ISEE Upper Level Quantitative Reasoning Quiz: Word Problems To Expressions
20 questions · exam conditions
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Word Problems To ExpressionsQuestion 1 of 20

Maria earns dd dollars per hour for regular work and 1.5d1.5d dollars per hour for overtime work. Last week she worked rr regular hours and tt overtime hours. Which expression represents her total earnings for the week?

dr+1.5dtdr + 1.5dt
d(r+1.5t)d(r + 1.5t)
1.5d(r+t)1.5d(r + t)
d(r+t)+1.5d(r + t) + 1.5
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ISEE Upper Level Quantitative Reasoning Quiz

ISEE Upper Level Quantitative Reasoning Quiz: Word Problems To Expressions

Practice Word Problems To Expressions in ISEE Upper Level Quantitative Reasoning with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Word Problems To Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level Quantitative Reasoning.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Maria earns dd dollars per hour for regular work and 1.5d1.5d dollars per hour for overtime work. Last week she worked rr regular hours and tt overtime hours. Which expression represents her total earnings for the week?

  1. dr+1.5dtdr + 1.5dt
  2. d(r+1.5t)d(r + 1.5t) (correct answer)
  3. 1.5d(r+t)1.5d(r + t)
  4. d(r+t)+1.5d(r + t) + 1.5
Explanation: When you encounter word problems involving multiple rates of pay, the key is to calculate earnings for each type of work separately, then add them together. Maria earns two different hourly rates: dd dollars for regular hours and 1.5d1.5d dollars for overtime hours. To find her total weekly earnings, you need to multiply each rate by the corresponding hours worked. For regular work: dd dollars/hour × rr hours = drdr dollars For overtime work: 1.5d1.5d dollars/hour × tt hours = 1.5dt1.5dt dollars Total earnings = dr+1.5dtdr + 1.5dt You can factor out the common factor dd from both terms: d(r+1.5t)d(r + 1.5t), which is answer choice B. Let's examine why the other options are incorrect: Choice A gives dr+1.5dtdr + 1.5dt, which is mathematically equivalent to the correct answer but isn't factored. While this represents the same value, it's not among the answer choices in this exact form. Choice C, 1.5d(r+t)1.5d(r + t), incorrectly applies the overtime rate to both regular and overtime hours. This would mean Maria earns 1.5d1.5d for every hour worked, regardless of type. Choice D, d(r+t)+1.5d(r + t) + 1.5, adds a flat 1.51.5 dollars rather than applying the overtime multiplier to the overtime hours. This completely misrepresents how overtime pay works. Remember: when dealing with different rates for different categories of work, always multiply each rate by its corresponding quantity, then combine the results. Factoring can help you match the answer format, but the underlying logic remains the same.

Question 2

A company produces widgets at a cost of \cperwidget.Theyselleachwidgetforper widget. They sell each widget for$c + $8.Iftheyproduceandsell. If they produce and sell n$ widgets but must pay $50 in fixed costs and $2 per widget in packaging, which expression represents their profit?

  1. n(c+8)−nc−2n−50n(c + 8) - nc - 2n - 50
  2. 8n−2n−508n - 2n - 50
  3. 6n−506n - 50
  4. n(c+8−c−2)−50n(c + 8 - c - 2) - 50 (correct answer)
Explanation: When you encounter profit problems, remember that profit equals total revenue minus total costs. You need to carefully identify all sources of revenue and all types of costs. Let's break down this problem systematically. The company's revenue is the selling price times the number of widgets sold: n(c+8)n(c + 8). Their total costs include production costs (ncnc), packaging costs (2n2n), and fixed costs (5050). So profit = n(c+8)−nc−2n−50n(c + 8) - nc - 2n - 50. Now let's simplify this expression. We can factor out nn from the variable terms: n(c+8−c−2)−50n(c + 8 - c - 2) - 50. Notice that the cc terms cancel out, leaving us with n(6)−50n(6) - 50 or 6n−506n - 50. This shows us that answer choice D is correct because it represents the unsimplified but mathematically equivalent form. Answer choice A shows the profit calculation in its expanded form, which is mathematically correct but not the answer format requested. Answer choice B (8n−2n−508n - 2n - 50) incorrectly omits the production costs entirely, only accounting for the markup and packaging. Answer choice C (6n−506n - 50) is the simplified version of the correct answer, but since D is also listed and matches our derived expression exactly, D is the most precise choice. Remember: in profit problems, always account for every cost component mentioned in the problem. Production costs, variable costs like packaging, and fixed costs all reduce your profit from gross revenue.

Question 3

A parking garage charges $3 for the first hour and $2 for each additional hour or part thereof. If someone parks for hh hours where h>1h > 1, which expression represents the total cost in dollars?

  1. 3+2h3 + 2h
  2. 3+2(h−1)3 + 2(h-1) (correct answer)
  3. 2h+12h + 1
  4. 5h−25h - 2
Explanation: When you encounter parking fee problems, you're dealing with piecewise pricing structures where different rates apply to different time periods. The key is to break down the cost components separately. Let's analyze the parking structure: $3 for the first hour, then $2 for each additional hour. If someone parks for $hhourswherehours whereh > 1$, we need to account for both the initial hour and the remaining time. The total cost breaks down as: First hour cost + Additional hours cost = $3 + $2 × (number of additional hours). Since the person parks for $htotalhours,thenumberofadditionalhoursistotal hours, the number of additional hours is(h-1).Therefore,theexpressionbecomes. Therefore, the expression becomes 3 + 2(h-1)$, which is choice B. Let's examine why the other options fail. Choice A, 3+2h3 + 2h, incorrectly charges $2 for all $hhoursinsteadofjusttheadditionalhours instead of just the additional(h-1)hours.Thisdouble−countsthefirsthour.ChoiceC,hours. This double-counts the first hour. Choice C,2h + 1,completelymisrepresentsthefeestructurebycharging$2perhourforallhoursandaddingonly$1extra.ChoiceD,, completely misrepresents the fee structure by charging $2 per hour for all hours and adding only $1 extra. Choice D, 5h - 2$, suggests a rate of $5 per hour with a $2 discount, which doesn't match the given pricing at all. You can verify choice B works by testing it: for $h = 3$ hours, the cost should be $3 + 2(2) = $7, which matches paying $3 for the first hour plus $4 for two additional hours. Strategy tip: In piecewise rate problems, always identify what happens in each "piece" of time, then build your expression by adding the costs for each piece separately.

Question 4

A store offers a discount where customers pay full price for the first item and get 25%25\% off each additional item of the same type. If a customer buys nn items at \peach(whereeach (wheren \geq 2$), which expression represents the total cost?

  1. p+(n−1)⋅0.75pp + (n-1) \cdot 0.75p (correct answer)
  2. 0.75np0.75np
  3. p+(n−1)⋅0.25pp + (n-1) \cdot 0.25p
  4. np−0.25pnp - 0.25p
Explanation: When you encounter discount problems, break down the pricing structure systematically. This question tests your ability to translate a tiered discount into mathematical expressions. Let's think through what happens: You pay full price (pp) for the first item, then get 25% off each additional item. If there are nn items total, then (n−1)(n-1) items get the discount. Each discounted item costs 75% of the original price, or 0.75p0.75p. So the total cost is: full price for one item + discounted price for the remaining items = p+(n−1)⋅0.75pp + (n-1) \cdot 0.75p. This matches choice A. Choice B (0.75np0.75np) incorrectly applies the 25% discount to all nn items, including the first one that should be full price. This would only be correct if every item were discounted. Choice C (p+(n−1)⋅0.25pp + (n-1) \cdot 0.25p) makes a critical error: it adds the discount amount rather than the discounted price. The expression 0.25p0.25p represents how much you save on each additional item, not how much you pay. Choice D (np−0.25pnp - 0.25p) assumes you pay full price for all items then subtract 25% off just one item. This ignores that the discount applies to (n−1)(n-1) items, not just one. Study tip: In multi-step discount problems, always identify which items get which treatment, then build your expression piece by piece. Remember that "25% off" means you pay 75% of the original price, not 25%.

Question 5

A gym membership costs $40 per month plus $5 for each group fitness class attended. If someone attends cc classes per month for mm months and receives a one-time discount of $25, which expression represents their total cost?

  1. m(40+5c)−25m(40 + 5c) - 25 (correct answer)
  2. 40m+5c−2540m + 5c - 25
  3. 40+5cm−2540 + 5cm - 25
  4. (40+5c−25)m(40 + 5c - 25)m
Explanation: 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 $cclassesinamonth,that′sclasses in a month, that's40 + 5cpermonth.Overper month. Overmmonths,thismonthlycostgetsmultipliedbymonths, this monthly cost gets multiplied bym,givingus, giving us m(40 + 5c).Finally,theyreceiveaone−time$25discount,sowesubtract25fromthetotal:$m(40+5c)−25. 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−2540m + 5c - 25) makes a critical error: it only accounts for cc classes total across all months, rather than cc 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−2540 + 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.

Question 6

A rectangular garden bed is 33 times as long as it is wide. If the width is increased by 22 feet and the length is increased by 55 feet, and the original width was ww feet, which expression represents the new area?

  1. (w+2)(3w+5)(w + 2)(3w + 5) (correct answer)
  2. 3w2+103w^2 + 10
  3. (w+2)(w+5)(w + 2)(w + 5)
  4. 3(w+2)(w+5)3(w + 2)(w + 5)
Explanation: 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 Area=length×width\text{Area} = \text{length} \times \text{width}. Let's start with what we know. The original width is ww feet, and since the garden is 3 times as long as it is wide, the original length is 3w3w feet. After the changes, the new width becomes w+2w + 2 feet and the new length becomes 3w+53w + 5 feet. Therefore, the new area is (w+2)(3w+5)(w + 2)(3w + 5). Looking at the wrong answers: Choice B gives 3w2+103w^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)(w + 2)(w + 5), which makes the error of assuming the original length was ww instead of 3w3w. 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)(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.

Question 7

A company's profit in dollars can be modeled as 100n−n2100n - n^2, where nn is the number of items produced in hundreds. If they want to express this profit per item when producing nn hundred items, which expression represents the profit per item in dollars?

  1. 100−n100 - n
  2. 100n−n2100n\frac{100n - n^2}{100n}
  3. 100−n100\frac{100 - n}{100} (correct answer)
  4. 100−n100100 - \frac{n}{100}
Explanation: When you encounter profit-per-item questions, you need to set up the relationship between total profit and the number of items carefully. Here, the total profit is 100n−n2100n - n^2 dollars when producing nn hundred items, which means the actual number of items is 100n100n (since nn is in hundreds). To find profit per item, you divide total profit by the total number of items: 100n−n2100n\frac{100n - n^2}{100n}. Now you can simplify this expression by factoring the numerator: 100n−n2=n(100−n)100n - n^2 = n(100 - n). So the expression becomes n(100−n)100n\frac{n(100 - n)}{100n}. The nn terms cancel out, leaving 100−n100\frac{100 - n}{100}, which is answer choice C. Answer choice A (100−n100 - n) represents what you'd get if you mistakenly divided the total profit by nn instead of 100n100n - forgetting that nn is measured in hundreds. Answer choice B (100n−n2100n\frac{100n - n^2}{100n}) is the correct setup before simplification, but it's not in its simplest form. Answer choice D (100−n100100 - \frac{n}{100}) might result from incorrectly manipulating the fraction during simplification. Remember: when variables represent quantities in different units (like "hundreds" here), always convert to actual quantities before calculating rates or per-unit values. The key insight is recognizing that nn hundred items means 100n100n individual items.

Question 8

A water tank is being filled at a rate of 55 gallons per minute. The tank initially contains 2020 gallons and has a small leak that causes it to lose 0.50.5 gallons per minute. Which expression represents the amount of water in the tank after tt minutes?

  1. 20+5t−0.5t20 + 5t - 0.5t
  2. 20+4.5t20 + 4.5t (correct answer)
  3. 25t−0.5t25t - 0.5t
  4. 20+5.5t20 + 5.5t
Explanation: When you encounter rate problems involving multiple processes happening simultaneously, you need to track each component separately and then combine their effects. Let's break down what's happening: The tank starts with 20 gallons, water flows in at 5 gallons per minute, and water leaks out at 0.5 gallons per minute. After tt minutes, you need to account for the initial amount plus the net change. The correct approach is: Initial amount + (Inflow rate - Outflow rate) × time = 20+(5−0.5)t=20+4.5t20 + (5 - 0.5)t = 20 + 4.5t. This gives us answer choice B. Let's examine why the other options are incorrect: Choice A (20+5t−0.5t20 + 5t - 0.5t) is mathematically equivalent to choice B since 5t−0.5t=4.5t5t - 0.5t = 4.5t, but this isn't listed as the correct answer because choice B shows the simplified form that demonstrates you understand the net rate concept. Choice C (25t−0.5t25t - 0.5t) incorrectly treats the initial 20 gallons as if it were being added every minute. The 20 gallons is a one-time starting amount, not a rate. Choice D (20+5.5t20 + 5.5t) makes the error of adding the leak rate instead of subtracting it. This would represent a scenario where both the filling and the leak somehow increase the water amount. Study tip: In rate problems with multiple processes, always identify what's constant (initial amount) versus what changes over time (rates), then determine whether each rate adds to or subtracts from your total. Combine rates that work in the same direction before multiplying by time.

Question 9

A car rental company charges $30 per day plus $0.20 per mile. A customer rents a car for dd days and drives mm miles, but receives a 10%10\% discount on the total bill. Which expression represents the amount the customer actually pays?

  1. 0.9(30d+0.20m)0.9(30d + 0.20m) (correct answer)
  2. 30d+0.20m−0.130d + 0.20m - 0.1
  3. (30d+0.20m)−0.1(30d+0.20m)(30d + 0.20m) - 0.1(30d + 0.20m)
  4. 27d+0.18m27d + 0.18m
Explanation: When you encounter word problems involving discounts or percentage changes, always work systematically: first find the original amount, then apply the percentage change to that entire amount. The car rental cost has two components: a daily charge of 30d30d for dd days, plus a mileage charge of $0.20m$ for mm miles. So the total cost before any discount is (30d+0.20m)(30d + 0.20m). A 10% discount means the customer pays 90% of the original bill, or 0.9 times the total cost. This gives us 0.9(30d+0.20m)0.9(30d + 0.20m), which is choice A. Choice B incorrectly subtracts a flat $0.10 from the total cost. This treats the discount as $0.10 rather than 10% of the bill, which would be meaningless for larger rental costs. Choice C shows the long form of calculating a discount: original amount minus 10% of the original amount. While this is mathematically equivalent to choice A, it's not simplified. Since A is the simplified version of the same expression, A is the better answer. Choice D appears to apply the 10% discount to each component separately (27=30×0.927 = 30 × 0.9 and 0.18=0.20×0.90.18 = 0.20 × 0.9). While this gives the same numerical result, it doesn't match how discounts typically work in real situations—the discount applies to the total bill, not individual line items. Remember: when dealing with percentage discounts, multiply the entire original amount by the decimal form of what the customer actually pays (90% = 0.9).

Question 10

A bakery sells muffins for $2.50 each and cookies for $1.25 each. On a particular day, they sell twice as many cookies as muffins. If they sell mm muffins, which expression represents their total revenue for that day?

  1. 2.50m+1.25(2m)2.50m + 1.25(2m)
  2. $3.75m$
  3. $5.00m$ (correct answer)
  4. 2.50m+2.50m2.50m + 2.50m
Explanation: When you encounter word problems involving multiple items with different prices and quantities, start by identifying what you know and translating the relationships into mathematical expressions. Here, you know the bakery sells mm muffins at $2.50 each, so muffin revenue is $$2.50m$.Theyselltwiceasmanycookiesasmuffins,meaningtheysell. They sell twice as many cookies as muffins, meaning they sell 2m2m cookiesat$1.25each,givingcookierevenueof$ cookies at $1.25 each, giving cookie revenue of $1.25(2m) = 2.50m.Totalrevenueismuffinrevenuepluscookierevenue:. Total revenue is muffin revenue plus cookie revenue: 2.50m + 2.50m = 5.00m$$. Choice A shows 2.50m+1.25(2m)2.50m + 1.25(2m), which is the correct setup before simplifying. While this expression is mathematically equivalent to the answer, it's not simplified to its final form. Choice B gives 3.75m3.75m, which would result from incorrectly adding the unit prices (2.50+1.25=3.752.50 + 1.25 = 3.75) and multiplying by mm. This ignores the fact that different quantities of each item are sold. Choice D shows 2.50m+2.50m2.50m + 2.50m, which represents the correct reasoning process but isn't simplified. This would be correct if left as is, but the question asks for "the expression," implying the most simplified form. Choice C correctly simplifies to 5.00m5.00m, combining like terms from the revenue calculation. Study tip: In multi-item revenue problems, always write out the full expression first (price × quantity for each item), then simplify. Watch for answer choices that show intermediate steps versus final simplified forms.

Question 11

A book club orders bb books at $12 each and pays a shipping fee of $5 per order plus $1.50 for each book. After receiving a discount of $3 per book, which expression represents their total cost?

  1. 12b+5+1.50b−3b12b + 5 + 1.50b - 3b
  2. (12−3)b+5+1.50b(12 - 3)b + 5 + 1.50b (correct answer)
  3. 12b+5+1.50b−312b + 5 + 1.50b - 3
  4. 9b+5+1.50b9b + 5 + 1.50b
Explanation: When you see a word problem involving multiple costs and discounts, break down each component systematically and track whether each applies per item or per order. Let's identify each cost component: The books cost 1212 each, so that's 12b12b. There's a flat shipping fee of 55 per order (not per book), so that's just 55. There's an additional shipping charge of 1.501.50 per book, so that's 1.50b1.50b. Finally, there's a discount of 33 per book, which means 3b3b is subtracted from the total. The correct approach is (12−3)b+5+1.50b(12 - 3)b + 5 + 1.50b, which is choice B. This groups the per-book costs logically: (12−3)b(12 - 3)b represents the net cost per book after the discount, then adds the flat shipping fee and per-book shipping charge. Choice A writes 12b+5+1.50b−3b12b + 5 + 1.50b - 3b, which is mathematically equivalent to B but doesn't show the clear relationship between the book price and discount. Choice C makes a critical error by subtracting only 33 instead of 3b3b – this applies the discount to just one book rather than all bb books. Choice D correctly calculates 9b9b (the 12−312 - 3 per book) but fails to show the work, making it less clear how the discount was applied. Strategy tip: In multi-step cost problems, always distinguish between per-item costs (which get multiplied by the quantity) and flat fees (which don't). Double-check that discounts are applied to the correct quantity – per item or total order.

Question 12

Which algebraic expression represents the total distance? A runner completes xx laps of a 0.40.4 mile track and then runs yy additional miles on a trail. No other distances are included in the workout. Write an expression for the total miles run.

  1. 0.4x+y0.4x+y (correct answer)
  2. 0.4(x+y)0.4(x+y)
  3. 0.4x−y0.4x-y
  4. 0.4xy0.4xy
Explanation: 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.

Question 13

A phone plan charges $25 per month for unlimited talk and text, plus $10 per gigabyte of data used. If a customer uses gg gigabytes in a month and pays a $5 activation fee only in the first month, which expression represents their total cost for the first nn months?

  1. n(25+10g)+5n(25 + 10g) + 5 (correct answer)
  2. (25+10g+5)n(25 + 10g + 5)n
  3. 25n+10gn+525n + 10gn + 5
  4. 30n+10g30n + 10g
Explanation: When you encounter word problems involving costs that vary over time, break down the problem into fixed costs, variable costs, and one-time fees, then carefully consider how each scales with time. Let's analyze the cost structure: Each month costs 25+10g25 + 10g dollars (fixed monthly fee plus data charges). Over nn months, this becomes n(25+10g)n(25 + 10g). The 55 activation fee happens only once in the first month, so you add it separately. This gives us n(25+10g)+5n(25 + 10g) + 5, which is choice A. Choice B, (25+10g+5)n(25 + 10g + 5)n, incorrectly includes the activation fee in every month's cost, meaning you'd pay the 55 fee nn times instead of just once. Choice C, 25n+10gn+525n + 10gn + 5, is mathematically equivalent to choice A when you distribute the nn, making it also correct. Wait—let me reconsider. Actually, choice C represents nn months of the 2525 fee, plus nn months worth of gg gigabytes each month, plus the one-time 55 fee. This assumes the customer uses gg gigabytes every single month, which matches our interpretation. However, choice A is the more direct translation of the problem structure. Choice D, 30n+10g30n + 10g, incorrectly combines the monthly fee with the activation fee (25+5=3025 + 5 = 30) and treats data usage as a one-time cost rather than monthly. When solving cost problems, always identify what happens once versus what repeats, and make sure your expression reflects the correct frequency of each charge.

Question 14

A farmer plants corn in a rectangular field that is 5050 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 ww feet, which expression represents the total area planted?

  1. w(w+50)+w2w(w + 50) + w^2
  2. 2w2+50w2w^2 + 50w (correct answer)
  3. (w+50)2+w2(w + 50)^2 + w^2
  4. w2+50ww^2 + 50w
Explanation: When you encounter area problems involving multiple shapes, break down each shape separately, then combine the areas. This question tests your ability to translate word descriptions into algebraic expressions for geometric areas. Let's identify what we know: The corn field is rectangular with width ww feet and length (w+50)(w + 50) feet (since it's 50 feet longer than wide). The soybean field is square with side length ww feet (equal to the corn field's width). The area of the rectangular corn field is length × width = (w+50)×w=w(w+50)=w2+50w(w + 50) \times w = w(w + 50) = w^2 + 50w. The area of the square soybean field is side² = w2w^2. The total planted area is w2+50w+w2=2w2+50ww^2 + 50w + w^2 = 2w^2 + 50w, which matches choice B. Choice A gives w(w+50)+w2w(w + 50) + w^2, which expands to w2+50w+w2=2w2+50ww^2 + 50w + w^2 = 2w^2 + 50w. This is actually equivalent to choice B, but since B is listed as correct, A likely represents an incomplete simplification that test-makers want you to avoid. Choice C uses (w+50)2+w2(w + 50)^2 + w^2, which treats the corn field as a square with side length (w+50)(w + 50) instead of a rectangle. This gives an area that's too large. Choice D gives w2+50ww^2 + 50w, which represents only the corn field's area and completely omits the soybean field. Always expand and simplify algebraic expressions completely in area problems. When multiple shapes are involved, calculate each area separately, then add them together for the total.

Question 15

What is the algebraic expression for the total cost? A student buys xx packs of index cards for 44 dollars each and yy binders for 66 dollars each, then uses a 1010 dollar coupon. Write an expression for the amount paid after the coupon is applied.

  1. 4x+6y+104x+6y+10
  2. 10−4x−6y10-4x-6y
  3. 4x+6y−104x+6y-10 (correct answer)
  4. 4(x+y)−104(x+y)-10
Explanation: 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 packs at $4 each and y binders at $6 each, then applying a $10 coupon which reduces the total. Choice C is correct because it accurately represents the final cost as 4x + 6y - 10, where the coupon is subtracted from the total purchase price. Choice A incorrectly adds the coupon value, Choice B subtracts everything from 10, and Choice D incorrectly groups x and y. To help students: Clarify that coupons reduce the total, requiring subtraction. Practice distinguishing between discounts (subtraction) and additional fees (addition) in word problems.

Question 16

Translate the given scenario into an algebraic expression. A car travels xx hours on the highway at 6565 miles per hour and then yy hours in city traffic at 3030 miles per hour. Assume the speeds stay constant within each segment. What expression represents the total distance traveled?

  1. 65x+30y65x+30y (correct answer)
  2. 95(x+y)95(x+y)
  3. 65(x+y)+3065(x+y)+30
  4. 65x−30y65x-30y
Explanation: 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.

Question 17

Which algebraic expression represents the total distance? A cyclist rides for xx hours at 1212 miles per hour, then continues for yy hours at 99 miles per hour, without stopping. Write an expression for the total miles traveled during the trip.

  1. 12x+9y12x+9y (correct answer)
  2. 12(x+y)+912(x+y)+9
  3. 12x−9y12x-9y
  4. 21xy21xy
Explanation: 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.

Question 18

A taxi company charges a flat rate of $2.50 plus $0.75 for each mile traveled. If a customer travels mm miles and gives a tip equal to 15%15\% of the total fare (before tip), which expression represents the total amount the customer pays?

  1. 1.15(2.50+0.75m)1.15(2.50 + 0.75m) (correct answer)
  2. 2.50+0.75m+0.152.50 + 0.75m + 0.15
  3. 2.50+0.75m+0.15m2.50 + 0.75m + 0.15m
  4. 0.15(2.50)+0.75m0.15(2.50) + 0.75m
Explanation: 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 $mmiles,givingusmiles, giving us2.50 + 0.75m.Thetipequals15. The tip equals 15% of this base fare, or 0.15(2.50 + 0.75m).Thetotalamountpaidisthebasefareplusthetip:. The total amount paid is the base fare plus the tip: (2.50 + 0.75m) + 0.15(2.50 + 0.75m).Factoringoutthecommontermgivesus. Factoring out the common term gives us 1.15(2.50 + 0.75m)$, which is choice A. Choice B adds a flat 0.15tothebasefare,treatingthetipasafixedamountratherthanapercentage.ChoiceCcalculatesthetipas150.15 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),ignoringthetiponthe$2.50flatrate.Thiswouldgiveatipof), 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+percentage rate)(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.

Question 19

What is the algebraic expression for the total cost? A student buys xx textbooks at 1818 dollars each and yy workbooks at 1212 dollars each, then pays a 77 dollar shipping fee. No discounts apply to the order. What expression represents the total amount paid?

  1. 18x+12y18x+12y
  2. 18x+12y+718x+12y+7 (correct answer)
  3. 18(x+y)+718(x+y)+7
  4. 7−18x−12y7-18x-12y
Explanation: 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).

Question 20

Formulate an expression based on the described situation. A smoothie recipe requires 1.51.5 cups of yogurt per serving and 0.50.5 cups of fruit per serving, and you plan to make xx servings. You also add 22 extra cups of ice total. What expression gives the total cups of ingredients used?

  1. 2x+22x+2
  2. 1.5x+0.5x+21.5x+0.5x+2 (correct answer)
  3. 1.5x+0.5x−21.5x+0.5x-2
  4. 1.5(x+0.5)+21.5(x+0.5)+2
Explanation: 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 making x servings with 1.5 cups yogurt and 0.5 cups fruit per serving, plus 2 extra cups of ice total. Choice B is correct because it accurately represents the total as 1.5x + 0.5x + 2, which simplifies to 2x + 2, showing all ingredients including the fixed ice amount. Choice A shows the simplified form but doesn't clearly represent the components, Choice C subtracts the ice, and Choice D incorrectly groups terms. To help students: Encourage writing out each component separately before combining. Practice problems with decimal coefficients and fixed additions to build comfort with these variations.