Word Problems to Expressions - SSAT Upper Level Quantitative

1. A subscription service offers two plans: Plan A costs $12 per month, and Plan B costs $8 per month plus $0.50 for each hour of usage. Which equation could be used to find the number of hours of usage, $h$, in a month for which Plan B would cost the same as Plan A?
   Answer: $8 + 0.50h = 12$ Write each plan's monthly cost in terms of $h$, the number of hours used. Plan A is a flat $12 per month, no matter how many hours are used, so its cost is $12$. Plan B has a base fee of $8 plus $0.50 for each hour, so its cost is $8 + 0.50h$. The plans cost the same when the two expressions are equal: $8 + 0.50h = 12$. (Solving it gives $h = 8$ hours, but the question asks only for the equation.) The equations with $12h$ treat Plan A's flat $12 as if it were charged every hour. The equation $8h + 0.50 = 12$ swaps Plan B's base fee and hourly rate, charging $8 per hour and a one-time $0.50. In cost problems, flat fees stay as constants and per-unit charges get multiplied by the variable.
2. At a farmer's market, apples cost $2.50 per pound and oranges cost $3.20 per pound. A customer buys $a$ pounds of apples and $o$ pounds of oranges, then uses a coupon for $1.75 off the total purchase. Which expression represents the amount the customer actually pays?
   Answer: $2.50a + 3.20o - 1.75$ The total before coupon is $2.50a + 3.20o$. After applying the $1.75 coupon, the amount paid is $2.50a + 3.20o - 1.75$. Choice B incorrectly applies the coupon only to apples. Choice C incorrectly reduces the pounds of apples by the coupon amount. Choice D incorrectly adds the coupon instead of subtracting it.
3. A car rental company charges a base fee plus a mileage charge. The total cost for renting a car for one day and driving $m$ miles is $C = 45 + 0.25m$ dollars. If a customer's total bill was $78.50, which equation could be used to find the number of miles driven?
   Answer: $45 + 0.25m = 78.50$ Given the cost formula $C = 45 + 0.25m$ and total bill of $78.50, we substitute to get $45 + 0.25m = 78.50$. Choice B incorrectly makes the base fee coefficient of $m$. Choice C incorrectly subtracts the base fee instead of adding it. Choice D incorrectly makes the total cost a coefficient rather than the result.
4. In a basketball league, teams earn 3 points for a win, 1 point for a tie, and 0 points for a loss. If a team has played $g$ games and has $w$ wins and $t$ ties, which expression represents the number of losses?
   Answer: $g - w - t$ Every game ends in a win, a tie, or a loss, so $g = w + t + L$, where $L$ is the number of losses. Solving for $L$ gives $L = g - w - t$. The point values (3 for a win, 1 for a tie) do not matter here, because the question counts games, not points. The expression $g - 3w - t$ mistakenly uses the 3 points for a win as if each win were 3 games. The expression $g - w + t$ adds the ties back instead of subtracting them. The expression $w + t - g$ reverses the subtraction, so it gives the negative of the number of losses.
5. A rectangular garden has length that is 8 feet more than twice its width. If the width is $w$ feet, and the gardener wants to install a fence around the entire perimeter plus an additional gate that costs the same as 5 feet of fencing, which expression represents the total feet of fencing needed?
   Answer: $2w + 2(2w + 8) + 5$ This problem tests your understanding of perimeter formulas and algebraic expressions. When you see a geometry problem involving perimeter plus additional costs, break it down into separate components. First, identify the dimensions. The width is $w$ feet, and the length is "8 feet more than twice the width," which translates to $2w + 8$ feet. The perimeter of any rectangle equals $2 \times \text{length} + 2 \times \text{width}$, so you need $2(2w + 8) + 2w$ feet of fencing for the perimeter. Additionally, the gate costs the same as 5 feet of fencing, so you add 5 to the total. The complete expression is $2(2w + 8) + 2w + 5$, which can be written as $2w + 2(2w + 8) + 5$. Looking at the wrong answers: Choice A gives $2w + (2w + 8) + 5$, which only accounts for one width and one length instead of the full perimeter—this would be half the perimeter plus the gate cost. Choice B shows $w(2w + 8) + 5$, which calculates the area of the rectangle (length times width) rather than the perimeter. Choice D gives $2w + 2(2w + 8)$ but forgets to include the additional 5 feet for the gate cost. Remember that perimeter problems require you to go around the entire shape, so you need two lengths and two widths. Always read carefully to catch additional costs beyond the basic perimeter calculation—these are common on the SSAT.
6. A school cafeteria sells lunch combinations where students choose one item from each category. There are $s$ sandwich options, $d$ drink options, and $c$ chips options. If the cafeteria removes 2 sandwich options but adds 3 new drink options, which expression represents the new total number of possible lunch combinations?
   Answer: $(s - 2)(d + 3)c$ The number of combinations is the product of choices in each category. After changes: $(s-2)$ sandwich options, $(d+3)$ drink options, and $c$ chips options. Total combinations = $(s-2)(d+3)c$. Choice B incorrectly adds instead of multiplying the categories. Choice C fails to group the changes properly and adds instead of multiplies. Choice D incorrectly modifies the original product by simple addition/subtraction.
7. A parking garage charges $3 for the first hour and $1.50 for each additional hour or fraction thereof. If someone parks for more than one hour, which expression correctly represents the total cost for parking $h$ hours, where $h > 1$?
   Answer: $3 + 1.5(h - 1)$ For $h$ hours where $h > 1$, you pay $3 for the first hour and $1.50 for each of the remaining $(h-1)$ hours. This gives $3 + 1.5(h-1)$. Choice A incorrectly charges $1.50 for all $h$ hours instead of just the additional hours. Choice C uses a flat rate per hour, ignoring the different first-hour rate. Choice D incorrectly adds both a rate of $3 per hour for all hours plus the additional hour charge.
8. A company produces widgets at a cost of $3.50 each, plus a daily fixed cost of $200 for equipment. If the company sells each widget for $7.25 and produces and sells $x$ widgets in a day, which expression represents the daily profit?
   Answer: $7.25x - 3.50x - 200$ When you encounter profit problems, remember that profit equals revenue minus all costs. You need to carefully identify each component and how they behave. Let's build the profit expression step by step. Revenue from selling $x$ widgets at $7.25 each is $7.25x$. Total costs have two parts: variable costs that depend on production quantity ($3.50 per widget, so $3.50x$ total) and fixed costs that remain constant regardless of quantity ($200 daily equipment cost). Therefore, total costs are $3.50x + 200$. Daily profit = Revenue - Total Costs = $7.25x - (3.50x + 200) = 7.25x - 3.50x - 200$ This matches answer choice D. Let's examine why the other options are incorrect: Choice A: $7.25x - (3.50x - 200)$ incorrectly subtracts the fixed cost from variable costs instead of adding them together. This would give you $7.25x - 3.50x + 200$, which unrealistically adds the fixed cost to profit. Choice B: $7.25x - 3.50x + 200$ makes the same error as A's simplified form—it treats the fixed cost as contributing to profit rather than reducing it. Choice C: $7.25x - 200$ completely ignores the variable production costs of $3.50 per widget, severely overestimating profit. Strategy tip: In profit problems, always write out "Profit = Revenue - Total Costs" first, then carefully identify all cost components. Fixed costs always reduce profit, while variable costs multiply by the quantity produced.
9. A movie theater has rows of seats where the first row has $f$ seats, and each subsequent row has 2 more seats than the previous row. Which expression represents the total number of seats in the first $n$ rows?
   Answer: $nf + n(n-1)$ The rows have $f$, $f+2$, $f+4$, ..., $f+2(n-1)$ seats. This is an arithmetic sequence with first term $f$, common difference 2, and $n$ terms. The sum is $nf + \frac{n(n-1)}{2} \cdot 2 = nf + n(n-1)$. Choice A gives seats in just the $n$th row, not the total. Choice B doubles the additional seats incorrectly. Choice D also gives seats in just one row.
10. A bookstore offers a membership plan where customers pay an annual fee of $25 and then receive a 15% discount on all purchases. If a member buys books worth $d$ dollars (before discount) during the year, which expression represents the total amount the member pays for the year including the membership fee?
    Answer: $25 + 0.85d$ The member pays the 25-dollar fee once, plus the discounted price of the books. A 15% discount means paying 85% of the original price, so the books cost $0.85d$. The total is $25 + 0.85d$. Writing $25 + 0.15d$ adds the discount amount instead of the amount actually paid. Writing $0.85(d + 25)$ wrongly applies the discount to the membership fee as well; the fee is paid in full and is never discounted. Writing $40 + 0.85d$ adds the 15 from the discount rate to the 25-dollar fee, which mixes a percent with a dollar amount.

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