Translating to Algebraic Expressions - Accuplacer Qas

1. A shirt costs $c$ dollars. The price is first increased by 20%, and then this new price is decreased by 10%. Which expression represents the final price of the shirt?
   Answer: $0.90(c + 0.20c)$ The initial price is $c$. An increase of 20% makes the new price $c + 0.20c$, or $1.20c$. This new price is then decreased by 10%. The discount amount is 10% of the new price, which is $0.10(c + 0.20c)$. The final price is the new price minus the discount: $(c + 0.20c) - 0.10(c + 0.20c)$. Factoring out $(c + 0.20c)$ gives $(1 - 0.10)(c + 0.20c)$, which is $0.90(c + 0.20c)$. Answer A incorrectly applies the 10% discount to the original price $c$. Answers C and D incorrectly treat percentages as flat amounts.
2. A plumber charges a flat fee of $50 for a house call, plus an hourly rate of $r$ dollars. For a job that takes $h$ hours, the plumber gives a 15% discount on the hourly charges only. Which expression represents the total cost?
   Answer: $50 + 0.85rh$ The total hourly charge is $rh$. A 15% discount on this amount means the customer pays 85% of it, which is $0.85rh$. The flat fee of $50 is not discounted. The total cost is the sum of the flat fee and the discounted hourly charges: $50 + 0.85rh$. Answer A incorrectly applies the discount to the flat fee as well. Answer B incorrectly subtracts a flat 15 cents instead of 15 percent. Answer D incorrectly adds 15% instead of subtracting it.
3. Which expression represents the difference between a number $n$ cubed and the cube of 5?
   Answer: $n^3 - 5^3$ The phrase 'a number $n$ cubed' is $n^3$. The phrase 'the cube of 5' is $5^3$. The 'difference between' these two quantities is $n^3 - 5^3$. Answer A fails to cube the 5. Answer B represents 'the cube of the difference between $n$ and 5'. Answer D reverses the order of subtraction.
4. A rope is $f$ feet and 8 inches long. Which expression represents the total length of the rope in inches?
   Answer: $12f + 8$ To find the total length in inches, all parts of the measurement must be in inches. There are 12 inches in 1 foot. Therefore, $f$ feet is equal to $12f$ inches. The additional 8 inches is then added to this amount. The total length in inches is $12f + 8$. Answer A incorrectly divides by 12. Answer B represents the total length in feet. Answer D uses an incorrect conversion factor of 3 (for yards to feet).
5. A parking garage charges a base fee plus an hourly rate. The total cost for parking 3 hours is $12, and for 7 hours is $20. If a customer parks for $h$ hours and receives a 10% discount when parking exceeds 5 hours, which expression represents their total cost?
   Answer: $\begin{cases} 6 + 2h & \text{if } h \leq 5 \\ 0.9(6 + 2h) & \text{if } h > 5 \end{cases}$ Let base fee = $b$ and hourly rate = $r$. From given info: $b + 3r = 12$ and $b + 7r = 20$. Solving: $4r = 8$, so $r = 2$ and $b = 6$. Normal cost = $6 + 2h$. For $h > 5$, 10% discount applies to total cost: $0.9(6 + 2h)$. Choice B is equivalent but unnecessarily complex. Choice C applies discount only to hourly portion. Choice D applies discount only to hourly rate, not total.
6. A manufacturer produces widgets where the cost per widget decreases as production volume increases. The cost per widget is $8 for the first 100 widgets, then decreases by $0.02 for each additional group of 50 widgets produced. If $x$ represents the total number of widgets produced (where $x > 100$), which expression represents the cost per widget?
   Answer: $8 - 0.02 \times \lfloor\frac{x-100}{50}\rfloor$ When you encounter piecewise pricing problems like this, you need to carefully track how the price changes occur and what triggers each change. Let's break down what happens: The first 100 widgets cost $8 each. After that, the cost decreases by $0.02 for each additional group of 50 widgets. The key insight is that the price reduction happens in discrete steps - only when you complete another full group of 50 widgets beyond the initial 100. For the correct answer D, $8 - 0.02 \times \lfloor\frac{x-100}{50}\rfloor$, let's trace through the logic. If you produce 175 widgets, you've made 75 widgets beyond the first 100. Dividing 75 by 50 gives 1.5, but the floor function $\lfloor 1.5 \rfloor = 1$ means you've completed exactly 1 full group of 50. So the cost per widget is $8 - 0.02(1) = 7.98$. Answer A treats every single widget beyond 100 as triggering a $0.02 reduction, which contradicts the "group of 50" requirement. Answer B uses the exact division without the floor function, so it would give partial reductions (like $0.03 for 75 widgets) rather than step-wise changes. Answer C incorrectly includes the first 100 widgets in the division, which would apply discounts to the base production volume. Remember: When you see "group of" or "every X units" in pricing problems, look for the floor function $\lfloor \rfloor$ to handle the step-wise changes correctly.
7. Let $k$ be the smallest of three consecutive odd integers. Which expression represents the sum of these three integers?
   Answer: $3k + 6$ Consecutive odd integers differ by 2. If the smallest is $k$, the next odd integer is $k + 2$, and the one after that is $k + 4$. The sum is $k + (k + 2) + (k + 4)$. Combining like terms gives $3k + 6$. Answer B, $3k + 3$, represents the sum of three consecutive integers (e.g., $k, k+1, k+2$). Answer A, (3k), would be the sum if the integers were $k-2, k, k+2$, but the stem specifies that $k$ is the smallest. Answer D, $3k - 6$, would be the sum if $k$ were the largest integer ($k-4, k-2, k$).
8. A student's scores on four quizzes are 88, 92, 85, and $x$. Which expression represents the average (arithmetic mean) of the student's quiz scores?
   Answer: $\frac{265 + x}{4}$ The average is the sum of the values divided by the number of values. There are 4 quiz scores. The sum is $88 + 92 + 85 + x = 265 + x$. To find the average, this sum must be divided by 4, giving $\frac{265 + x}{4}$. Answer B represents the sum, not the average. Answer C incorrectly divides by 3, the number of known scores, instead of the total number of scores. Answer D is an order-of-operations error, dividing only $x$ by 4.
9. A person buys $x$ apples at $0.75 each and $y$ oranges at $0.90 each. Which expression represents the total change received from a $20 bill?
   Answer: $20 - (0.75x + 0.90y)$ First, calculate the total cost. The cost of apples is $0.75x$ and the cost of oranges is $0.90y$. The total cost is their sum, $0.75x + 0.90y$. The change from a $20 bill is the initial amount minus the total cost. This requires subtracting the entire cost expression, so parentheses are necessary: $20 - (0.75x + 0.90y)$. Answer A represents only the total cost. Answer B reverses the subtraction. Answer C is a common error where the subtraction is not distributed to the second term.
10. A number, $y$, is 8 less than the product of 3 and another number, $x$. Which expression represents the value of $y$?
    Answer: $3x - 8$ The phrase 'the product of 3 and another number, $x$' translates to $3x$. The phrase '8 less than' indicates that 8 should be subtracted from this product. Therefore, the correct expression is $3x - 8$. Answer A, $8 - 3x$, is a common reversal error for 'less than'. Answer B, $3(x - 8)$, incorrectly subtracts 8 from $x$ before multiplying by 3. Answer D, $3x + 8$, incorrectly uses addition instead of subtraction.
11. Three consecutive odd integers have the property that when the largest is divided by the smallest, the quotient is $\frac{1}{3}$ more than when the middle integer is divided by the smallest. If $n$ represents the smallest odd integer, which equation correctly models this relationship?
    Answer: $\frac{n+4}{n} - \frac{n+2}{n} = \frac{1}{3}$ Three consecutive odd integers: $n$, $n+2$, $n+4$. The condition states: $\frac{\text{largest}}{\text{smallest}} = \frac{\text{middle}}{\text{smallest}} + \frac{1}{3}$. This gives $\frac{n+4}{n} = \frac{n+2}{n} + \frac{1}{3}$. Rearranging: $\frac{n+4}{n} - \frac{n+2}{n} = \frac{1}{3}$. Choice A is the unrearranged form. Choice C uses wrong denominator for largest/middle. Choice D interprets 'more than' as multiplication.
12. Maria is $m$ years old. Her brother, David, is 3 years older than her. Which expression represents the sum of their ages 5 years from now?
    Answer: $2m + 13$ Currently, Maria's age is $m$ and David's age is $m + 3$. In 5 years, Maria's age will be $m + 5$ and David's age will be $(m + 3) + 5$, which is $m + 8$. The sum of their ages in 5 years will be $(m + 5) + (m + 8)$. Combining like terms gives $2m + 13$. Answer A is the sum of their current ages. Answer C represents adding 5 years only once to their combined current age ($2m+3+5$). Answer B would be correct if David were 3 years younger than Maria.
13. The length of a rectangular garden is 5 feet less than twice its width. If the width is represented by $w$, which expression represents the perimeter of the garden?
    Answer: $2w + 2(2w - 5)$ First, translate the length. 'Twice its width' is $2w$. '5 feet less than' that is $2w - 5$. The perimeter of a rectangle is $2L + 2W$, where $L$ is length and $W$ is width. Substituting the expressions gives $2(2w - 5) + 2w$. Answer A represents the area, not the perimeter. Answer B represents the sum of one width and one length (half the perimeter). Answer C makes a reversal error for 'less than', writing the length as $5 - 2w$.
14. A person has $q$ quarters, $d$ dimes, and $n$ nickels. The number of dimes is twice the number of quarters. Which expression represents the total value of the coins in cents, in terms of $q$ and $n$?
    Answer: $45q + 5n$ The value of the coins in cents is $25q + 10d + 5n$. The problem states that the number of dimes is twice the number of quarters, so $d = 2q$. We substitute $2q$ for $d$ in the value expression: $25q + 10(2q) + 5n$. This simplifies to $25q + 20q + 5n$, which equals $45q + 5n$. Answer A represents the number of coins, not their value. Answer B represents the value in dollars, not cents. Answer C results from incorrectly setting up the relationship as $q = 2d$.
15. Which expression represents one-half of the number that is 10 more than the square of $x$?
    Answer: $\frac{x^2 + 10}{2}$ Working from the inside out, 'the square of $x$' is $x^2$. '10 more than' that is $x^2 + 10$. 'One-half of' this entire quantity means dividing the whole expression by 2, which gives $\frac{x^2 + 10}{2}$. Answer A, $\frac{x^2}{2} + 10$, represents '10 more than one-half of the square of $x$', an order-of-operations error. Answer C represents 'one-half of the square of the sum of $x$ and 10'. Answer D incorrectly uses subtraction for 'more than'.
16. A container holds $G$ gallons of water. A portion of the water, $p$ percent, is poured out. Then, 5 gallons are added. Which expression represents the final amount of water in the container, in gallons?
    Answer: $G(1 - \frac{p}{100}) + 5$ The initial amount is $G$. The amount poured out is $p$ percent of $G$, which is $(\frac{p}{100})G$. The amount remaining is $G - (\frac{p}{100})G$, which can be factored as $G(1 - \frac{p}{100})$. Finally, 5 gallons are added, so the final amount is $G(1 - \frac{p}{100}) + 5$. Answer A treats the percentage $p$ as a flat amount. Answer B incorrectly applies the percentage decrease after the 5 gallons are added. Answer D incorrectly subtracts 5 instead of adding it.
17. The value of a car, initially $V$ dollars, decreases by a total of $D$ dollars over a period of $Y$ years at a constant rate. Which expression represents the value of the car after $t$ years, where $t < Y$?
    Answer: $V - t(\frac{D}{Y})$ First, find the rate of decrease per year. If the value decreases by $D$ dollars in $Y$ years, the constant rate is $\frac{D}{Y}$ dollars per year. The total decrease after $t$ years is the rate multiplied by the time, which is $t(\frac{D}{Y})$. The value after $t$ years is the initial value $V$ minus this total decrease: $V - t(\frac{D}{Y})$. Answer A forgets to divide by $Y$ to find the yearly rate. Answer B incorrectly adds the decrease, representing appreciation. Answer D omits the total decrease amount $D$.
18. Which expression represents the result when the sum of a number $x$ and 6 is subtracted from triple the number?
    Answer: $3x - (x + 6)$ The phrase 'subtracted from' means the first quantity is taken away from the second. 'The sum of a number $x$ and 6' is $(x + 6)$. 'Triple the number' is $3x$. Subtracting the sum from triple the number means starting with $3x$ and subtracting the entire quantity $(x+6)$, which yields $3x - (x + 6)$. Answer A is a common reversal error for 'subtracted from'. Answer B makes a common error by failing to distribute the negative sign to both terms in the sum. Answer C misinterprets 'triple the number' as 'triple the sum'.
19. Which expression represents the quotient of 10 and the quantity 3 less than a number $x$?
    Answer: $\frac{10}{x-3}$ The phrase 'the quotient of A and B' translates to $\frac{A}{B}$. Here, A is 10 and B is 'the quantity 3 less than a number $x$'. The quantity '3 less than $x$' is $x - 3$. Therefore, the full expression is $\frac{10}{x-3}$. Answer A reverses the numerator and denominator. Answer B makes the common reversal error for 'less than'. Answer C is an order-of-operations error, representing 'the quotient of 10 and $x$, minus 3'.
20. A rectangular swimming pool is surrounded by a walkway of uniform width. The pool dimensions are 20 feet by 30 feet. If the walkway width is $w$ feet and the total area of the pool plus walkway exceeds the pool area by more than 440 square feet, which inequality represents this constraint?
    Answer: $4w^2 + 100w > 440$ Pool area = $20 \times 30 = 600$ sq ft. Total area including walkway = $(20 + 2w)(30 + 2w) = 600 + 40w + 60w + 4w^2 = 600 + 100w + 4w^2$. Difference = $600 + 100w + 4w^2 - 600 = 4w^2 + 100w$. Given this exceeds 440: $4w^2 + 100w > 440$. Choice B is unreduced form. Choice C uses wrong threshold. Choice D assumes walkway only on two sides.

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