giving $y=-2x+20$; checking, $x=12$ produces $-2(12)+20=-4$ as required. The equation $y=2x+20$ has the correct starting value but a positive rate, which would warm the freezer instead of cooling it. The equation $y=-4x+12$ mistakenly uses the given data values themselves as slope and intercept, and $y=-12x+4$ does the same thing with the roles further scrambled. Identify the value at time zero as the constant first, then compute the rate as change in output over change in input, and confirm the model by testing the second data point.

Question 3

1. $T = 15p + 100$ (correct answer)
2. $T = 100p + 15$
3. $T = 15 + 100p$
4. $T = 100 - 15p$

Explanation: This is a linear cost model, where a per-unit rate multiplies the variable and a one-time fee stays constant. The $15 charge applies once for each passenger, so it becomes $15p$, while the $100 flat fee is charged once per trip no matter how many passengers ride, so it stays as $+100$, giving$T = 15p + 100$. Both$T = 100p + 15$and$T = 15 + 100p$swap those two roles, charging $100 for every passenger and treating the $15 as the one-time trip fee. $T = 100 - 15p$ makes every extra passenger lower the total, which turns a charge into a discount. Decide which quantity repeats with the variable and which happens only once: the repeating amount gets the variable attached, and the one-time amount is the constant.

Question 4

1. $w = 100 - 10t$
2. $w = 10 + 100t$
3. $w = 100 + 10t$ (correct answer)
4. $w = 100t + 10$

Explanation: This is a linear accumulation model, where the starting amount is the constant and the filling rate multiplies the time. At $t = 0$ the tank already holds 100 gallons, so 100 is the constant, and each minute adds 10 more gallons, so the accumulated water is $10t$, giving $w = 100 + 10t$. Both $w = 10 + 100t$ and $w = 100t + 10$ reverse the roles, starting the tank at 10 gallons and filling it at 100 gallons per minute. $w = 100 - 10t$ has the right numbers in the right places but subtracts, which would model a tank draining rather than being filled. Check the sign against the story first: filling and growing mean addition, while draining and shrinking mean subtraction.

Question 5

1. $C = 30.15t$
2. $C = 0.15 + 30t$
3. $C = 30t + 0.15$
4. $C = 30 + 0.15t$ (correct answer)

Explanation: This is a linear cost model, where the fixed monthly charge is the constant and the per-text rate multiplies the number of texts. The $30 is paid once each month regardless of texting, and each of the $t$messages costs $0.15, so the message cost is$0.15t$, giving$C = 30 + 0.15t$. The expression$C = 30.15t$adds the two amounts together and then multiplies everything by$t$, which wrongly charges the $30 monthly fee once per text message. Both $C = 0.15 + 30t$ and $C = 30t + 0.15$ swap the roles, making $30 the per-text rate and $0.15 the monthly fee. Only the amount that repeats with each unit belongs next to the variable, so never fold a one-time fee into the coefficient.

Question 6

1. $y = 2 + 3x$
2. $y = 3x + 2$
3. $y = 3 - 2x$
4. $y = 2x + 3$ (correct answer)

Explanation: This is a linear cost model in which the base fare is the constant and the per-mile rate multiplies the number of miles. The $2 charge repeats for every mile, so it becomes $2x$, and the $3 base fare is paid once regardless of distance, so it is the constant, giving $y = 2x + 3$. Both$y = 2 + 3x$and$y = 3x + 2$reverse those roles, charging $3 per mile with a $2 base fare. $y = 3 - 2x$ makes the fare shrink as the trip gets longer, which contradicts a charge that accumulates. Ask what happens at $x = 0$: the correct model must return the base fare alone, which is a fast way to confirm the constant term.

Question 7

1. $h = 5 + 2d$ (correct answer)
2. $h = 2 + 5d$
3. $h = 5d + 2$
4. $h = 5 - 2d$

Explanation: This is a linear growth model, where the initial height is the constant and the growth rate multiplies the number of days. The plant starts at 5 cm, so 5 is the constant, and it gains 2 cm each day, so the growth after $d$ days is $2d$, giving $h = 5 + 2d$. Both $h = 2 + 5d$ and $h = 5d + 2$ swap the two numbers, starting the plant at 2 cm and growing it 5 cm per day. $h = 5 - 2d$ uses the right numbers in the right roles but subtracts, describing a plant that shrinks 2 cm per day. Test the model at time zero: whatever remains when the variable is 0 must be the starting value.

Question 8

1. $T = 50c + 10$
2. $T = 10 + 50c$
3. $T = 50 + 10c$ (correct answer)
4. $T = 50 - 10c$

Explanation: This is a linear cost model where the fixed monthly fee is the constant and the per-class rate multiplies the number of classes. The $50 is charged once a month whether or not any classes are attended, and each class costs $10, so the class cost is $10c$, giving $T = 50 + 10c$. Both $T = 50c + 10$ and $T = 10 + 50c$ reverse the roles, charging $50 for every class and treating the $10 as the monthly fee. $T = 50 - 10c$ keeps the numbers in the right roles but subtracts, so attending classes would reduce the bill instead of raising it. Confirm the direction of change before choosing: if the total should rise as the variable grows, the rate must be added, not subtracted.

Question 9

1. 0.10
2. $y$
3. $x$ (correct answer)
4. 25

Explanation: In the phone plan equation y = 0.10x + 25, we need to identify what each variable represents based on the context. The equation models total monthly cost where y represents the total cost, 25 represents the fixed monthly fee, and 0.10 represents the cost per text message. Therefore, x must represent the number of text messages sent, since it's the variable being multiplied by the per-text rate. The structure follows the pattern: total cost = (rate per text)(number of texts) + fixed fee.

Question 10

1. $y = 200 - x$
2. $y = 5x + 200$
3. $y = 200 + 5x$
4. $y = 200 - 5x$ (correct answer)

Explanation: This is a linear altitude model where y represents altitude and x represents time in minutes. The slope of -5 means the altitude decreases by 5 meters per minute (negative because it's descending). The y-intercept of 200 represents the initial altitude when x = 0 minutes. The equation y = 200 - 5x correctly models this decreasing relationship. Choice B incorrectly uses a positive slope, which would represent ascending rather than descending. Choice A uses the wrong rate of change.

Question 11

1. $y = 6x + 1.5$
2. $y = 1.5x + 6$ (correct answer)
3. $y = 6 - 1.5x$
4. $y = 1.5 - 6x$

Explanation: This is a linear growth model where y is plant height in inches and x is weeks since purchase. The plant starts at 6 inches (when x = 0), making 6 the y-intercept. The plant grows 1.5 inches per week, making 1.5 the slope. The equation is y = 1.5x + 6. Choice A incorrectly reverses the slope and intercept, putting 6 as the growth rate and 1.5 as the starting height. Choices C and D use negative slopes, which would mean the plant is shrinking.

Question 12

1. $y = 2x + 18$
2. $y = -2x + 18$ (correct answer)
3. $y = -6x + 18$
4. $y = 2x - 18$

Explanation: This models a candle burning at constant rate where y is height in cm and x is time in hours. At x = 0, the candle is 18 cm tall (y-intercept = 18). At x = 3, it's 12 cm tall. The candle lost 6 cm in 3 hours, so the rate is -2 cm per hour (slope = -2). The equation is y = -2x + 18. Choice A uses positive slope, meaning the candle would grow taller. Choice C has slope -6, which would mean the candle loses 6 cm per hour instead of per 3 hours.

Question 13

1. The runner starts 0.25 meters ahead of the starting line.
2. The runner runs 0.25 meters per second. (correct answer)
3. The runner runs 4 seconds per meter.
4. The runner runs 0.25 meters in total.

Explanation: The model y = 0.25x represents distance (y in meters) versus time (x in seconds) for a runner. The slope 0.25 means the runner's distance increases by 0.25 meters for each second that passes - this is the runner's speed of 0.25 meters per second. There is no y-intercept term, meaning the runner starts at the starting line (0 meters when x = 0). Choice C incorrectly inverts the units to seconds per meter. Choice A misinterprets the slope as a starting position.

Question 14

1. The gym charges $80 each month.
2. The gym charges a sign-up fee of $30.
3. The gym charges $30 per month. (correct answer)
4. The gym charges a sign-up fee of $80 each month.

Explanation: In the linear model y = 30x + 80 for gym costs, y is total cost and x is months. The slope (coefficient of x) is 30, which represents the rate of change - the cost increases by $30 for each additional month. This is the monthly fee. The y-intercept 80 is the initial cost when x = 0, representing the one-time sign-up fee. Choice B incorrectly identifies the slope as the sign-up fee instead of recognizing it as the monthly rate.

Question 15

1. The $-5x^2$ term
2. The $20x$ term
3. The constant term $2$ (correct answer)
4. The coefficient $-5$

Explanation: This quadratic model y = -5x² + 20x + 2 represents height in meters after x seconds. To find initial height, evaluate at x = 0: y = -5(0)² + 20(0) + 2 = 0 + 0 + 2 = 2 meters. The constant term 2 represents the initial height when time equals zero. The -5x² term affects the curved trajectory, and the 20x term contributes to the initial upward velocity. Choice A or D might be confused as representing starting conditions, but only the constant term gives the height at x = 0.

Question 16

1. 133 (correct answer)
2. 108
3. 43
4. 143

Explanation: The linear model y = 18x + 25 shows earnings where y is total dollars and x is hours tutored. The slope 18 means the student earns $18 per hour, and the intercept 25 represents a base payment of $25. To find earnings after 6 hours, substitute x = 6: y = 18(6) + 25 = 108 + 25 = 133. Choice B shows the common error of forgetting to add the y-intercept, calculating only 18(6) = 108.

Question 17

1. 63
2. 78 (correct answer)
3. 72
4. 93

Explanation: The linear model y = 7x + 15 represents water in a tank where y is liters and x is minutes. The slope 7 means the tank fills at 7 liters per minute, and the intercept 15 means there were already 15 liters in the tank at the start. After 9 minutes: y = 7(9) + 15 = 63 + 15 = 78 liters. Choice A shows the error of forgetting the initial 15 liters, calculating only 7(9) = 63.

Question 18

1. The taxi charges $2.50 for each mile driven.
2. The taxi charges a starting fee of $4 before any miles are driven. (correct answer)
3. The taxi charges $4 for each mile driven.
4. The taxi ride always costs $2.50 total.

Explanation: The linear model y = 2.50x + 4 represents taxi cost where y is total cost and x is miles driven. The y-intercept is the value of y when x = 0, which occurs at the point (0, 4). This means when zero miles are driven, the cost is still $4 - this is the starting fee charged before any distance is traveled. The slope 2.50 represents the cost per mile. Choice C confuses the y-intercept (4) with the slope (2.50), incorrectly stating the per-mile charge is $4.

Question 19

1. $120$
2. $125$ (correct answer)
3. $115$
4. $135$

Explanation: This is a linear model y = 15x + 5 where y represents total amount saved and x represents weeks. The slope 15 means the student saves $15 each week. The intercept 5 represents $5 already saved at the start (week 0). To find the total after 8 weeks, substitute x = 8: y = 15(8) + 5 = 120 + 5 = 125 dollars. The model predicts the student will have saved $125 total after 8 weeks. Choice A would result from forgetting the initial $5, while other choices involve calculation errors.

Question 20

1. $7000 (correct answer)
2. $5200
3. $5000
4. $2000

Explanation: This profit model P = 5000 + 200x shows profit P for x units sold. The y-intercept 5000 represents baseline profit, and slope 200 means profit increases by $200 per unit sold. For 10 units: P = 5000 + 200(10) = 5000 + 2000 = $7000. Choice B incorrectly subtracts instead of adds the variable component.