Variables in Context - ISEE Lower Level Quantitative Reasoning

1. A painter charges a flat fee of $100 plus an hourly rate for his work. The total cost for a job that takes h hours is represented by the expression $100 + 30h$. What does the number 30 most likely represent in this context?
   Answer: The cost per hour of work in dollars. In the expression $100 + 30h$, the total cost is calculated by adding a fixed amount (100) to an amount that changes with the number of hours worked. The term $30h$ means that for every hour h, the cost increases by $30. Therefore, 30 represents the cost per hour, or the hourly rate.
2. At a carnival, a game costs 3 tickets to play. A student starts with t tickets and plays the game 4 times. Which expression shows the number of tickets the student has left?
   Answer: t - 3(4) The total number of tickets used is the cost per game (3) multiplied by the number of times played (4), which is $3 \times 4$. To find the remaining tickets, this total must be subtracted from the starting number of tickets, t. The correct expression is $t - 3(4)$.
3. A recipe calls for 3 cups of flour for every 2 cups of sugar. If a baker uses s cups of sugar, which expression represents the amount of flour needed?
   Answer: (3/2)s The ratio of flour to sugar is 3 to 2, which can be written as the fraction 3/2. This means for every 1 cup of sugar, you need 3/2 (or 1.5) cups of flour. To find the amount of flour needed for s cups of sugar, you multiply s by this ratio. The correct expression is $(3/2)s$.
4. Mariah starts with a full bag of c cat treats. She gives her cat 5 treats in the morning and 3 treats in the evening. Which expression shows how many treats are left in the bag at the end of one day?
   Answer: c - (5 + 3) To find the number of treats left, we must subtract the total number of treats given away from the starting amount, c. The total number of treats given away is the sum of the morning treats (5) and evening treats (3), which is $(5 + 3)$. Therefore, the expression representing the remaining treats is $c - (5 + 3)$.
5. Ms. Davis bought b boxes of crayons. Each box contains 24 crayons. She wants to give 2 crayons to each of her 30 students. Which expression shows the number of crayons she will have left?
   Answer: 24b - 30(2) First, find the total number of crayons Ms. Davis has, which is the number of boxes b times 24 crayons per box, or (24b). Next, find the number of crayons she gives away, which is 30 students times 2 crayons each, or $30(2)$. To find the number left, subtract the amount given away from the total amount she has: $24b - 30(2)$.
6. A box contains p pencils. The pencils are to be shared equally among 7 students, and there will be 4 pencils left over. Which equation could be used to find the number of pencils each student receives, s?
   Answer: p = 7s + 4 The total number of pencils, p, must account for the pencils given to the students and the pencils left over. If each of the 7 students receives s pencils, the total number of pencils given out is $7 \times s$, or $7s$. Since there are 4 pencils left over, the original total p must be the sum of those given out and those left over: $p = 7s + 4$.
7. A baker made a batch of d dozen muffins. He sold 40 individual muffins. Since there are 12 muffins in a dozen, which expression represents the number of muffins left?
   Answer: 12d - 40 First, find the total number of individual muffins the baker made. Since d represents the number of dozens, the total number of muffins is $12 \times d$, or $12d$. He sold 40 muffins, so we subtract 40 from the total. The correct expression is $12d - 40$.
8. Leo is y years old. His aunt is 5 years younger than twice Leo's age. Which expression represents the aunt's age?
   Answer: 2y - 5 First, find twice Leo's age, which is $2 \times y$, or $2y$. Then, to find the aunt's age, subtract 5 from that amount. The correct expression is $2y - 5$.
9. A delivery truck starts the day with 150 packages. At the first stop, the driver delivers p packages. At the second stop, the driver picks up 12 new packages. Which expression represents the number of packages on the truck after the second stop?
   Answer: 150 - p + 12 The truck begins with 150 packages. Delivering p packages means subtracting p from the total, resulting in $150 - p$. Picking up 12 new packages means adding 12 to the new total. Therefore, the expression is $150 - p + 12$.
10. Jamal had m dollars in his wallet. He spent half of his money on a book and then received $5 from his friend. Which expression represents the amount of money Jamal has now?
    Answer: m/2 + 5 First, calculate the amount of money Jamal had after buying the book. Spending half of his money means dividing his initial amount m by 2, which is $m/2$. Then, he received $5, so we add 5 to this new amount. The correct expression is $m/2 + 5$.
11. A rectangular field has a length of 50 meters. If the perimeter of the field is P meters, which expression can be used to find the width of the field?
    Answer: (P - 100) / 2 The formula for the perimeter of a rectangle is $P = 2l + 2w$. We know the length l is 50, so $P = 2(50) + 2w$, which simplifies to $P = 100 + 2w$. To find the width w, we first subtract 100 from the perimeter: $P - 100 = 2w$. Then, we divide by 2: $w = (P - 100) / 2$.
12. A school cafeteria has a total of T chairs. There are 12 large tables that each have 8 chairs, and a few small tables that each have 4 chairs. If s represents the number of small tables, which equation correctly represents the total number of chairs?
    Answer: T = 12(8) + 4s The total number of chairs, T, is the sum of the chairs at the large tables and the chairs at the small tables. The number of chairs at the 12 large tables is $12 \times 8$. The number of chairs at the s small tables is $4 \times s$, or $4s$. Adding these together gives the total: $T = 12(8) + 4s$.
13. A bus travels at a constant speed of 55 miles per hour. If h is the number of hours the bus has been traveling, which expression represents the remaining distance to a city that is 400 miles away?
    Answer: 400 - 55h The distance the bus has traveled is its speed (55) multiplied by the number of hours (h), which is $55h$. The remaining distance is the total distance (400) minus the distance already traveled. Therefore, the expression is $400 - 55h$.
14. A phone plan costs $25 per month, plus $2 for every gigabyte of data used. If g represents the number of gigabytes used, which expression represents the total cost for 3 months?
    Answer: 3(25 + 2g) The cost for one month is the base fee ($25) plus the data cost ($2g$), which is represented by the expression $25 + 2g$. To find the total cost for 3 months, this entire monthly cost must be multiplied by 3. The correct expression is $3(25 + 2g)$.
15. A bakery had 200 cupcakes at the start of the day. They sold c cupcakes and threw away 8 that were dropped. Which equation would be used to find c if there were 24 cupcakes left at the end of the day?
    Answer: 200 - c - 8 = 24 The number of cupcakes at the end of the day is the starting amount minus the cupcakes sold and minus the cupcakes thrown away. This can be represented as: Starting Amount - Sold - Thrown Away = Left. Plugging in the numbers gives the equation $200 - c - 8 = 24$.
16. A video game store sells new games for $40 and used games for $15. If a customer buys n new games and 3 used games, which expression represents the total cost of the purchase?
    Answer: 40n + 15(3) The total cost is the sum of the cost of the new games and the cost of the used games. The cost of n new games at $40 each is $40 \times n$, or $40n$. The cost of 3 used games at $15 each is $15 \times 3$. Therefore, the total cost is represented by the expression $40n + 15(3)$.
17. There are 4 quarts in a gallon. A family drinks g gallons of milk and 3 additional quarts of milk in one week. Which expression represents the total number of quarts of milk they drank?
    Answer: 4g + 3 To find the total number of quarts, we must first convert the gallons to quarts. Since there are 4 quarts in a gallon, g gallons is equal to $4 \times g$, or $4g$ quarts. Then, we add the 3 additional quarts. The correct expression for the total number of quarts is $4g + 3$.
18. Sam earns money by mowing lawns for $20 each and washing cars for $10 each. He wants to earn exactly $100. If he mows m lawns, which expression shows how many cars he must wash to reach his goal?
    Answer: (100 - 20m) / 10 First, calculate the money Sam earns from mowing lawns, which is (20m). The remaining money he needs to earn is $100 - 20m$. Since he earns $10 for each car wash, we divide the remaining amount by 10 to find the number of cars he must wash. The correct expression is $(100 - 20m) / 10$.
19. A group of s students is divided into 4 equal teams for a game. Two students leave before the game begins. Which expression represents the number of students on each team now?
    Answer: (s - 2) / 4 First, find the number of students remaining. If 2 students leave from the original group of s students, there are $s - 2$ students left. This new total is then divided into 4 equal teams. Therefore, the number of students on each team is represented by $(s - 2) / 4$.
20. A library has f fiction books and n non-fiction books. If the library donates 50 fiction books and buys 75 new non-fiction books, which expression represents the new total number of books in the library?
    Answer: f - 50 + n + 75 The new number of fiction books is $f - 50$. The new number of non-fiction books is $n + 75$. The new total number of books is the sum of these two amounts, which is $(f - 50) + (n + 75)$. By the commutative property of addition, this is the same as $f - 50 + n + 75$. The other options incorrectly group the operations.

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