Situations to Expressions - ISEE Middle Level Quantitative Reasoning

On a science test, a student earns 5 points for each correct answer, loses 2 points for each incorrect answer, and gets 0 points for unanswered questions. The test has 40 questions in total. If a student answers c questions correctly and i questions incorrectly, which expression represents the student's score?
Answer: $5c - 2i$ The student's score is based on the points gained from correct answers minus the points lost from incorrect answers. The points gained are 5c. The points lost are 2i. The total score is $5c - 2i$ . The total number of questions (40) is extra information not needed to write the expression for the score, as the number of unanswered questions does not affect the score.
A rectangular garden has a length that is 5 feet longer than twice its width. If the width is represented by w feet, which expression represents the perimeter of the garden?
Answer: $6w + 10$ First, express the length in terms of the width w. "Twice its width" is 2w. "5 feet longer than that" is 2w + 5. The perimeter of a rectangle is 2(length + width). Substituting the expressions gives $2((2w + 5) + w)$ . Simplifying inside the parentheses gives $2(3w + 5)$ . Distributing the 2 gives $6w + 10$ .
A bouncing ball is dropped from a height of H feet. On each bounce, it returns to a height that is r times its previous height. Which expression represents the height of the ball after its third bounce?
Answer: $Hr^3$ We need to track the height after each bounce. The starting height is H. After the first bounce, the height is $H \times r = Hr$ . After the second bounce, the ball reaches a height of r times the previous height, so $(Hr) \times r = Hr^2$ . After the third bounce, the height is again r times the previous height, so $(Hr^2) \times r = Hr^3$ .
Maria is y years old. Her brother, Sam, is 4 years younger. Their father is 1 year more than twice the sum of Maria's and Sam's ages. Which expression represents the father's age?
Answer: $4y - 7$ First, find the ages of Maria and Sam: Maria is y, Sam is y - 4. The sum of their ages is $y + (y - 4) = 2y - 4$ . Twice the sum of their ages is $2(2y - 4) = 4y - 8$ . The father is 1 year more than this, so his age is $(4y - 8) + 1 = 4y - 7$ .
A car is rented for d days at a rate of $35 per day. This rate includes 100 free miles per day. For each mile driven over the total number of free miles, there is a charge of $0.20. If the total miles driven is m, and m is greater than 100d, which expression shows the total cost?
Answer: $35d + 0.20(m - 100d)$ The cost has two parts: the daily fee and the mileage charge. The total daily fee is $35d$ . The total number of free miles is $100d$ . The number of miles charged is the total miles minus the free miles, which is $m - 100d$ . The cost for these miles is $0.20(m - 100d)$ . The total cost is the sum of these two parts: $35d + 0.20(m - 100d)$ .
The original price of a jacket is p dollars. It is on sale for 30% off. A sales tax of 7% is then applied to the discounted price. Which expression represents the final cost of the jacket?
Answer: $1.07(0.70p)$ First, calculate the discounted price. A 30% discount means the price is 70% of the original, so the discounted price is 0.70p. Next, calculate the 7% sales tax on this new price. An increase of 7% is equivalent to multiplying by 1.07. So, the final cost is $1.07 \times (0.70p)$ . Choice D is equivalent, but the order in A better reflects the sequence of operations described.
A fundraiser sells $t$ tickets for $7 each, then pays $12$ for posters. How would you express the total earnings in this scenario?
Answer: $7t-12$ This question tests middle school quantitative reasoning skills by translating situations into algebraic expressions. The core concept involves understanding how real-world quantitative data can be represented using variables and operations in mathematics. In this specific stimulus, a fundraiser sells t tickets at $7 each, generating revenue, then pays $12 for posters, which is an expense to subtract. Choice C is correct because it accurately models the situation by calculating the ticket revenue (7t) and then subtracting the poster cost ($12), giving 7t-12. Choice A is incorrect because it adds 12 instead of subtracting it, which would incorrectly increase rather than decrease the earnings by the poster expense. To help students, encourage practice in identifying key quantitative relationships and expressing them algebraically. Teach them to distinguish between revenue (money coming in) and expenses (money going out), ensuring expenses are subtracted from revenue.
A museum sells tickets for a special exhibit. The first ticket costs $25. Each additional ticket purchased in the same transaction costs $18. If a group buys t tickets, where t > 1, which expression represents the total cost for the group?
Answer: $25 + 18(t - 1)$ The total cost is the price of the first ticket plus the price of the additional tickets. The first ticket is $25. If there are t tickets in total, there are $t - 1$ additional tickets. The cost of these additional tickets is $18(t - 1)$ . So, the total cost is $25 + 18(t - 1)$ .
A family drives $h$ hours at 55 miles per hour. Which expression correctly models the travel distance?
Answer: $55h$ This question tests middle school quantitative reasoning skills by translating situations into algebraic expressions. The core concept involves understanding how real-world quantitative data can be represented using variables and operations in mathematics. In this specific stimulus, a family drives for h hours at a constant speed of 55 miles per hour, requiring us to calculate the total distance traveled. Choice C is correct because it accurately models the situation using the distance formula (distance = rate × time), multiplying the hours driven (h) by the speed (55 mph), giving 55h. Choice A is incorrect because it adds hours to miles per hour, which combines incompatible units and doesn't represent the multiplicative relationship between time and rate. To help students, encourage practice in identifying key quantitative relationships and expressing them algebraically. Teach them to recognize rate problems and apply the fundamental relationship that distance equals rate times time.
A family phone plan costs $90 per month for the first two lines. For each additional line, the cost is $15. The family uses n lines, where n is greater than 2. Which expression represents the total monthly cost for the lines?
Answer: $90 + 15(n - 2)$ The total cost is the sum of the base cost for the first two lines and the cost for the additional lines. The base cost is a flat $90. The number of additional lines is n - 2. The cost for these additional lines is $15(n - 2)$ . Therefore, the total monthly cost is $90 + 15(n - 2)$ .
Let k be the smallest of three consecutive even integers. Which expression represents the sum of these three integers?
Answer: $3k + 6$ If k is the smallest of three consecutive even integers, the next even integer is k + 2, and the one after that is k + 4. The sum of these three integers is $k + (k + 2) + (k + 4)$ . Combining like terms gives $3k + 6$ .
Mia has $m dollars and buys 3 notebooks for $2 each. How can the remaining balance be expressed algebraically?
Answer: $m-(3\cdot2)$ This question tests middle school quantitative reasoning skills by translating situations into algebraic expressions. The core concept involves understanding how real-world quantitative data can be represented using variables and operations in mathematics. In this specific stimulus, Mia starts with m dollars and purchases 3 notebooks at $2 each, requiring us to calculate her remaining balance. Choice B is correct because it accurately models the situation by subtracting the total cost of notebooks (3×2 = $6) from her initial amount m, giving m-(3·2). Choice A is incorrect because it adds instead of subtracts the cost, which would increase rather than decrease Mia's money. To help students, encourage practice in identifying key quantitative relationships and expressing them algebraically. Teach them to analyze scenarios for relevant details and operations, and watch for common pitfalls such as using addition when subtraction is needed for spending scenarios.
For a party, you buy $c$ cups at $1 each and 2 pizzas at $8 each. What expression represents the total cost of items?
Answer: $c+2\cdot8$ This question tests middle school quantitative reasoning skills by translating situations into algebraic expressions. The core concept involves understanding how real-world quantitative data can be represented using variables and operations in mathematics. In this specific stimulus, you buy c cups at $1 each and 2 pizzas at $8 each, requiring us to express the total cost. Choice C is correct because it accurately models the situation by calculating c×1 for the cups (which simplifies to c) and 2×8 for the pizzas, giving c+2·8 or c+16. Choice D is incorrect because it groups (c+2) and multiplies by 8, which would mean buying (c+2) items at $8 each, rather than c cups at $1 each plus 2 pizzas at $8 each. To help students, encourage practice in identifying key quantitative relationships and expressing them algebraically. Teach them to calculate costs for different item types separately before combining them, and to be careful about order of operations.
A salesperson earns a monthly salary of $2,500 plus a 6% commission on all sales. For sales exceeding $40,000 in a month, an additional bonus of 3% is paid on the portion of sales above $40,000. If total monthly sales are S dollars, and S > 40,000, which expression represents the total monthly earnings?
Answer: $2,500 + 0.06S + 0.03(S - 40,000)$ Total earnings are the sum of the base salary, the regular commission, and the bonus commission. The base salary is $2,500. The regular commission is 6% of total sales, or $0.06S$ . The bonus is 3% on the amount of sales over $40,000, which is represented by $S - 40,000$ . So the bonus is $0.03(S - 40,000)$ . Adding all parts gives $2,500 + 0.06S + 0.03(S - 40,000)$ .
A class sells $s$ bracelets for $4 each at a fundraiser. How would you express the total earnings in this scenario?
Answer: $4s$ This question tests middle school quantitative reasoning skills by translating situations into algebraic expressions. The core concept involves understanding how real-world quantitative data can be represented using variables and operations in mathematics. In this specific stimulus, the class sells s bracelets at $4 each, requiring us to find the total earnings from this fundraiser. Choice C is correct because it accurately models the situation by multiplying the number of bracelets sold (s) by the price per bracelet ($4), giving 4s. Choice A is incorrect because it adds 4 to s instead of multiplying, which would only increase the count by 4 rather than calculating total revenue. To help students, encourage practice in identifying key quantitative relationships and expressing them algebraically. Teach them to recognize when multiplication is needed (for repeated addition or rate problems) versus when simple addition or subtraction applies.
A teacher buys $p$ packs of pencils at $6 each, plus $5 shipping. What expression represents the total cost of items?
Answer: $6p+5$ This question tests middle school quantitative reasoning skills by translating situations into algebraic expressions. The core concept involves understanding how real-world quantitative data can be represented using variables and operations in mathematics. In this specific stimulus, a teacher purchases p packs of pencils at $6 each, plus an additional $5 shipping fee, requiring us to express the total cost. Choice B is correct because it accurately models the situation by multiplying the number of packs (p) by the cost per pack ($6) and then adding the shipping fee ($5), giving 6p+5. Choice A is incorrect because it groups (p+5) and multiplies by 6, which would mean adding 5 to the number of packs before calculating cost, rather than adding shipping after. To help students, encourage practice in identifying key quantitative relationships and expressing them algebraically. Teach them to carefully distinguish between costs that are per-item (requiring multiplication) and fixed costs (requiring addition).
A taxi company charges an initial fee of $3.00, plus $1.80 per mile traveled. There is also a surcharge of $0.50 for each passenger after the first one. Which expression represents the cost for a group of p people traveling m miles, where p > 1?
Answer: $3.00 + 1.80m + 0.50(p - 1)$ The total cost is composed of three parts: the initial fee, the mileage cost, and the passenger surcharge. The initial fee is $3.00. The mileage cost is (1.80m). The surcharge applies to passengers after the first, so there are $p-1$ such passengers. The surcharge cost is $0.50(p-1)$ . The total cost is $3.00 + 1.80m + 0.50(p - 1)$ .
A student has $b$ dollars and buys 5 pens for $p$ dollars each. How can the remaining balance be expressed algebraically?
Answer: $b-(5p)$ This question tests middle school quantitative reasoning skills by translating situations into algebraic expressions. The core concept involves understanding how real-world quantitative data can be represented using variables and operations in mathematics. In this specific stimulus, a student starts with b dollars and buys 5 pens at p dollars each, requiring us to express the remaining balance. Choice B is correct because it accurately models the situation by calculating the total cost of pens (5×p = 5p) and subtracting this from the initial amount b, giving b-(5p) or b-5p. Choice A is incorrect because it factors out 5 from (b-p), which would mean multiplying 5 by the difference between b and p, rather than subtracting the cost of 5 pens from b. To help students, encourage practice in identifying key quantitative relationships and expressing them algebraically. Teach them to calculate the total cost of multiple items first (quantity × price) before subtracting from the initial amount.
A baker sells cupcakes for $3 each and cookies for $2 each. The cost to make each cupcake is c dollars, and the cost to make each cookie is k dollars. On a given day, the baker sells 70 cupcakes and 120 cookies. Which expression represents the baker's total profit for that day?
Answer: $70(3-c) + 120(2-k)$ Profit is calculated as (Revenue - Cost). A more direct way is to find the profit per item and multiply by the number of items sold. The profit on one cupcake is $(3-c)$ . The profit on one cookie is $(2-k)$ . For 70 cupcakes, the profit is $70(3-c)$ . For 120 cookies, the profit is $120(2-k)$ . The total profit is the sum of the profit from cupcakes and cookies: $70(3-c) + 120(2-k)$ .
On a road trip, you drive 120 miles, then $d$ more miles to the hotel. Which expression correctly models the travel distance?
Answer: $120+d$ This question tests middle school quantitative reasoning skills by translating situations into algebraic expressions. The core concept involves understanding how real-world quantitative data can be represented using variables and operations in mathematics. In this specific stimulus, you drive 120 miles initially, then drive d additional miles to reach the hotel, requiring us to find the total distance traveled. Choice D is correct because it accurately models the situation by adding the initial distance (120 miles) to the additional distance (d miles), giving 120+d. Choice A is incorrect because it multiplies 120 by d, which would mean driving 120 miles d times rather than adding d miles to the initial 120 miles. To help students, encourage practice in identifying key quantitative relationships and expressing them algebraically. Teach them to recognize when distances or quantities should be added together (cumulative total) versus when multiplication is appropriate (repeated groups or rates).

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