Writing and Modeling with Equations

A local bakery sells cupcakes for $2 each and cakes for $12 each. Which of the following equations represents the total cost, C, of buying x cupcakes and y cakes?

• C = 2x + 12

• C = 2x + 12y

• C = x + y

• C = 12x + 2y

Answer explanation

The total cost, C, can be found by multiplying the number of cupcakes (x) by their price ($2) and the number of cakes (y) by their price ($12), then adding the two products together. The correct equation is C = 2x + 12y.

A gym membership costs a one-time fee of $50 and then $30 per month. If m represents the number of months, which equation gives the total cost, T, of the membership for m months?

• T = 50m + 30

• T = 30m + 50

• T = 50 + 30m

• T = 80m

Answer explanation

The question asks for the total cost of a gym membership for m months. The one-time fee is $50, and the monthly fee is $30. To find the total cost, we add the one-time fee to the product of the monthly fee and the number of months. The correct equation is T = 50 + 30m.

A fruit vendor sells oranges for $1.50 each and apples for $1 each. If you bought o oranges and a apples, which equation represents the total cost, P?

• P = o + a

• P = 1.50o + a

• P = 0 + 1.5a

• P = 1.50 + a

Answer explanation

The question asks for the total cost of buying o oranges and a apples. Oranges cost $1.50 each, and apples cost $1 each. To find the total cost, P, we need to multiply the number of oranges by their price and add it to the number of apples multiplied by their price. The correct equation is P = 1.50o + a.

A company produces gadgets. The production cost is $500 plus $10 per gadget. If g is the number of gadgets and C is the total cost, which equation represents this situation?

• C = 500g + 10

• C = 10g + 500

• C = 500 + 10g

• C = 510g

Answer explanation

The question states that the production cost is $500 plus $10 per gadget. To represent this situation, we need an equation that combines the fixed cost ($500) and the variable cost ($10 per gadget). The correct choice is C = 500 + 10g, where C is the total cost and g is the number of gadgets.

A subscription service charges a $20 setup fee and then $5 for every book purchased. If b represents the number of books and S the total cost, which equation models this scenario?

• S = 5b + 20

• S = 20b + 5

• S = 25b

• S = b + 25

Answer explanation

In this question, the subscription service charges a $20 setup fee and then $5 for every book purchased. To model this scenario, we need to add the setup fee to the cost of each book multiplied by the number of books (b). The correct equation is S = 5b + 20, which represents the total cost (S) as a function of the number of books (b).

A venue rents out spaces for events. They charge a booking fee of $300 and then an additional $15 per guest. If g is the number of guests and R is the total rental cost, which of the following equations is correct?

• R = 300 + 15

• R = 15g + 300

• R = 315g

• R = g + 315

Answer explanation

The total rental cost (R) consists of a fixed booking fee of $300 and an additional charge of $15 per guest. To find the total cost, we need to multiply the number of guests (g) by the additional charge per guest ($15) and then add the fixed booking fee ($300). Therefore, the correct equation is R = 15g + 300.

A coffee shop charges a flat fee of $5 for a coffee tasting event, plus $2 for every pastry bought. If x represents the number of pastries and y represents the total cost, which of the following equations represents this situation?

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A mobile data plan has a base cost of $15 and charges $10 for every additional gigabyte of data used. If x represents the gigabytes of data and y represents the total cost, which equation corresponds to this situation?

• y = 15x

• y = 10x

• y = 10x + 15

• y = 15x + 10

Answer explanation

The question states that the base cost is $15 and there is an additional charge of $10 for every gigabyte of data used. Therefore, the total cost (y) can be represented as the base cost plus the additional charge per gigabyte (10x). The correct equation is y = 10x + 15.

A car rental service has a base fee and then charges a certain amount for every mile driven. If x is the number of miles driven and y is the total cost, which equation best models this scenario?

• y = 5x + 20

• y = 20x + 5

• y = 25x

• y = x + 25

Answer explanation

In this question, the car rental service charges a base fee and an additional amount for every mile driven. The equation y = 5x + 20 best models this scenario, as it represents the total cost (y) as a combination of the base fee (20) and the cost per mile (5) multiplied by the number of miles driven (x).

A band charges a fixed fee of $100 to perform and then an additional $50 for each song requested. If x is the number of songs and y is the total cost, which of the following equations is correct?

• y = 100x + 50

• y = 50x + 100

• y = 150x

• y = x + 150

Answer explanation

The question asks for the total cost (y) of a band's performance, which includes a fixed fee of $100 and an additional $50 for each song (x). The correct equation representing this relationship is y = 50x + 100, as it accounts for both the fixed fee and the additional cost per song.