- The measure of an angle is 18 degrees more than one-third the measure of its supplementary angle. If x is the measure of the angle, which equation correctly represents this relationship?
Answer: x = (1/3)(180 - x) + 18
Let the angle be x. Its supplement is 180 - x. 'One-third the measure of its supplement' is (1/3)(180 - x). '18 degrees more than' this quantity is (1/3)(180 - x) + 18. Since this is equal to the measure of the angle x, the equation is x = (1/3)(180 - x) + 18.
- A runner moves at 7 miles per hour for t hours. Which equation gives distance d?
Answer: d=7t
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes a constant speed of 7 miles per hour multiplied by time t to find distance d, requiring an equation of the form d = speed × time. Choice C is correct because it accurately models the relationship using multiplication of the given speed and time variable. Choice B is incorrect because it divides instead of multiplies, a common error when inverting the formula. To help students, teach identifying key parts of scenarios like pace and duration, and translating them into mathematical terms. Encourage practice with varied contexts such as running or walking to build flexibility in model creation. - A car rental company charges a fee of $30 per day plus $0.25 per mile driven. If a car is rented for d days and driven for m miles, which equation calculates the total rental cost, C, in dollars?
Answer: C = 30d + 0.25m
The total cost is the sum of the daily cost and the mileage cost. The daily cost is $30 multiplied by the number of days, d, which is 30d. The mileage cost is $0.25 multiplied by the number of miles, m, which is 0.25m. The total cost C is the sum of these two amounts: C = 30d + 0.25m.
- Noah deposits 35 dollars, then saves 15 dollars each month for m months. Identify S.
Answer: S=15m+35
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes an initial deposit of 35 dollars plus 15 dollars added each month for m months, requiring an equation of the form S = rate × months + initial. Choice C is correct because it accurately models the relationship using multiplication for the monthly savings and addition for the initial deposit. Choice B is incorrect because it subtracts with swapped coefficients, a common error when misordering terms. To help students, teach identifying key parts of scenarios like initials and rates, and translating them into mathematical terms. Encourage practice with varied contexts such as finances or increments to build flexibility in model creation. - Ethan deposits 60 dollars, then saves 12 dollars each month for m months. Choose equation for S.
Answer: S=12m+60
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes an initial deposit of 60 dollars plus 12 dollars added each month for m months, requiring an equation of the form S = rate × months + initial. Choice A is correct because it accurately models the relationship using multiplication for the monthly savings and addition for the initial deposit. Choice B is incorrect because it subtracts with swapped order, a common error when confusing addition with subtraction. To help students, teach identifying key parts of scenarios like starting amounts and monthly adds, and translating them into mathematical terms. Encourage practice with varied contexts such as budgeting or saving plans to build flexibility in model creation. - A jacket is on sale for 20% off its original price. An additional 5% sales tax is calculated on the discounted price. If the final price after the discount and tax is $84, which equation can be used to find the original price, p?
Answer: 1.05(0.80p) = 84
The original price is p. A 20% discount means the price becomes p - 0.20p = 0.80p. The 5% sales tax is applied to this discounted price, so the total cost is the discounted price plus 5% of the discounted price: 0.80p + 0.05(0.80p). This can be factored as (1 + 0.05)(0.80p), which simplifies to 1.05(0.80p). The final price is $84, so 1.05(0.80p) = 84.
- A store pays 200 dollars fixed costs and earns 6 dollars profit per item sold. For x items, choose P.
Answer: P=200+6x
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes a store with $200 fixed costs that earns $6 profit per item, requiring an equation of the form P = fixed costs + (profit per item × items). Choice A is correct because P = 200 + 6x accurately models starting with $200 in costs and adding $6 profit for each of x items sold. Choice B incorrectly subtracts the profit, Choice C reverses the coefficients, and Choice D omits the per-item profit entirely. To help students, emphasize that profit adds to the total while costs might be subtracted in net profit scenarios. Teaching students to identify whether values increase or decrease the total helps avoid sign errors. - Leo has $150 in his savings account. He plans to deposit $25 each week. Which equation can be used to find the number of weeks, w, it will take for him to have exactly $750 in his account?
Answer: 150 + 25w = 750
The total amount in the account will be the initial amount ($150) plus the total amount deposited over w weeks. The amount deposited is $25 per week, so for w weeks, he deposits 25w. The equation for the total amount to reach $750 is 150 + 25w = 750.
- A lab uses 8 grams of Chemical A plus 3 grams per trial of Chemical B. For t trials, write G.
Answer: G=8+3t
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes using 8 grams of Chemical A (fixed amount) plus 3 grams per trial of Chemical B, requiring an equation of the form G = fixed amount + (amount per trial × trials). Choice A is correct because G = 8 + 3t accurately models using 8 grams initially plus 3 grams for each of t trials. Choice B incorrectly groups the constants before multiplying, Choice C subtracts instead of adds, and Choice D reverses the coefficients. To help students, teach them to distinguish between fixed amounts (used once) and variable amounts (used repeatedly). Practice identifying keywords like 'per trial' or 'each time' that signal multiplication. - Jada has a collection of dimes and nickels worth a total of $4.15. She has 7 more dimes than nickels. Which equation can be used to find n, the number of nickels Jada has, if all values are expressed in cents?
Answer: 5n + 10(n + 7) = 415
Let n be the number of nickels. The number of dimes is n + 7. In cents, the value of the nickels is 5n and the value of the dimes is 10(n + 7). The total value is 415 cents. The equation is the sum of the values of the nickels and dimes: 5n + 10(n + 7) = 415.
- Maya deposits 80 dollars, then saves 25 dollars each month for m months. Identify S.
Answer: S=25m+80
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes an initial deposit of 80 dollars plus 25 dollars added each month for m months, requiring an equation of the form S = rate × months + initial. Choice A is correct because it accurately models the relationship using multiplication for the monthly savings and addition for the initial deposit. Choice B is incorrect because it swaps the coefficients, a common error when students misread the roles of constants and variables. To help students, teach identifying key parts of scenarios like fixed amounts and rates, and translating them into mathematical terms. Encourage practice with varied contexts such as banking or accumulation to build flexibility in model creation. - Liam deposits 150 dollars, then saves 40 dollars each month for m months. Which equation fits?
Answer: S=40m+150
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes an initial deposit of 150 dollars plus 40 dollars added each month for m months, requiring an equation of the form S = rate × months + initial. Choice B is correct because it accurately models the relationship using multiplication for the monthly savings and addition for the initial deposit. Choice A is incorrect because it subtracts instead of adds, a common error when students misinterpret accumulation as depletion. To help students, teach identifying key parts of scenarios like starting points and increments, and translating them into mathematical terms. Encourage practice with varied contexts such as savings or growth to build flexibility in model creation. - Maya starts with 120 dollars and saves 25 dollars monthly for m months, totaling T. Choose the equation.
Answer: T=120+25m
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes starting with $120 and adding $25 each month for m months, requiring an equation of the form T = initial amount + (monthly amount × months). Choice A is correct because T = 120 + 25m accurately models starting with $120 and adding $25 for each of m months. Choice B incorrectly reverses the constants, Choice C subtracts instead of adds, and Choice D doesn't include the variable m. To help students, emphasize identifying the starting value and the repeated action, then translating 'each month' or 'per month' into multiplication. Regular practice with savings and accumulation problems builds this skill. - A company has fixed costs of 500 dollars and earns 20 dollars per unit sold. Choose P.
Answer: P=20u−500
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes earnings of 20 dollars per unit minus fixed costs of 500 dollars for profit P, requiring an equation of the form P = rate × units - fixed. Choice B is correct because it accurately models the relationship using multiplication for earnings and subtraction for costs. Choice A is incorrect because it adds the fixed cost, a common error when confusing profit calculation. To help students, teach identifying key parts of scenarios like incomes and outflows, and translating them into mathematical terms. Encourage practice with varied contexts such as manufacturing or trade to build flexibility in model creation. - The temperature at 6:00 a.m. was -8°F. The temperature increased at a constant rate of 3°F per hour. Which equation can be used to find h, the number of hours it took for the temperature to reach 13°F?
Answer: -8 + 3h = 13
The final temperature (13°F) is equal to the initial temperature (-8°F) plus the total increase. The total increase is the rate (3°F per hour) multiplied by the number of hours (h), which is 3h. So, the equation is -8 + 3h = 13.
- A reaction needs 12 mL of Solution A and 4 mL per test of Solution B. For k tests, choose V.
Answer: V=12+4k
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes needing 12 mL of Solution A (fixed amount) and 4 mL per test of Solution B, requiring an equation of the form V = fixed volume + (volume per test × tests). Choice A is correct because V = 12 + 4k accurately models using 12 mL initially plus 4 mL for each of k tests. Choice B incorrectly groups the constants (12+4) before multiplying by k, Choice C reverses the relationship making it 12k + 4, and Choice D subtracts instead of adds. To help students, emphasize distinguishing between one-time amounts and repeated amounts, and teach that 'per test' signals multiplication. Laboratory contexts provide excellent real-world applications for these equation-building skills. - A car drives at 55 miles per hour for t hours, traveling d miles. Which equation models this?
Answer: d=55t
This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes a car traveling at a constant rate of 55 miles per hour for t hours, requiring an equation that shows distance equals rate times time (d = rt). Choice B is correct because d = 55t accurately models the relationship where distance equals the rate (55 mph) multiplied by time (t hours). Choice A incorrectly adds instead of multiplying, Choice C reverses the variables, and Choice D uses an incorrect rate. To help students, teach them to identify the key relationship (distance = rate × time) and translate verbal descriptions into mathematical operations. Practice with various motion problems helps students recognize when to multiply versus add. - A student's scores on the first three of four tests are 78, 91, and 82. Which equation can be used to find the score x the student must earn on the fourth test to have an average score of exactly 85?
Answer: (78 + 91 + 82 + x) / 4 = 85
The average of a set of numbers is their sum divided by the count of the numbers. To find the average of the four test scores, we sum them (78 + 91 + 82 + x) and divide by 4. To achieve an average of 85, this expression must equal 85. So, the equation is (78 + 91 + 82 + x) / 4 = 85.
- In a basketball game, a player scored 31 points by making a combination of 2-point shots and 3-point shots. The player made 5 more 2-point shots than 3-point shots. If x represents the number of 3-point shots made, which equation models this situation?
Answer: 3x + 2(x + 5) = 31
Let x be the number of 3-point shots. The number of 2-point shots is '5 more than' x, which is x + 5. The points from 3-pointers are 3x. The points from 2-pointers are 2(x + 5). The total points are the sum of these, which is 31. Therefore, the equation is 3x + 2(x + 5) = 31.
- Town A has a population of 12,000 and is growing by 250 people per year. Town B has a population of 15,000 and is decreasing by 150 people per year. Which equation can be used to find y, the number of years it will take for the two towns to have the same population?
Answer: 12000 + 250y = 15000 - 150y
The population of Town A after y years will be its initial population plus the growth: 12000 + 250y. The population of Town B after y years will be its initial population minus the decrease: 15000 - 150y. To find when the populations are equal, we set these two expressions equal to each other: 12000 + 250y = 15000 - 150y.
