Words to Equations - ISEE Lower Level Quantitative Reasoning

A tank contains 200 gallons of water. A pump removes water at a rate of 5 gallons per minute, while a hose adds water at a rate of 3 gallons per minute. Which equation can be used to find m, the number of minutes it will take until the tank contains exactly 150 gallons of water?
Answer: 200 - 5m + 3m = 150 The tank starts with 200 gallons. Each minute (m), 5 gallons are removed, so we subtract 5m. At the same time, 3 gallons are added, so we add 3m. The amount of water in the tank after m minutes is the starting amount plus the changes: $200 - 5m + 3m$ . We want this amount to be 150 gallons. Therefore, the equation is $200 - 5m + 3m = 150$ .
Maya has a jar of coins containing only dimes and quarters. She has 3 more quarters than she has dimes. The total value of the coins is $2.75. If d represents the number of dimes, which equation represents the total value of the coins in cents?
Answer: 10d + 25(d + 3) = 275 The number of dimes is d. The number of quarters is '3 more than dimes', which is $d + 3$ . To find the total value in cents, multiply the number of each coin by its value in cents. The value of the dimes is 10d. The value of the quarters is $25(d + 3)$ . The total value is $2.75, which is 275 cents. The equation is the sum of the values: $10d + 25(d + 3) = 275$ .
A school library starts the week with 1,200 books. On Monday, 30 books are checked out. On Tuesday, a donation of d new books arrives. On Wednesday, twice as many books are checked out as on Monday. At the end of Wednesday, the library has 1,190 books. Which equation represents the number of books in the library?
Answer: 1200 - 30 + d - (2 * 30) = 1190 Start with the initial number of books, 1,200. Subtract the books checked out on Monday (30). Add the donated books on Tuesday (d). On Wednesday, 'twice as many books...as on Monday' were checked out, which is $2 \times 30$ . This amount is also subtracted. The final count is 1,190. The equation showing all these changes is $1200 - 30 + d - (2 \times 30) = 1190$ .
The number of fish in a pond, 60, is 12 less than three times the number of fish, f, that were in the pond last year. Which equation correctly represents the relationship between the current and previous number of fish?
Answer: 3f - 12 = 60 The phrase 'three times the number of fish, f' translates to $3f$ . The phrase '12 less than' means you subtract 12 from the previous quantity. So, '12 less than three times the number of fish' is $3f - 12$ . This amount is equal to 60. Therefore, the correct equation is $3f - 12 = 60$ .
A math puzzle says: 'I am thinking of a number, n. If you divide it by 4 and then add 7, the result is 15.' Which equation matches this puzzle?
Answer: n/4 + 7 = 15 The puzzle describes a sequence of operations. First, 'divide it by 4' gives $n/4$ . Second, 'then add 7' means you add 7 to that result: $n/4 + 7$ . The final result 'is 15', so the correct equation is $n/4 + 7 = 15$ . Choice A is incorrect because it adds before dividing.
Leo bought 4 notebooks that each cost the same price and a single pen for $2. The total cost for all the items before tax was $14. If n is the price of one notebook, which equation represents the total cost of his purchase?
Answer: 4n + 2 = 14 The cost of 4 notebooks at price n each is $4 \times n$ , or 4n. He also bought a pen for $2. The total cost is the sum of the cost of the notebooks and the cost of the pen, which is $4n + 2$ . This total is equal to $14. The correct equation is $4n + 2 = 14$ .
Five less than the product of a number x and three is the same as the number increased by nine. Which equation correctly represents this complex statement?
Answer: 3x - 5 = x + 9 Let's break down the statement. 'The product of a number x and three' is 3x. 'Five less than' this product means we subtract 5 from it, giving $3x - 5$ . 'The number increased by nine' is $x + 9$ . 'Is the same as' means these two expressions are equal. Therefore, the equation is $3x - 5 = x + 9$ .
A class of 25 students is going on a trip that costs a total of $500. The students held a bake sale and raised $50, which will be used to reduce the total cost of the trip. The remaining cost will be split equally among the students. If c represents the new, lower cost for each student, which equation shows this situation?
Answer: 25c = 500 - 50 The total cost of the trip is $500. The fundraiser reduces this cost by $50, so the new total cost is $500 - 50$ . This new total is split equally among 25 students, with each paying c. Therefore, the total amount paid by the students is $25 \times c$ , or 25c. The correct equation is $25c = 500 - 50$ .
A teacher starts with a supply of p pencils. After giving 3 pencils to each of her 20 students, she has 8 pencils remaining. Which equation correctly represents this situation?
Answer: p - (20 * 3) = 8 The initial number of pencils is p. The number of pencils given away is the number of students (20) times the number of pencils each received (3), which is $20 \times 3$ . The number remaining (8) is the starting amount minus the amount given away. Therefore, the equation is $p - (20 \times 3) = 8$ .
Which of the following situations can be represented by the equation $50 - 4n = 10$ ?
Answer: A girl had $50. She bought 4 identical shirts for n dollars each and received $10 in change. The equation $50 - 4n = 10$ represents starting with a quantity of 50, subtracting a group of 4 equal items of size n, and having a result of 10. Story C describes paying with $50, where the cost of 4 shirts at n dollars each (4n) is subtracted from the 50, resulting in $10 change. This matches the equation.
At a book fair, paperback books cost $4 each and hardcover books cost $7 each. A school bought h hardcover books. They bought twice as many paperback books as hardcover books. The total cost for all the books was $225. Which equation represents the school's purchase?
Answer: 7h + 4(2h) = 225 The number of hardcover books is h, and their cost is $7h$ . The number of paperback books is 'twice as many', which is 2h. The cost of the paperback books is $4(2h)$ . The total cost is the sum of the cost of hardcovers and paperbacks. So, the equation is $7h + 4(2h) = 225$ .
A pizza was cut into 12 slices. Mark ate m slices and his friend ate twice as many slices as Mark. After they both finished eating, there were 3 slices left. Which equation shows this situation?
Answer: 12 - m - 2m = 3 The starting number of slices is 12. The number of slices eaten must be subtracted from this total. Mark ate m slices. His friend ate 'twice as many', which is 2m slices. The total number of slices eaten is $m + 2m$ . The remaining slices are found by subtracting the eaten slices from the start: $12 - m - 2m$ . This is equal to 3. So, the equation is $12 - m - 2m = 3$ .
Maria scored 85, 90, and x on three science tests. Her average score for the three tests was exactly 88. Which equation represents this situation?
Answer: (85 + 90 + x) / 3 = 88 To find the average of a set of numbers, you add the numbers together and then divide by the count of the numbers. In this case, the sum of Maria's three test scores is $85 + 90 + x$ . To find the average, this sum is divided by 3. The problem states this average is 88. Therefore, the correct equation is $(85 + 90 + x) / 3 = 88$ .
A baker makes a large batch of c cookies. He packs them into boxes of 12. After filling 10 full boxes, he has 5 cookies left over. Which equation correctly represents the total number of cookies, c, that he made?
Answer: c = (12 * 10) + 5 The total number of cookies is the number of cookies packed in boxes plus the number of cookies left over. The number of cookies in the boxes is the number of boxes (10) multiplied by the number of cookies per box (12), which is $12 \times 10$ . There are 5 cookies left over, which are added to this total. Therefore, the total number of cookies c is given by the equation $c = (12 \times 10) + 5$ .
Jason and Ken are collecting cans for a recycling drive. Ken collected 10 fewer than three times the number of cans Jason collected. Together, they collected a total of 90 cans. If j represents the number of cans Jason collected, which equation represents their combined total?
Answer: j + (3j - 10) = 90 Jason collected j cans. 'Three times the number of cans Jason collected' is 3j. '10 fewer than' that amount is $3j - 10$ . This is the number of cans Ken collected. 'Together, they collected a total of 90' means you add Jason's cans and Ken's cans to get 90. So, the equation is $j + (3j - 10) = 90$ .
A group of 4 friends buys a gift that costs $48. One friend, Amy, contributes $12 toward the cost. The remaining cost is split equally among all 4 friends. If p represents the equal amount that each of the 4 friends pays in addition to Amy's initial contribution, which equation shows this situation?
Answer: 4p + 12 = 48 The total cost of the gift is $48. The total money collected must equal this amount. Amy contributes $12. Then, each of the 4 friends contributes an additional p dollars. The total from the four friends is $4 \times p$ , or 4p. The sum of Amy's contribution and the friends' contribution is $12 + 4p$ . This total must equal the gift's cost, so $12 + 4p = 48$ .
A rectangular garden has a perimeter of 50 feet. The length of the garden is 5 feet longer than its width. If w represents the width in feet, which equation can be used to find the dimensions of the garden?
Answer: 2w + 2(w + 5) = 50 The width is w. The length is '5 feet longer than its width', which is $w + 5$ . The formula for the perimeter of a rectangle is $2 \times \text{length} + 2 \times \text{width}$ . Substituting the expressions for length and width gives $2(w + 5) + 2w$ . The total perimeter is 50, so the equation is $2w + 2(w + 5) = 50$ .
Which of the following stories can be represented by the equation $3x + 5 = 23$ ?
Answer: Sam bought 3 movie tickets for x dollars each. He also spent $5 on popcorn. His total spending was $23. Let's analyze the equation $3x + 5 = 23$ . It shows a quantity x being multiplied by 3, then 5 being added to the result, with a final total of 23. Story C matches this perfectly: the cost of 3 tickets at x dollars each is 3x, and adding the $5 popcorn cost gives a total of $3x + 5 = 23$ .
A father is 25 years older than his son. In 5 years, the sum of their ages will be 55. If s is the son's current age, which equation represents the sum of their ages in 5 years?
Answer: (s + 5) + (s + 25 + 5) = 55 The son's current age is s. The father's current age is $s + 25$ . In 5 years, the son's age will be $s + 5$ . In 5 years, the father's age will be his current age plus 5, which is $(s + 25) + 5$ . The sum of their ages in 5 years is 55. So, the equation is the sum of their future ages: $(s + 5) + (s + 25 + 5) = 55$ .
Liam starts with $20 in his savings account. Each week for several weeks, he adds exactly $5 to the account. Which equation could be used to find w, the number of weeks it will take for him to have a total of $75 in the account?
Answer: 20 + 5w = 75 Liam begins with a starting amount of $20. He then adds a recurring amount each week. The money added after w weeks is $5 \times w$ , or 5w. The total amount in the account is the starting amount plus the weekly savings, which is $20 + 5w$ . This total equals $75, so the correct equation is $20 + 5w = 75$ .

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