Algebra 1 practice test

A car rental company charges a flat fee of $20 plus $0.15 per mile driven. Write a linear equation representing the total cost C for m miles driven.

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Answer explanation

The total cost C can be represented by the equation C = 20 + 0.15m, where 20 is the flat fee and 0.15 is the cost per mile driven (m). This linear equation captures the relationship between miles driven and total cost.

The sum of two consecutive integers is 51. Find the integers.

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Answer explanation

Let the integers be x and x+1. Their sum is x + (x + 1) = 51. Simplifying gives 2x + 1 = 51. Solving for x, we find x = 25. Thus, the integers are 25 and 26.

A store sells pens for $1.50 each and notebooks for $2.00 each. If a customer buys a total of 10 items for $15.00, how many pens and how many notebooks did they purchase?

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Answer explanation

Let p be the number of pens and n be the number of notebooks. We have the equations: p + n = 10 and 1.5p + 2n = 15. Solving these, we find p = 5 and n = 5. Thus, the customer bought 5 pens and 5 notebooks.

The perimeter of a rectangle is 50 meters. If the length is 5 meters more than twice the width, find the dimensions of the rectangle.

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Answer explanation

Let width = w. Then length = 2w + 5. The perimeter is 2(length + width) = 50. Solving 2((2w + 5) + w) = 50 gives w = 5. Thus, length = 15. Dimensions are 15m (length) and 5m (width).

A taxi charges a base fare of $3.00 plus $0.50 per mile. Write a linear equation representing the total fare F for m miles traveled.

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Answer explanation

The total fare F can be represented by the equation F = 3.00 + 0.50m, where 3.00 is the base fare and 0.50m accounts for the cost per mile traveled.

A population of bacteria doubles every 3 hours. If the initial population is 500, express the population P as a function of time t in hours.

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Answer explanation

The population doubles every 3 hours, so after t hours, the number of 3-hour intervals is t/3. The population function is P(t) = 500 * 2^(t/3). This represents exponential growth.

The value of a car depreciates by 15% each year. If the car's initial value is $20,000, write an exponential function representing its value after t years.

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Answer explanation

The car's value after t years can be modeled by the function V(t) = 20000 * (0.85)^t, where 0.85 represents the remaining value after a 15% depreciation each year.

A savings account earns 5% interest compounded annually. If the initial deposit is $1,000, write an exponential function representing the account balance after t years.

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Answer explanation

The account balance after t years can be modeled by the function A(t) = 1000(1 + 0.05)^t, which simplifies to A(t) = 1000(1.05)^t, representing the initial deposit growing at 5% interest compounded annually.

The half-life of a certain radioactive substance is 10 years. If the initial amount is 100 grams, write an exponential function representing the amount remaining after t years.

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Answer explanation

The exponential function for radioactive decay is A(t) = A0 * (1/2)^(t/T), where A0 is the initial amount, t is time, and T is the half-life. Here, A(t) = 100 * (1/2)^(t/10).

A population of rabbits triples every 4 months. If the initial population is 200, express the population P as a function of time t in months.

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Answer explanation

The population triples every 4 months, so after t months, the population P can be expressed as P(t) = 200 * 3^(t/4). This accounts for the growth factor of 3 every 4 months.