Word Problems to Expressions - SSAT Middle Level Quantitative

1. Maya buys 4 notebooks that each cost n dollars at the school store. She also buys 2 pens that each cost p dollars. Then she uses a coupon worth c dollars off her total. What is the algebraic expression for the total cost of the items?
   Answer: $4\mathit{n} + 2\mathit{p} - \mathit{c}$ This question tests middle school algebraic skills: translating word problems into algebraic expressions. Algebraic expressions use variables to represent quantities and operations from real-world situations. Understanding requires identifying quantities and operations in the text. In this scenario, the problem involves buying 4 notebooks at n dollars each, 2 pens at p dollars each, and subtracting a coupon of c dollars, leading to the expression 4n + 2p - c. Choice B is correct because it accurately translates the quantities and operations described in the problem. Choice A is incorrect because it uses addition for the coupon instead of subtraction, which is a common mistake when misinterpreting discounts as additional costs. Teaching strategies: Encourage students to identify keywords indicating operations (e.g., 'off' for subtraction), practice translating simple scenarios, and verify expressions by checking against the word problem logic. Watch for: students confusing operation keywords or misidentifying variables.
2. Sofia buys 3 shirts that each cost s dollars at the mall. She also buys p pairs of socks that each cost d dollars. Then she subtracts a discount of k dollars from her total. What is the algebraic expression for the total cost of the items?
   Answer: $3\mathit{s} + \mathit{p} * \mathit{d} - \mathit{k}$ This question tests middle school algebraic skills: translating word problems into algebraic expressions. Algebraic expressions use variables to represent quantities and operations from real-world situations. Understanding requires identifying quantities and operations in the text. In this scenario, the problem involves buying 3 shirts at s dollars each, p pairs of socks at d dollars each, and subtracting a discount of k dollars, leading to the expression 3s + p*d - k. Choice A is correct because it accurately translates the quantities and operations described in the problem. Choice C is incorrect because it subtracts the sock cost instead of adding it, which is a common mistake when misreading totals. Teaching strategies: Encourage students to identify keywords indicating operations (e.g., 'each' for multiplication, 'subtracts' for subtraction), practice translating simple scenarios, and verify expressions by checking against the word problem logic. Watch for: students confusing operation keywords or misidentifying variables.
3. Riley cooks pasta and buys ingredients for dinner. She buys n boxes of noodles at x dollars each. She also buys j jars of sauce at y dollars each. What expression shows the total cost of the meal?
   Answer: $\mathit{n} * \mathit{x} + \mathit{j} * \mathit{y}$ This question tests middle school algebraic skills: translating word problems into algebraic expressions. Algebraic expressions use variables to represent quantities and operations from real-world situations. Understanding requires identifying quantities and operations in the text. In this scenario, the problem involves buying n boxes of noodles at x dollars each and j jars of sauce at y dollars each, leading to the expression n*x + j*y. Choice A is correct because it accurately translates the quantities and operations described in the problem. Choice C is incorrect because it subtracts the sauce cost instead of adding it, which is a common mistake when misinterpreting total costs. Teaching strategies: Encourage students to identify keywords indicating operations (e.g., 'each' for multiplication), practice translating simple scenarios, and verify expressions by checking against the word problem logic. Watch for: students confusing operation keywords or misidentifying variables.
4. At a bakery, muffins cost $m$ dollars each and cookies cost half as much as muffins. If Sarah buys 4 muffins and 6 cookies, which expression represents her total cost in dollars?
   Answer: $7m$ When you encounter word problems involving multiple items with related costs, start by identifying the cost of each item, then build your expression step by step. Here, muffins cost $m$ dollars each, and cookies cost "half as much as muffins," meaning cookies cost $\frac{m}{2}$ dollars each. For Sarah's purchase: - 4 muffins at $m$ dollars each = $4m$ dollars - 6 cookies at $\frac{m}{2}$ dollars each = $6 \cdot \frac{m}{2} = \frac{6m}{2} = 3m$ dollars Total cost = $4m + 3m = 7m$ dollars Choice A ($7m$) is correct because it represents the simplified sum of both purchases. Choice B ($10m$) incorrectly assumes both muffins and cookies cost $m$ dollars each, giving $4m + 6m = 10m$. This ignores that cookies cost half the price of muffins. Choice C ($4m + 3$) makes the error of thinking cookies cost $\frac{1}{2}$ dollar each (a fixed amount) rather than $\frac{m}{2}$ dollars each. This gives $6 \times \frac{1}{2} = 3$ instead of $3m$. Choice D ($4m + 6m$) fails to account for the price difference entirely, treating cookies as if they also cost $m$ dollars each. Strategy tip: In multi-item cost problems, always express each item's cost in terms of the given variable first, then multiply by quantities. Watch for relationships like "half as much" or "twice as expensive" — these create fractional or multiple relationships, not addition or subtraction.
5. A swimming pool is being filled with water. Initially, it contains 500 gallons. Water flows in at a rate of 25 gallons per minute for $t$ minutes, but due to a small leak, 2 gallons per minute flow out. Which expression represents the amount of water in the pool after $t$ minutes?
   Answer: $500 + 23t$ This is a rate problem where you need to track water flowing both in and out of the pool simultaneously. When dealing with rates in opposite directions, you always combine them by finding the net rate. Start with what you know: the pool begins with 500 gallons. Water flows in at 25 gallons per minute, but 2 gallons per minute leak out. The net rate of water entering the pool is $25 - 2 = 23$ gallons per minute. After $t$ minutes, the total amount of water added will be $23t$ gallons. Adding this to the initial 500 gallons gives you $500 + 23t$, which is answer choice B. Let's examine why the other options are incorrect. Choice A ($500 + 25t$) only accounts for the water flowing in and completely ignores the leak—a common oversight when students focus on just one part of the problem. Choice C ($500 + 27t$) makes the error of adding the inflow and outflow rates instead of subtracting them, treating both as if they increase the water level. Choice D ($525t - 2t$) incorrectly treats the initial 500 gallons as a rate rather than a starting amount, and it doesn't properly represent how the rates combine over time. Study tip: In rate problems involving opposite directions (filling vs. draining, moving toward vs. away from), always identify the net rate first. Look for keywords like "flows out," "leaks," or "drains" that signal you need to subtract rates rather than add them.
6. Noah buys x movie tickets that cost t dollars each. He also buys y snacks that cost s dollars each. Then he pays a one-time service fee of f dollars. Translate the word problem into an algebraic expression.
   Answer: $\mathit{x} * \mathit{t} + \mathit{y} * \mathit{s} + \mathit{f}$ This question tests middle school algebraic skills: translating word problems into algebraic expressions. Algebraic expressions use variables to represent quantities and operations from real-world situations. Understanding requires identifying quantities and operations in the text. In this scenario, the problem involves buying x tickets at t dollars each, y snacks at s dollars each, and adding a fee of f dollars, leading to the expression x*t + y*s + f. Choice A is correct because it accurately translates the quantities and operations described in the problem. Choice D is incorrect because it subtracts the fee instead of adding it, which is a common mistake when misreading additional costs. Teaching strategies: Encourage students to identify keywords indicating operations (e.g., 'each' for multiplication, 'fee' for addition), practice translating simple scenarios, and verify expressions by checking against the word problem logic. Watch for: students confusing operation keywords or misidentifying variables.
7. Ben makes a salad for his family dinner. He buys l heads of lettuce at a dollars each. He also buys t tomatoes at b dollars each. Then he uses a store coupon for c dollars off. What expression shows the total cost of the meal?
   Answer: $\mathit{l} * \mathit{a} + \mathit{t} * \mathit{b} - \mathit{c}$ This question tests middle school algebraic skills: translating word problems into algebraic expressions. Algebraic expressions use variables to represent quantities and operations from real-world situations. Understanding requires identifying quantities and operations in the text. In this scenario, the problem involves buying l heads of lettuce at a dollars each, t tomatoes at b dollars each, and subtracting a coupon of c dollars, leading to the expression l*a + t*b - c. Choice A is correct because it accurately translates the quantities and operations described in the problem. Choice B is incorrect because it adds the coupon instead of subtracting it, which is a common mistake when misreading discounts. Teaching strategies: Encourage students to identify keywords indicating operations (e.g., 'each' for multiplication, 'off' for subtraction), practice translating simple scenarios, and verify expressions by checking against the word problem logic. Watch for: students confusing operation keywords or misidentifying variables.
8. A taxi charges $4 for the first mile and $2.50 for each additional mile. If a trip is $m$ miles long (where $m \geq 1$), which expression represents the total fare?
   Answer: $4 + 2.50(m - 1)$ When you encounter taxi fare problems, you're dealing with a two-part pricing structure: a fixed initial charge plus a variable charge based on distance. The key is identifying what gets charged at each rate. Let's break down this taxi's pricing: $4 for the first mile, then $2.50 for each additional mile beyond that first one. If your trip is $m$ miles long, you pay $4 for mile #1, then $2.50 for each of the remaining $(m-1)$ miles. The total fare is: $\$4 + \$2.50 \times (m-1) = 4 + 2.50(m-1)$, which is choice B. Let's see why the other options miss the mark. Choice A ($4 + 2.50m$) incorrectly charges the $2.50 rate for all $m$ miles, including the first mile that should only cost $4. This double-charges for the first mile. Choice C ($$6.50m$$) treats every mile the same at $6.50 each, ignoring the two-tier pricing structure entirely. Choice D ($$4m + 2.50$$) flips the logic, charging $4 per mile plus a flat $2.50, which doesn't match the given pricing at all. To verify: for a 3-mile trip, you'd pay $4 + 2.50(3-1) = 4 + 2.50(2) = 4 + 5 = \$9$. That's $4 for the first mile and $2.50 each for the remaining 2 miles. Strategy tip: In multi-tier pricing problems, always identify which quantity gets charged at which rate. Watch for the phrase "additional" or "beyond the first" — it signals you need $(m-1)$ rather than $m$.
9. Maria has some stickers. She gives away 8 stickers and then buys 3 packs of stickers, with each pack containing $n$ stickers. If she ends up with 25 stickers, which expression represents the number of stickers Maria had originally?
   Answer: $25 + 8 - 3n$ When you encounter word problems involving multiple steps and unknown quantities, the key strategy is to work backwards from the final result to find what you started with. Let's trace Maria's sticker journey in reverse. She ends with 25 stickers. Before that final count, she had bought 3 packs of $n$ stickers each, adding $3n$ stickers to her collection. So before buying those packs, she must have had $25 - 3n$ stickers. Going further back, she had given away 8 stickers before buying the packs. If she had $25 - 3n$ stickers after giving away 8, then originally she must have had $25 - 3n + 8$ stickers. Rearranging this expression gives us $25 + 8 - 3n$, which matches choice A. Let's examine why the other options don't work. Choice B ($25 - 8 + 3n$) incorrectly adds the packs she bought instead of subtracting them when working backwards. Choice C ($25 + 8 + 3n$) makes the mistake of adding both the given-away stickers and the bought packs, as if both increased her original amount. Choice D ($25 - 8 - 3n$) subtracts both quantities, which would mean she somehow started with fewer stickers than she ended with despite buying more. Remember this backwards-working technique: when a word problem describes a sequence of changes leading to a final amount, reverse each operation to find the starting point. Additions become subtractions, and subtractions become additions.
10. A library charges a late fee that depends on how many days a book is overdue. For the first 3 days, there is no fee. After that, the fee is $0.50 per day. If a book is $t$ days late (where $t > 3$), which expression represents the total late fee?
    Answer: $0.50(t - 3)$ Only the days after the 3-day grace period are charged. A book that is $t$ days late has $t - 3$ charged days, so the fee is $0.50(t - 3)$ dollars. For example, a book 7 days late has 4 charged days and a fee of 2 dollars. $0.50t$ charges every late day and ignores the grace period. $0.50t - 3$ subtracts 3 dollars instead of 3 days; for 7 days it gives only 50 cents. $0.50(t - 2)$ allows only 2 free days instead of 3, so it charges one day too many. $0.50(t + 3)$ adds the free days instead of removing them. Strategy tip: Test an expression with a real number. Try $t = 7$: the fee should be 4 days times 50 cents, or 2 dollars, and only one expression gives that.
11. Tom has a collection of baseball cards. He gives $\frac{1}{3}$ of his cards to his brother and then buys 12 more cards. If he originally had $c$ cards, which expression represents the number of cards he has now?
    Answer: $\frac{2c}{3} + 12$ When you encounter word problems involving fractions and operations, the key is to translate each step of the story into mathematical expressions, working in the order events happen. Let's trace through Tom's card collection step by step. Tom starts with $c$ cards. First, he gives away $\frac{1}{3}$ of his cards to his brother. This means he gives away $\frac{1}{3} \cdot c = \frac{c}{3}$ cards. After giving these away, he has $c - \frac{c}{3}$ cards remaining. To simplify this, think of $c$ as $\frac{3c}{3}$, so $c - \frac{c}{3} = \frac{3c}{3} - \frac{c}{3} = \frac{2c}{3}$ cards. Next, Tom buys 12 more cards. Adding these to what he already has: $\frac{2c}{3} + 12$. This matches choice A. Choice B ($\frac{c}{3} + 12$) represents only the cards Tom gave away plus 12, ignoring the cards he kept. Choice C ($c - \frac{1}{3} + 12$) incorrectly subtracts just $\frac{1}{3}$ instead of $\frac{1}{3}$ of his total cards—this treats the fraction as a fixed number rather than a fraction of $c$. Choice D ($\frac{c + 12}{3}$) suggests he somehow divided his final total by 3, which doesn't match the story at all. Study tip: In fraction word problems, always convert "giving away a fraction" into subtraction: if you give away $\frac{1}{3}$, you keep $\frac{2}{3}$. Write out each step algebraically before trying to match answer choices.
12. Harper buys 2 graphic novels that each cost g dollars at a book fair. She also buys m mystery books that each cost b dollars. Then she adds a donation of d dollars to support the library. What is the algebraic expression for the total cost of the items?
    Answer: $2\mathit{g} + \mathit{m} * \mathit{b} + \mathit{d}$ This question tests middle school algebraic skills: translating word problems into algebraic expressions. Algebraic expressions use variables to represent quantities and operations from real-world situations. Understanding requires identifying quantities and operations in the text. In this scenario, the problem involves buying 2 graphic novels at g dollars each, m mystery books at b dollars each, and adding a donation of d dollars, leading to the expression 2g + m*b + d. Choice A is correct because it accurately translates the quantities and operations described in the problem. Choice C is incorrect because it subtracts the book cost instead of adding it, which is a common mistake when misinterpreting totals. Teaching strategies: Encourage students to identify keywords indicating operations (e.g., 'each' for multiplication, 'adds' for addition), practice translating simple scenarios, and verify expressions by checking against the word problem logic. Watch for: students confusing operation keywords or misidentifying variables.
13. A train travels at a constant speed. In the first hour, it travels $d$ miles. Due to track maintenance, its speed is reduced by 10 mph for the next 2 hours. Which expression represents the total distance traveled in 3 hours?
    Answer: $3d - 20$ Because the train travels $d$ miles in the first hour, its speed is $d$ mph. For the next 2 hours its speed is $d - 10$ mph, so it covers $2(d - 10) = 2d - 20$ miles. The total is $d + 2d - 20 = 3d - 20$. The expression $d + 2d - 10$ forgets to multiply the 10 mph reduction by the 2 hours. The expression $d + (d - 10)$ counts the slower speed for only 1 hour instead of 2. The expression $3d - 30$ applies the reduction to all 3 hours, including the first hour at full speed. The expression $d - 20$ leaves out the distance covered at the slower speed.
14. For a fundraiser, students sell raffle tickets for t dollars each and cupcakes for c dollars each. They sell r raffle tickets and u cupcakes after school. Which expression represents the total earnings from the fundraiser?
    Answer: $\mathit{t} * \mathit{r} + \mathit{c} * \mathit{u}$ This question tests middle school algebraic skills: translating word problems into algebraic expressions. Algebraic expressions use variables to represent quantities and operations from real-world situations. Understanding requires identifying quantities and operations in the text. In this scenario, the problem involves selling tickets at t dollars each with r sold and cupcakes at c dollars each with u sold, leading to the expression t*r + c*u. Choice A is correct because it accurately translates the quantities and operations described in the problem. Choice D is incorrect because it switches the prices and quantities, which is a common mistake when misassigning variables. Teaching strategies: Encourage students to identify keywords indicating operations (e.g., 'each' for multiplication, 'total' for addition), practice translating simple scenarios, and verify expressions by checking against the word problem logic. Watch for: students confusing operation keywords or misidentifying variables.
15. Leah makes tacos and buys ingredients at the market. She buys m pounds of meat at p dollars per pound. She also buys s bags of shredded cheese at c dollars per bag. What expression shows the total cost of the meal?
    Answer: $\mathit{m} * \mathit{p} + \mathit{s} * \mathit{c}$ This question tests middle school algebraic skills: translating word problems into algebraic expressions. Algebraic expressions use variables to represent quantities and operations from real-world situations. Understanding requires identifying quantities and operations in the text. In this scenario, the problem involves buying m pounds of meat at p dollars per pound and s bags of cheese at c dollars per bag, leading to the expression m*p + s*c. Choice A is correct because it accurately translates the quantities and operations described in the problem. Choice B is incorrect because it uses addition of all variables instead of multiplication for per-unit costs, which is a common mistake when ignoring rates. Teaching strategies: Encourage students to identify keywords indicating operations (e.g., 'per' for multiplication), practice translating simple scenarios, and verify expressions by checking against the word problem logic. Watch for: students confusing operation keywords or misidentifying variables.
16. A farmer plants corn in a square field. If each side of the square is $x$ feet long, and he plants corn in all but a square border that is 2 feet wide around the entire perimeter, which expression represents the area planted with corn?
    Answer: $(x - 4)^2$ A 2-foot border runs along every edge, so each side of the planted square loses 2 feet at one end and 2 feet at the other: $x - 2 - 2 = x - 4$. The planted area is $(x - 4)^2$. The expression $(x - 2)^2$ subtracts the border width only once per side. The expression $x^2 - 16$ subtracts $4^2$ from $x^2$, but $(x - 4)^2 = x^2 - 8x + 16$, which is not the same. The expression $x^2 - 4x$ removes only part of the border area, and $x^2 - 4$ removes just 4 square feet from the whole field.
17. Josh earns $d$ dollars per day working at a bookstore. On weekdays, he works full days, but on Saturday he only works half a day. If he doesn't work on Sunday, which expression represents his total earnings for one week?
    Answer: $5.5d$ This is a word problem asking you to translate a real-world scenario into an algebraic expression. When you see questions like this, break down the information systematically and identify what happens each day of the week. Let's work through Josh's weekly schedule. He earns $d$ dollars per full day. Monday through Friday (5 weekdays), he works full days, earning $5d$ total. On Saturday, he works half a day, earning $0.5d$. On Sunday, he doesn't work, earning $0d$. Adding these together: $5d + 0.5d + 0d = 5.5d$. Looking at the wrong answers: Choice B ($6.5d$) likely comes from incorrectly counting 6 full days plus Saturday's half day, perhaps forgetting that Sunday is a day off. Choice C ($7d$) assumes Josh works a full day every day of the week, ignoring both that Saturday is half time and Sunday is off. Choice D ($5d + 0.5$) makes a common algebraic error—it correctly identifies 5 full weekdays but writes Saturday's earnings as $0.5$ instead of $0.5d$. Remember, Josh earns 0.5 times his daily rate, not just 0.5 dollars. The correct answer is A: $5.5d$. Strategy tip: In word-to-algebra problems, always double-check your variables. When the problem states someone earns "$d$ dollars per day," any fractional work must be written as a fraction times $d$, not just the fraction alone. This variable consistency mistake appears frequently on the SSAT.
18. A school is planning a field trip. The bus rental costs $300 plus $2 per student. The museum charges $8 per student for admission. If $s$ students go on the trip, which expression represents the cost per student?
    Answer: $\frac{300}{s} + 10$ Find the total cost, then divide by the number of students. The bus costs $300 + 2s$ and admission costs $8s$, so the total is $300 + 10s$. The cost per student is $\frac{300 + 10s}{s} = \frac{300}{s} + 10$. The expression $300 + 10s$ is the total cost for the whole group, not the cost per student. The expression $\frac{300}{s} + 2$ spreads the bus cost correctly but forgets the $8 admission. The expression $\frac{300 + 2s}{s} + 8s$ adds the admission for the whole group to one student's share. The expression $\frac{310}{s}$ adds 300, 2 and 8 as if every charge were a single fixed fee.
19. Lisa buys $x$ notebooks at $2 each and $y$ pens at $3 each. She pays with a $20 bill. Which expression represents the amount of change she should receive?
    Answer: $20 - (2x + 3y)$ When you encounter word problems involving purchases and change, you need to set up the relationship between what's spent and what's paid. The key is understanding that change equals the amount paid minus the total cost of items purchased. Let's break down Lisa's purchase: she buys $x$ notebooks at $2 each, costing $2x$ dollars total, and $y$ pens at $3 each, costing $3y$ dollars total. Her total purchase is $2x + 3y$ dollars. Since she pays with a $20 bill, her change is $20 - (2x + 3y)$, which matches answer choice B. Looking at the wrong answers: Choice A gives $2x + 3y - 20$, which would represent how much more money Lisa needs if her purchase exceeds $20, but it doesn't represent change from a $20 payment. Choice C shows $20 - 5xy$, which incorrectly multiplies the variables $x$ and $y$ together instead of treating them as separate quantities—this represents a fundamental misunderstanding of how to calculate total cost. Choice D gives $20 + 2x + 3y$, which adds the payment to the purchase cost, representing the total amount of money involved rather than change. Remember this pattern: change problems always follow the formula "amount paid minus total cost." When you see parentheses like in choice B, that's often a good sign—it shows the total cost is being calculated first, then subtracted from the payment amount.
20. A school cafeteria serves lunch to $n$ students on Monday. On Tuesday, 15 fewer students eat lunch. On Wednesday, twice as many students as Tuesday eat lunch. Which expression represents the total number of students who ate lunch over these three days?
    Answer: $4n - 45$ Monday has $n$ students. Tuesday has 15 fewer, $n - 15$. Wednesday has twice Tuesday's count, $2(n - 15)$. The total is $n + (n - 15) + 2(n - 15) = n + n - 15 + 2n - 30 = 4n - 45$. $4n - 30$ comes from distributing the 2 to $n$ only, writing Wednesday as $2n - 15$. $n + (n - 15) + 2n$ doubles Monday instead of Tuesday and simplifies to $4n - 15$. $n + (n - 15) + 2(n + 15)$ writes Tuesday correctly but adds 15 inside Wednesday's parentheses, giving $4n + 15$. $3n - 30$ is $n + 2(n - 15)$, which leaves out Tuesday. Strategy tip: Write one expression per day before combining, and distribute carefully: $2(n - 15) = 2n - 30$, not $2n - 15$.

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