Linear Equation Word Problems - Pre Algebra

1. A taxi company charges an initial fee when you get in the cab, plus a rate per mile traveled. The total cost for a 8-mile trip is $19.20, and the total cost for a 15-mile trip is $30.30. What is the cost per mile?
   Answer: $1.58 Let $f$ = initial fee and $r$ = rate per mile. We have: $f + 8r = 19.20$ and $f + 15r = 30.30$. Subtracting the first equation from the second: $7r = 30.30 - 19.20 = 11.10$. Therefore $r = 11.10 ÷ 7 = 1.585...$ which rounds to $1.58. Choice A ($1.45) might result from incorrectly calculating $11.10 ÷ 8$. Choice B ($1.50) might come from rounding $11.10 ÷ 7$ incorrectly. Choice D ($1.62) might result from a calculation error in the subtraction step.
2. A tutoring service charges a registration fee plus a rate per hour of tutoring. The total cost for 8 hours of tutoring is $156, and the total cost for 12 hours is $204. A student wants to prepay for exactly enough hours so that their average cost per hour (including the registration fee) is $15.00. How many hours should they purchase?
   Answer: 20 hours Let $r$ = registration fee and $h$ = hourly rate. We have: $r + 8h = 156$ and $r + 12h = 204$. Subtracting: $4h = 48$, so $h = 12$. Then $r = 156 - 96 = 60$. For average cost of $15 per hour over $n$ hours: $\frac{60 + 12n}{n} = 15$. Solving: $60 + 12n = 15n$, so $60 = 3n$, giving $n = 20$. Choice A (16 hours) gives average cost $\frac{60 + 192}{16} = 15.75$. Choice B (18 hours) gives $\frac{60 + 216}{18} = 15.33$. Choice D (22 hours) gives $\frac{60 + 264}{22} = 14.73$.
3. A gym membership has a one-time enrollment fee plus a monthly charge. After 4 months, the total amount paid is $290. After 9 months, the total amount paid is $490. What will be the total amount paid after exactly 1 year of membership?
   Answer: $610 Let $e$ = enrollment fee and $m$ = monthly charge. We have: $e + 4m = 290$ and $e + 9m = 490$. Subtracting: $5m = 200$, so $m = 40$. Then $e = 290 - 160 = 130$. After 12 months: $130 + 12(40) = 130 + 480 = 610$. Choice B ($630) might result from miscounting months or calculation errors. Choice C ($650) might come from using an incorrect monthly rate. Choice D ($670) might result from adding the enrollment fee twice.
4. Marcus has been saving money for a new bike. He started with some initial amount and saves the same amount each week. After 6 weeks, he had $127. After 10 weeks, he had $167. How much money did he start with?
   Answer: $67 Let $s$ = initial amount and $w$ = weekly savings. We have: $s + 6w = 127$ and $s + 10w = 167$. Subtracting the first from the second: $4w = 40$, so $w = 10$. Substituting back: $s + 6(10) = 127$, so $s = 127 - 60 = 67$. Choice B ($70) might result from a calculation error. Choice C ($73) might come from using the wrong equation setup. Choice D ($77) might result from adding instead of subtracting when solving for the initial amount.

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