Question

Keith plans to buy a car when he is 16. He currently has $3,000 saved up to buy the car. If his parents plan to give him $2,500 as a birthday gift when he turns 16, and Keith wants to spend at least $2,000 of it on his $15,000 car. What is the most amount of money he would need to save each month now (for the next two years) now that he is anticipating his birthday gift? (read it closely, how do you add in the 2,500 amount into the equation?)

Answer 1

$416.67

Explanation:

The problem statement gives us a range of values that Keith might use from his birthday gift. It asks for the maximum monthly amount that Keith would need to save.

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savings

If x is the amount Keith must save monthly, then his savings in two years will be 2(12x) = 24x. Added to the 3000 he has already saved, the amount he must spend from his birthday gift will be the difference between the car price and his savings:

15000 -(3000 +24x) = 12000 -24x

birthday gift

We are told this will be at least 2000, but it can be no more than 2500 (the amount of the gift), so we have ...

2000 ≤ 12000 -24x ≤ 2500

monthly amount

Solving this, we find ...

-10000 ≤ -24x ≤ -9500 . . . . . . subtract 12000

416.67 ≥ x ≥ 395.83 . . . . . . . . divide by -24 (reverses the inequality)

The most Keith would have to save each month is $416.67.

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Additional comment

The 2500 amount does not actually come into play when computing the maximum he must save each month. It sets the minimum Keith must save.