Video: US-SAT04S3-Q12-293183064714

Car company one charges its customers according to the formula 𝐢₁ = 40 + 20𝑑, where 𝑑 is the number of days of car rental and 𝐢₁ is the total cost. Car company 2 charges its customers according to the formula 𝐢₂ = 30 + 22𝑑, where 𝑑 is the number of days of car rental and 𝐢₂ is the total cost of rental. For a particular number of days, the cost in dollars, is the same for both companies. Calculate this cost in dollars.

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Video Transcript

Car company one charges its customers according to the formula, 𝐢 one is equal to 40 plus 20𝑑, where 𝑑 is the number of days of car rental and 𝐢 one is the total cost. Car company two charges its customers according to the formula, 𝐢 two is equal to 30 plus 22𝑑, where 𝑑 is the number of days of car rental and 𝐢 two is the total cost of rental. For a particular number of days, the cost in dollars is the same for both companies. Calculate this cost in dollars.

In this question, we’re given the formula for two car companies, 𝐢 one is equal to 40 plus 20𝑑 and 𝐢 two is equal to 30 plus 22𝑑. We need to work out when the cost for both companies is the same. This means that the cost of 𝐢 one must be equal to the cost of 𝐢 two. We can set up an equation by putting the two expressions equal to each other. 40 plus 20𝑑 is equal to 30 plus 22𝑑. In order to solve this, we need to balance the equation to calculate the value of 𝑑. We can firstly subtract 30 from both sides of the equation.

Subtracting 30 from the left-hand side gives us 10 plus 20𝑑 as 40 minus 30 is equal to 10. Subtracting 30 from the right-hand side just leaves us with 22𝑑. Our next step is to subtract 20𝑑 from both sides of this new equation. On the left-hand side, the 20𝑑 is cancel and we’re left with 10. On the right-hand side, 22𝑑 minus 20𝑑 is equal to two 𝑑. Finally, we can divide both sides of this equation by two. This gives us a value of 𝑑 equal to five. We have therefore proved that the cost will be the same with car company one and car company two, if the car is rented for five days. This isn’t the end of the question, though, as we were asked to calculate the cost in dollars.

We need to substitute 𝑑 equals five into one of the two equations. Substituting 𝑑 equals five into the formula for car company one gives us 𝐢 one is equal to 40 plus 20 multiplied by five. 20 multiplied by five is equal to 100. So we have 40 plus 100. This is equal to 140 which suggests that the cost in dollars would be 140 dollars. We can check this answer by substituting 𝑑 equals five into the formula for car company two. This gives us 𝐢 two is equal to 30 plus 22 multiplied by five. 22 multiplied by five is equal to 110. And 30 plus 110 once again gives us an answer of 140. The cost in dollars after five days, which is the same for both companies, is 140 dollars.

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