Word Problems to Expressions - ISEE Upper Level Quantitative Reasoning

1. Tom has $n$ nickels and $q$ quarters. He spends half of his quarters and receives $3$ more nickels. Which expression represents the total value of his coins in cents after these transactions?
   Answer: $5(n + 3) + 25(\frac{q}{2})$ When you encounter word problems involving coins and transactions, break them down step by step: identify the initial amounts, track what changes, then calculate the final value using each coin's worth. Tom starts with $n$ nickels and $q$ quarters. After the transactions, he has: - Nickels: $n + 3$ (he receives 3 more) - Quarters: $\frac{q}{2}$ (he spends half, so half remain) To find the total value in cents, multiply each coin type by its value: nickels are worth 5 cents each, quarters are worth 25 cents each. This gives us: $5(n + 3) + 25\left(\frac{q}{2}\right)$ Choice A correctly represents this calculation. Choice B ($5n + 25q + 15$) incorrectly keeps all the original quarters ($25q$) instead of just half, and adds 15 cents as if it's separate value rather than the value of 3 additional nickels. Choice C ($5n + 3 + 25q$) makes two errors: it adds 3 cents instead of the value of 3 nickels (15 cents), and again keeps all original quarters. Choice D ($5n + 15 + \frac{25q}{2}$) correctly handles the quarters and adds 15 for the extra nickels, but treats the original nickels and new nickels separately instead of combining them as $5(n + 3)$. Strategy tip: In coin problems, always track quantities first, then convert to values by multiplying by each coin's worth. Don't mix quantities and values in your expressions.
2. A rectangle has length $3y + 2$ and width $y - 1$. If the area of the rectangle is equal to the area of a square with side length $s$, which equation relates $y$ and $s$?
   Answer: $(3y + 2)(y - 1) = s^2$ When you see a problem comparing areas of different shapes, focus on setting up the correct area formulas for each shape involved. The rectangle has length $3y + 2$ and width $y - 1$, so its area is length × width = $(3y + 2)(y - 1)$. The square has side length $s$, so its area is $s^2$. Since the problem states these areas are equal, you need the equation $(3y + 2)(y - 1) = s^2$, which is choice A. Let's examine why the other choices are incorrect. Choice B, $3y + 2 + y - 1 = 4s$, adds the length and width of the rectangle (giving you the semi-perimeter) and sets it equal to the perimeter of the square. This confuses area with perimeter completely. Choice C, $(3y + 2) + (y - 1) = s^2$, makes a similar error by adding the rectangle's dimensions but then setting this sum equal to the square's area—mixing perimeter and area concepts. Choice D, $3y^2 + y - 2 = s$, appears to attempt expanding $(3y + 2)(y - 1)$ but makes algebraic errors and sets the result equal to $s$ instead of $s^2$. Remember that area problems require you to match the correct area formulas. For rectangles, always use length × width, and for squares, use side². Don't let the algebraic expressions distract you from applying the fundamental geometric formulas correctly.
3. A gym membership costs $40 per month plus $5 for each group fitness class attended. If someone attends $c$ classes per month for $m$ months and receives a one-time discount of $25, which expression represents their total cost?
   Answer: $m(40 + 5c) - 25$ When you encounter word problems involving costs over time, you need to carefully identify what happens each month versus what happens only once, then build your expression step by step. Let's break down the costs: Each month, someone pays $40 for membership plus $5 per class. If they attend $c$ classes in a month, that's $40 + 5c$ per month. Over $m$ months, this monthly cost gets multiplied by $m$, giving us $m(40 + 5c)$. Finally, they receive a one-time $25 discount, so we subtract 25 from the total: $m(40 + 5c) - 25$. Choice A correctly captures this structure and is the right answer. Choice B ($40m + 5c - 25$) makes a critical error: it only accounts for $c$ classes total across all months, rather than $c$ classes per month. This would mean someone attends the same number of classes whether they're a member for 1 month or 12 months. Choice C ($40 + 5cm - 25$) incorrectly treats the $40 membership fee as a one-time cost rather than a monthly charge. This expression suggests you only pay the base membership fee once. Choice D ($(40 + 5c - 25)m$) applies the discount every month instead of just once. This would give someone $25 off each month, totaling $25m in discounts rather than the single $25 discount described. Remember: In multi-step cost problems, identify what repeats each period versus what happens only once, then carefully apply the time multiplier to only the repeating elements.
4. A rectangular garden bed is $3$ times as long as it is wide. If the width is increased by $2$ feet and the length is increased by $5$ feet, and the original width was $w$ feet, which expression represents the new area?
   Answer: $(w + 2)(3w + 5)$ When you encounter problems about changing dimensions of rectangles, the key is to carefully track how each dimension changes and then use the area formula $\text{Area} = \text{length} \times \text{width}$. Let's start with what we know. The original width is $w$ feet, and since the garden is 3 times as long as it is wide, the original length is $3w$ feet. After the changes, the new width becomes $w + 2$ feet and the new length becomes $3w + 5$ feet. Therefore, the new area is $(w + 2)(3w + 5)$. Looking at the wrong answers: Choice B gives $3w^2 + 10$, which incorrectly treats this as if you're just adding the increases to some version of the original area—this completely ignores how area actually works. Choice C uses $(w + 2)(w + 5)$, which makes the error of assuming the original length was $w$ instead of $3w$. This misses the crucial "3 times as long" relationship. Choice D multiplies the correct expression from choice C by 3, which suggests someone remembered the "3 times" relationship but applied it incorrectly to the entire new area rather than just to the length dimension. The correct answer is A: $(w + 2)(3w + 5)$. Study tip: In rectangle problems with changing dimensions, always identify each original dimension first, then carefully add the changes to each dimension separately, and finally multiply the new dimensions together. Don't try to shortcut by manipulating areas directly—work with the individual dimensions.
5. A company's profit in dollars can be modeled as $100n - n^2$, where $n$ is the number of items produced in hundreds. If they want to express this profit per item when producing $n$ hundred items, which expression represents the profit per item in dollars?
   Answer: $\frac{100 - n}{100}$ The profit is $100n - n^2$ dollars, but $n$ counts items in hundreds, so the company actually produces $100n$ items. Profit per item is the total profit divided by the number of items: $\frac{100n - n^2}{100n} = \frac{n(100 - n)}{100n} = \frac{100 - n}{100}$ dollars per item. The expression $100 - n$ divides the profit by $n$ only, forgetting that $n$ is measured in hundreds. The expression $\frac{100n - n^2}{100}$ divides by 100 instead of by the full item count $100n$. The expression $100 - \frac{n}{100}$ divides the two terms by different amounts: $100n$ by $n$, but $n^2$ by $100n$.
6. A taxi company charges a flat rate of $2.50 plus $0.75 for each mile traveled. If a customer travels $m$ miles and gives a tip equal to $15\%$ of the total fare (before tip), which expression represents the total amount the customer pays?
   Answer: $1.15(2.50 + 0.75m)$ When you encounter word problems involving percentages and multiple charges, break down the problem step by step to build the correct expression. First, identify what the customer pays: the base fare plus a 15% tip on that fare. The base fare consists of a $2.50 flat rate plus $0.75 per mile for $m$ miles, giving us $2.50 + 0.75m$. The tip equals 15% of this base fare, or $0.15(2.50 + 0.75m)$. The total amount paid is the base fare plus the tip: $(2.50 + 0.75m) + 0.15(2.50 + 0.75m)$. Factoring out the common term gives us $1.15(2.50 + 0.75m)$, which is choice A. Choice B adds a flat $0.15 to the base fare, treating the tip as a fixed amount rather than a percentage. Choice C calculates the tip as 15% of only the mileage charge ($0.15m$), ignoring the tip on the $2.50 flat rate. This would give a tip of $0.15(0.75m) = 0.1125m$, not $0.15m$. Choice D calculates 15% of only the flat rate and adds the mileage charge without any tip on the mileage portion. Remember that when a percentage applies to a total amount, you multiply the entire base amount by $(1 + \text{percentage rate})$. The factor 1.15 represents paying 100% of the original fare plus 15% tip. This pattern appears frequently in real-world problems involving taxes, tips, and markups.
7. Formulate an expression based on the described situation. A smoothie recipe requires $1.5$ cups of yogurt per serving and $0.5$ cups of fruit per serving, and you plan to make $x$ servings. You also add $2$ extra cups of ice total. What expression gives the total cups of ingredients used?
   Answer: $1.5x+0.5x+2$ Build the expression one ingredient at a time. Yogurt is 1.5 cups per serving, so $x$ servings use $1.5x$ cups. Fruit is 0.5 cups per serving, so it adds $0.5x$ cups. The ice is 2 cups in total, not per serving. The total is $1.5x + 0.5x + 2$, which equals $2x + 2$. The expression $2(x+2)$ equals $2x + 4$, which counts the ice twice by multiplying it along with the servings. The expression $1.5x + 0.5x - 2$ subtracts the ice instead of adding it. The expression $1.5(x + 0.5) + 2$ multiplies the fruit amount by the yogurt amount instead of by the number of servings. Tip: give each per-serving amount its own $x$, and add a one-time amount once.
8. Which algebraic expression represents the total distance? A cyclist rides for $x$ hours at $12$ miles per hour, then continues for $y$ hours at $9$ miles per hour, without stopping. Write an expression for the total miles traveled during the trip.
   Answer: $12x+9y$ This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes a cyclist traveling at two different speeds for different durations, requiring the distance formula (distance = rate × time) for each segment. Choice A is correct because it accurately represents the total distance as 12x + 9y, where 12x is the distance for the first segment and 9y is the distance for the second segment. Choice B incorrectly groups the hours before applying rates, Choice C subtracts instead of adds, and Choice D multiplies all values together. To help students: Remind them that distance equals rate times time for each segment separately. Practice identifying when to add versus multiply components in multi-part journey problems.
9. A triangle has sides of length $x$, $x+3$, and $2x-1$. Which expression represents the perimeter of the triangle?
   Answer: $4x + 2$ When you encounter a triangle with given side lengths, questions about perimeter are testing your understanding of what perimeter means and your ability to combine algebraic expressions. The perimeter of any polygon is simply the sum of all its side lengths. For this triangle with sides of length $x$, $x+3$, and $2x-1$, you need to add these three expressions together: Perimeter = $x + (x+3) + (2x-1)$ Combining like terms: $x + x + 2x + 3 - 1 = 4x + 2$ Let's examine why the other answers are incorrect. Choice B gives $3x + 3$, which would result from incorrectly combining the variable terms (getting $3x$ instead of $4x$) or making an arithmetic error with the constants. Choice C, $2x^2 + 2x - 1$, represents a fundamental misunderstanding—this looks like someone tried to multiply terms together or confused perimeter with area calculations. Choice D, $x(x+3)(2x-1)$, is clearly wrong as it shows the product of all three side lengths, which has no geometric meaning for a triangle and would yield a cubic expression. The correct answer is A: $4x + 2$. Remember that perimeter problems are always about addition, never multiplication. When you see algebraic expressions for side lengths, simply add them together and combine like terms. Don't let the algebra intimidate you—treat each expression as you would any number when finding perimeter.
10. A water tank is being filled at a rate of $5$ gallons per minute. The tank initially contains $20$ gallons and has a small leak that causes it to lose $0.5$ gallons per minute. Which expression represents the amount of water in the tank after $t$ minutes?
    Answer: $20 + 4.5t$ The water in the tank equals the starting amount plus the net rate times the time. The tank starts with 20 gallons. Water flows in at 5 gallons per minute and leaks out at 0.5 gallons per minute, so the net rate is $5 - 0.5 = 4.5$ gallons per minute. After $t$ minutes the tank holds $20 + 4.5t$ gallons. The expression $20t + 4.5$ swaps the roles of the numbers, treating the one-time 20 gallons as a rate and the 4.5 as a fixed amount. The expression $25t - 0.5t$ also multiplies the starting 20 gallons by $t$, adding it every minute. The expression $20 + 5.5t$ adds the leak instead of subtracting it. Tip: a starting amount is a constant term; only rates get multiplied by time.
11. How can the given quantities be expressed algebraically? A student saves $x$ dollars each week for $4$ weeks, then saves $y$ dollars each week for $3$ more weeks. No money is spent during this time. What expression represents total savings after seven weeks?
    Answer: $4x+3y$ This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes saving x dollars per week for 4 weeks (giving 4x total) and y dollars per week for 3 weeks (giving 3y total). Choice B is correct because it accurately represents the total savings as 4x + 3y, combining both saving periods. Choice A incorrectly adds x and y before multiplying by 7, Choice C reverses the coefficients, and Choice D incorrectly adds 7 as a separate term. To help students: Emphasize calculating each saving period separately before combining. Practice problems where different rates apply to different time periods to build pattern recognition.
12. A rectangular garden has length $3$ meters more than twice its width. If the width is $w$ meters, which expression represents the perimeter of the garden?
    Answer: $6w + 6$ When you encounter a word problem about perimeter, you need to first translate the given relationships into algebraic expressions, then apply the perimeter formula. Let's break down the information: the width is $w$ meters, and the length is "3 meters more than twice the width." This means length = $2w + 3$ meters. For a rectangle, perimeter equals 2 times length plus 2 times width: $P = 2l + 2w$. Substituting our expressions: $P = 2(2w + 3) + 2w = 4w + 6 + 2w = 6w + 6$. Looking at the wrong answers: Choice B ($4w + 6$) represents what you'd get if you forgot to double the width term—this comes from incorrectly calculating $2(2w + 3) + w$ instead of $2(2w + 3) + 2w$. Choice C ($6w + 3$) results from forgetting to double the constant term, calculating the perimeter as $2(2w + 3) + 2w$ but making an error with the 6. Choice D ($2w + 3$) is simply the expression for the length alone, not the perimeter—a common trap for students who might confuse what the question is asking for. The correct answer is A: $6w + 6$. Strategy tip: In perimeter problems, always write out the formula $P = 2l + 2w$ explicitly and substitute carefully. Double-check that you've doubled both the length and width terms, and verify your answer makes sense by testing with simple numbers.
13. Which algebraic expression represents the total distance? A runner completes $x$ laps of a $0.4$ mile track and then runs $y$ additional miles on a trail. No other distances are included in the workout. Write an expression for the total miles run.
    Answer: $0.4x+y$ This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes running x laps of a 0.4-mile track (giving 0.4x miles) and then y additional miles on a trail. Choice A is correct because it accurately represents the total distance as 0.4x + y, combining the track distance with the trail distance. Choice B incorrectly groups x and y before multiplying, Choice C subtracts the trail miles, and Choice D multiplies all terms together. To help students: Clarify that the y miles are already in miles, not laps. Practice problems where units differ between parts to avoid confusion about what needs conversion.
14. Translate the given scenario into an algebraic expression. A car travels $x$ hours on the highway at $65$ miles per hour and then $y$ hours in city traffic at $30$ miles per hour. Assume the speeds stay constant within each segment. What expression represents the total distance traveled?
    Answer: $65x+30y$ This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes a car traveling at 65 mph for x hours and 30 mph for y hours, requiring the distance formula for each segment. Choice A is correct because it accurately represents the total distance as 65x + 30y, where each term is the product of speed and time for that segment. Choice B incorrectly adds the times before multiplying, Choice C incorrectly groups variables, and Choice D subtracts the city distance. To help students: Reinforce that distance = rate × time must be calculated separately for each segment with different speeds. Practice multi-segment journey problems to build confidence with this common pattern.
15. What is the algebraic expression for the total cost? A student buys $x$ textbooks at $18$ dollars each and $y$ workbooks at $12$ dollars each, then pays a $7$ dollar shipping fee. No discounts apply to the order. What expression represents the total amount paid?
    Answer: $18x+12y+7$ This question tests ISEE Upper Level quantitative reasoning: translating word problems into algebraic expressions. Understanding how to derive expressions from verbal descriptions is a key skill in quantitative reasoning. In this scenario, the word problem describes buying x textbooks at $18 each, y workbooks at $12 each, and paying a $7 shipping fee. Choice B is correct because it accurately represents the total cost as 18x + 12y + 7, combining all purchase costs with the shipping fee. Choice A omits the shipping fee, Choice C incorrectly groups x and y, and Choice D subtracts everything from 7. To help students: Identify all cost components including fees that apply to the entire order. Practice distinguishing between per-item costs (that multiply) and fixed fees (that add once).
16. A phone plan charges $25 per month for unlimited talk and text, plus $10 per gigabyte of data used. If a customer uses $g$ gigabytes in a month and pays a $5 activation fee only in the first month, which expression represents their total cost for the first $n$ months?
    Answer: $n(25 + 10g) + 5$ Separate the charges that repeat every month from the charge that happens once. Each month costs the 25-dollar plan plus 10 dollars per gigabyte, or $25 + 10g$. Over $n$ months that is $n(25 + 10g)$. The 5-dollar activation fee is paid only once, so the total is $n(25 + 10g) + 5$. The expression $(25 + 10g + 5)n$ charges the activation fee every month. The expression $25n + 10g + 5$ charges the monthly plan $n$ times but the data only once, as if data were used in just one month. The expression $30n + 10g$ folds the one-time fee into every month and also charges data only once. Tip: ask of each charge "how many times is this paid?"; multiply only the repeating charges by $n$.
17. A farmer plants corn in a rectangular field that is $50$ feet longer than it is wide. He also plants soybeans in a square field whose side length equals the width of the corn field. If the width of the corn field is $w$ feet, which expression represents the total area planted?
    Answer: $2w^2 + 50w$ Find each field's area, then add. The corn field is a rectangle $w$ feet wide and $w + 50$ feet long, so its area is $w(w + 50) = w^2 + 50w$. The soybean field is a square with side $w$, so its area is $w^2$. The total is $w^2 + 50w + w^2 = 2w^2 + 50w$. The expression $2w^2 + 50$ forgets to multiply the extra 50 feet by the width, adding just 50 square feet. The expression $(w + 50)^2 + w^2$ treats the corn field as a square with side $w + 50$, which is too large. The expression $w^2 + 50w$ is the corn field alone and leaves out the soybean field. Tip: when you distribute, every term inside the parentheses gets multiplied; $w(w + 50)$ gives $50w$, not 50.
18. A car rental company charges $30 per day plus $0.20 per mile. A customer rents a car for $d$ days and drives $m$ miles, but receives a $10\%$ discount on the total bill. Which expression represents the amount the customer actually pays?
    Answer: $0.9(30d + 0.20m)$ Before the discount, the bill is the daily charge plus the mileage charge: $30d + 0.20m$. A 10% discount means the customer pays 100% − 10% = 90% of that whole bill, so the amount paid is $0.9(30d + 0.20m)$, which also equals $27d + 0.18m$. The expression $0.1(30d + 0.20m)$ is the amount of the discount itself, not what the customer pays. The expression $30d + 0.20m - 0.1$ subtracts a flat 0.1 dollar instead of 10% of the bill. The expression $27d + 0.20m$ takes 10% off the daily charge but forgets to take it off the mileage charge; the discount applies to the entire bill.
19. A school cafeteria sells lunch combinations. Each combination includes one main dish costing $\$m$, one side dish costing $\$s$, and a drink. Drinks cost $1.50 for small, $2.00 for medium, and $2.50 for large. If students buy $a$ combinations with small drinks, $b$ combinations with medium drinks, and $c$ combinations with large drinks, which expression represents the total revenue?
    Answer: $(a + b + c)(m + s) + 1.50a + 2.00b + 2.50c$ Every combination includes a main dish and a side dish, so all $a + b + c$ combinations bring in $m + s$ dollars each: $(a + b + c)(m + s)$. The drinks add 1.50 dollars for each of the $a$ small-drink combinations, 2.00 dollars for each of the $b$ medium ones and 2.50 dollars for each of the $c$ large ones: $1.50a + 2.00b + 2.50c$. The total revenue is $(a + b + c)(m + s) + 1.50a + 2.00b + 2.50c$. The expression $(m + s)(a + b + c) + 6.00(a + b + c)$ charges every combination for all three drinks. The expression $a(m + s) + b(m + s) + c(m + s) + 1.50 + 2.00 + 2.50$ adds each drink price only once instead of once per combination. The expression $(a + b + c)(m + s + 2.00)$ charges every combination the medium-drink price.
20. A student has test scores of $85$, $92$, and $x$. To have an average score of at least $90$, which inequality represents the constraint on $x$?
    Answer: $\frac{85 + 92 + x}{3} \geq 90$ When you encounter problems about averages and minimum requirements, you're dealing with inequality setup. The key is translating "at least" into mathematical language and remembering that average equals the sum divided by the number of values. To find the constraint on $x$, start with the average formula. The student has three test scores: 85, 92, and $x$. The average of these three scores is $\frac{85 + 92 + x}{3}$. Since this average must be "at least 90," you need the inequality $\frac{85 + 92 + x}{3} \geq 90$. This directly matches choice A. Choice B ($85 + 92 + x \geq 90$) makes a critical error—it sets up an inequality for the sum of the scores rather than the average. This would only require a total of 90 points across three tests, giving an average of just 30, which clearly doesn't match the problem. Choice C ($\frac{85 + 92 + x}{3} = 90$) uses the correct average formula but with an equals sign instead of "greater than or equal to." This would only work if the student wanted exactly a 90 average, not "at least 90." Choice D ($x \geq 90$) ignores the other test scores entirely. While getting 90 or higher on the third test might help, it's not the complete constraint—the student could score lower than 90 and still achieve the desired average. Remember: "at least" translates to ≥, and average problems require setting up the sum divided by count, not just the sum alone.

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