Expressions - GED Math

1. The first term of an arithmetic sequence is $(3y-5)$ and the common difference is $(1-2y)$. The formula for the $n$-th term of an arithmetic sequence is $a_n = a_1 + (n-1)d$. Which expression represents the 11th term of this sequence?
   Answer: $5 - 17y$ We use the formula $a_n = a_1 + (n-1)d$ with $n=11$, $a_1 = 3y-5$, and $d = 1-2y$. Substitute these into the formula: $a_{11} = (3y-5) + (11-1)(1-2y)$. Simplify the expression: $a_{11} = (3y-5) + 10(1-2y)$. Distribute the 10: $a_{11} = 3y - 5 + 10 - 20y$. Combine like terms: $(3y - 20y) + (-5 + 10) = -17y + 5$. This is equivalent to $5 - 17y$.
2. A company's profit $P(x)$ from selling $x$ units of a product is given by $P(x) = R(x) - C(x)$, where $R(x)$ is revenue and $C(x)$ is cost. The revenue is modeled by $R(x) = x(200 - 0.5x)$. The cost is modeled by a fixed cost of $5000 plus a variable cost of $20 per unit. Which expression represents the company's profit, $P(x)$?
   Answer: $-0.5x^2 + 180x - 5000$ First, write the expressions for revenue $R(x)$ and cost $C(x)$. The revenue expression is given: $R(x) = x(200 - 0.5x) = 200x - 0.5x^2$. The cost expression is a sum of the fixed and variable costs: $C(x) = 5000 + 20x$. The profit is $P(x) = R(x) - C(x)$. Substitute the expressions for $R(x)$ and $C(x)$: $P(x) = (200x - 0.5x^2) - (5000 + 20x)$. It is crucial to use parentheses around the cost function to ensure the negative sign distributes correctly. $P(x) = 200x - 0.5x^2 - 5000 - 20x$. Now, combine like terms: $P(x) = -0.5x^2 + (200x - 20x) - 5000 = -0.5x^2 + 180x - 5000$.
3. Which of the following expressions is equivalent to the simplified form of $(3n + 2)^2 - (3n - 2)^2$?
   Answer: $24n$ This problem can be solved in two ways. Method 1: Expand each square. $(3n + 2)^2 = (3n+2)(3n+2) = 9n^2 + 6n + 6n + 4 = 9n^2 + 12n + 4$. And $(3n - 2)^2 = (3n-2)(3n-2) = 9n^2 - 6n - 6n + 4 = 9n^2 - 12n + 4$. Now subtract the second expansion from the first: $(9n^2 + 12n + 4) - (9n^2 - 12n + 4) = 9n^2 + 12n + 4 - 9n^2 + 12n - 4$. The $9n^2$ terms cancel, and the 4s cancel. We are left with $12n + 12n = 24n$. Method 2: Use the difference of squares formula, $a^2 - b^2 = (a+b)(a-b)$. Here, $a = 3n+2$ and $b = 3n-2$. So, $a+b = (3n+2) + (3n-2) = 6n$. And $a-b = (3n+2) - (3n-2) = 3n+2-3n+2 = 4$. The expression is $(a+b)(a-b) = (6n)(4) = 24n$.
4. The surface area of a rectangular box with length $l$, width $w$, and height $h$ is given by $2(lw + lh + wh)$. If the width is 3 units less than the length, and the height is twice the width, which expression represents the surface area in terms of length $l$ only?
   Answer: $6l^2 - 18l + 18$ The correct answer is B. Given: w = l - 3 and h = 2w = 2(l - 3) = 2l - 6. Substituting into the surface area formula: 2[l(l-3) + l(2l-6) + (l-3)(2l-6)] = 2[l² - 3l + 2l² - 6l + (2l² - 6l - 6l + 18)] = 2[l² - 3l + 2l² - 6l + 2l² - 12l + 18] = 2[5l² - 21l + 18] = 10l² - 42l + 36. Wait, let me recalculate: 2[l² - 3l + 2l² - 6l + 2l² - 12l + 18] = 2[3l² - 9l + 9] = 6l² - 18l + 18.
5. A silo is constructed from a cylinder with a hemisphere on top. The radius of the cylinder's base is $r$, and the height of the cylindrical portion is three times the radius. The volume of a cylinder is $\pi r^2 h$ and the volume of a sphere is $\frac{4}{3}\pi r^3$. Which expression represents the total volume of the silo in terms of its radius $r$?
   Answer: $\frac{11}{3}\pi r^3$ The total volume is the sum of the volume of the cylinder and the volume of the hemisphere. The cylinder's height $h$ is given as $3r$. The volume of the cylinder is $V_{cyl} = \pi r^2 h = \pi r^2 (3r) = 3\pi r^3$. A hemisphere is half of a sphere. The volume of the hemisphere is $V_{hemi} = \frac{1}{2} V_{sphere} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$. The total volume is $V_{total} = V_{cyl} + V_{hemi} = 3\pi r^3 + \frac{2}{3}\pi r^3$. To add these, find a common denominator: $3\pi r^3 = \frac{9}{3}\pi r^3$. So, $V_{total} = \frac{9}{3}\pi r^3 + \frac{2}{3}\pi r^3 = \frac{11}{3}\pi r^3$.
6. The expression $2x^2 - 3x + 1$ is evaluated when $x = -2$. What is the value?
   Answer: $15$ The correct answer is D. Substituting x = -2: 2(-2)² - 3(-2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15. Choice A results from incorrectly calculating (-2)² = -4. Choice B comes from the error 2(-2)² - 3(-2) + 1 = -8 + 6 + 1. Choice C results from multiple sign errors throughout the calculation.
7. The cost of producing $n$ items is given by $C(n) = 50n + 1200$. The revenue from selling $n$ items is $R(n) = 80n$. Which expression represents the profit when $n$ items are produced and sold?
   Answer: $30n - 1200$ The correct answer is A. Profit = Revenue - Cost = R(n) - C(n) = 80n - (50n + 1200) = 80n - 50n - 1200 = 30n - 1200. Choice B incorrectly adds the constant term instead of subtracting it. Choice C incorrectly adds the coefficients of n instead of subtracting. Choice D makes both errors: adding coefficients and adding the constant.
8. Translate the following verbal phrase into an algebraic expression: "The quotient of the square of a number $n$ and 5, decreased by the product of 3 and the absolute difference between the number and 8."
   Answer: $\frac{n^2}{5} - 3|n - 8|$ The phrase is broken down into two main parts joined by "decreased by" (subtraction). The first part is "The quotient of the square of a number $n$ and 5", which translates to $\frac{n^2}{5}$. The second part is "the product of 3 and the absolute difference between the number and 8". The "absolute difference between n and 8" is $|n - 8|$. The "product of 3 and" that quantity is $3|n - 8|$. Combining the two parts, "decreased by" means we subtract the second part from the first: $\frac{n^2}{5} - 3|n - 8|$.
9. A store offers a discount where customers pay 85% of the original price, then an additional $5 off the discounted amount. If the original price is $p$ dollars, which expression represents the final amount a customer pays?
   Answer: $0.85p - 5$ The correct answer is A. First, the customer pays 85% of the original price: 0.85p. Then $5 is subtracted from this discounted amount: 0.85p - 5. Choice B incorrectly treats the $5 as a percentage discount. Choice C is mathematically equivalent to A but unnecessarily complicated. Choice D incorrectly applies the percentage discount to (p - 5) instead of applying the discounts sequentially.
10. Juan earns $p$ dollars per hour at his job and works 6 hours on Saturday and 4 hours on Sunday. Which expression represents the total amount, in dollars, Juan earns over the weekend?
    Answer: $6p+4p$ This question tests your ability to translate a word problem into an algebraic expression involving multiplication and addition. When Juan works for a certain number of hours at $p$ dollars per hour, his earnings equal the hours worked times his hourly rate. On Saturday, he works 6 hours, so he earns $6p$ dollars. On Sunday, he works 4 hours, so he earns $4p$ dollars. To find his total weekend earnings, you add these amounts together: $6p + 4p$. Choice A ($6p + 4p$) correctly represents Juan's total weekend earnings by multiplying his hourly rate by the hours worked each day, then adding those daily totals. Choice B ($6 + p + 4$) incorrectly adds the hours and hourly rate instead of multiplying them. This would give you $10 + p$, which doesn't represent earnings at all—you can't just add hours to dollars per hour. Choice C ($6p - 4p$) uses subtraction instead of addition, which would calculate the difference between Saturday's and Sunday's earnings ($2p$) rather than the total weekend earnings. Choice D ($p(6 - 4)$) also uses subtraction within parentheses, giving you $p \times 2 = 2p$, which again represents the difference in daily earnings, not the total. Remember this pattern: when calculating total pay, multiply rate times hours for each period, then add all the periods together. Watch out for expressions that subtract when you should add, or that add quantities with different units (like hours plus hourly rate).
11. A solution is formed by mixing two substances. Substance A has a saline concentration of $c_1$ and Substance B has a saline concentration of $c_2$. If $V_A$ liters of Substance A are mixed with $V_B$ liters of Substance B, which expression represents the saline concentration of the final mixture?
    Answer: $\frac{c_1V_A + c_2V_B}{V_A + V_B}$ The concentration of the final mixture is the total amount of saline divided by the total volume of the mixture. The amount of saline from Substance A is its concentration times its volume: $c_1V_A$. The amount of saline from Substance B is $c_2V_B$. The total amount of saline is the sum of these two amounts: $c_1V_A + c_2V_B$. The total volume of the mixture is the sum of the individual volumes: $V_A + V_B$. Therefore, the final concentration is the total saline divided by the total volume, which is $\frac{c_1V_A + c_2V_B}{V_A + V_B}$.
12. A rectangular garden has a length that is 5 feet greater than twice its width, $w$. A paved walkway of uniform width $x$ is built around the outside of the garden. Which of the following expressions represents the area of the walkway only?
    Answer: $4x^2 + 6wx + 10x$ The area of the walkway is the area of the large rectangle (garden plus walkway) minus the area of the small rectangle (garden only). First, define the dimensions. Garden width = $w$. Garden length = $2w + 5$. The walkway adds width $x$ to both sides of the garden. So, the total width is $w + 2x$ and the total length is $(2w + 5) + 2x$. The total area is $(w + 2x)(2w + 5 + 2x) = 2w^2 + 5w + 2wx + 4wx + 10x + 4x^2 = 2w^2 + 6wx + 5w + 10x + 4x^2$. The garden area is $w(2w + 5) = 2w^2 + 5w$. Subtract the garden area from the total area: $(2w^2 + 6wx + 5w + 10x + 4x^2) - (2w^2 + 5w) = 4x^2 + 6wx + 10x$.
13. An item's price $P$ is first discounted by $d$ percent. Then, a coupon for $C$ dollars is applied. Finally, a sales tax of $t$ percent is calculated on the new price. Which expression represents the final cost of the item?
    Answer: $(1 + \frac{t}{100})(P(1 - \frac{d}{100}) - C)$ Let's build the expression step-by-step following the order of operations. First, discount the price $P$ by $d$ percent. The discounted price is $P - P(\frac{d}{100}) = P(1 - \frac{d}{100})$. Second, apply the coupon for $C$ dollars. This is a flat subtraction from the current price: $P(1 - \frac{d}{100}) - C$. Third, calculate the sales tax of $t$ percent on this new price. The total amount including tax is the new price plus the tax amount, which is $1 + \frac{t}{100}$ times the new price. Therefore, the final cost is $(P(1 - \frac{d}{100}) - C) \times (1 + \frac{t}{100})$, which matches choice A.
14. A chemical solution's temperature, initially at $T_0$ degrees Celsius, decreases by $k$ degrees per minute for $m$ minutes. It is then rapidly heated, causing its temperature to increase by a total amount equal to one-fifth of its temperature after the cooling phase. Which expression represents the final temperature in degrees Celsius?
    Answer: $1.2(T_0 - km)$ Let's model the temperature changes step-by-step. The initial temperature is $T_0$. It decreases by $k$ degrees per minute for $m$ minutes, so the total decrease is $km$. The temperature after the cooling phase is $T_{cooled} = T_0 - km$. Next, the temperature increases by an amount equal to one-fifth (or 0.2) of this cooled temperature. The increase is $0.2 \times T_{cooled} = 0.2(T_0 - km)$. The final temperature is the cooled temperature plus the increase: $T_{final} = T_{cooled} + 0.2 \times T_{cooled} = 1 \times T_{cooled} + 0.2 \times T_{cooled} = 1.2 \times T_{cooled}$. Substituting the expression for $T_{cooled}$, we get $T_{final} = 1.2(T_0 - km)$.
15. A company's profit can be modeled by the expression $500n - 15n^2 - 2000$, where $n$ is the number of units sold in hundreds. If the company sells 800 units, what is their profit in dollars?
    Answer: $2040$ The correct answer is C. Since n represents hundreds of units, 800 units = 8 hundreds, so n = 8. Substituting: 500(8) - 15(8)² - 2000 = 4000 - 15(64) - 2000 = 4000 - 960 - 2000 = 2040. Choice A results from using n = 800 instead of n = 8. Choice B makes a sign error with the constant term. Choice D combines both the wrong value of n and sign errors.
16. A specialized consulting firm calculates the total cost for a project, C(h), based on the number of hours, h. The cost includes a fixed initiation fee of $1,500, a tiered hourly labor charge, and a materials surcharge. The labor charge is $200 per hour for the first 40 hours, and 1.5 times that rate for each hour thereafter. The materials surcharge is 5% of the total labor charge. If a project takes h hours and h > 40, which expression represents the total cost?
    Answer: $315h - 2700$ First, calculate the total labor cost. The cost for the first 40 hours is $40 \times \$200 = \$8000$. The rate for hours beyond 40 is $1.5 \times \$200 = \$300$ per hour. The number of hours charged at this rate is $(h - 40)$. The cost for these additional hours is $300(h - 40)$. The total labor cost $L$ is $8000 + 300(h - 40) = 8000 + 300h - 12000 = 300h - 4000$. The materials surcharge is 5% of this total labor cost, which is $0.05(300h - 4000)$. The total project cost is the sum of the initiation fee, the total labor cost, and the surcharge: $C(h) = 1500 + (300h - 4000) + 0.05(300h - 4000)$. This can be simplified as $C(h) = 1500 + 1.05(300h - 4000) = 1500 + 315h - 4200 = 315h - 2700$.
17. A parking garage charges a flat rate of $3 for the first hour, then $1.50 for each additional hour or fraction thereof. If $h$ represents the total number of hours parked (where $h > 1$), which expression represents the total cost in dollars?
    Answer: $3 + 1.50(h - 1)$ The correct answer is B. The cost is $3 for the first hour plus $1.50 for each additional hour. Since there are (h-1) additional hours beyond the first hour, the expression is 3 + 1.50(h-1). Choice A incorrectly charges $1.50 for all h hours including the first. Choice C treats it as a simple rate problem. Choice D incorrectly multiplies the flat rate by h and adds the hourly rate once.
18. A ball is thrown upward from a height of 6 feet. Its height $h$ (in feet) after $t$ seconds is given by $h = -16t^2 + 32t + 6$. What is the height of the ball after 1.5 seconds?
    Answer: $18$ feet When you encounter a quadratic function describing motion, you're dealing with a substitution problem where you plug in the given time value to find the height. Given the height function $h = -16t^2 + 32t + 6$, you need to substitute $t = 1.5$ seconds and calculate step by step. First, calculate $t^2$: $(1.5)^2 = 2.25$ Now substitute into the equation: $h = -16(2.25) + 32(1.5) + 6$ $h = -36 + 48 + 6$ $h = 18$ feet Let's examine why the other answers are incorrect. Choice A (42 feet) likely comes from incorrectly adding all terms without proper signs: $16 + 32 + 6 = 54$, then making another error. Choice B (22 feet) might result from calculation errors in the multiplication steps. Choice C (30 feet) could come from forgetting the negative sign on the first term, giving $16(2.25) + 32(1.5) + 6 = 36 + 48 + 6 = 90$, then making another mistake. The correct answer is D (18 feet). Study tip: For quadratic motion problems, always work systematically: square the time value first, then multiply each term carefully, paying close attention to negative signs. The $-16t^2$ term represents gravity's effect and is always negative in these problems, so double-check that you're subtracting this value in your final calculation.
19. The expression $\frac{3x - 6}{x + 2}$ is undefined when $x = -2$. For what value of $x$ does this expression equal zero?
    Answer: $x = 2$ The correct answer is C. A rational expression equals zero when its numerator equals zero (and the denominator is non-zero). Setting 3x - 6 = 0: 3x = 6, so x = 2. When x = 2, the denominator x + 2 = 4 ≠ 0, so the expression is defined and equals zero. Choice A is where the expression is undefined. Choice B makes the denominator equal 2, not zero. Choice D comes from incorrectly solving 3x - 6 = 0 as x = 6.
20. An investor deposits a principal amount, $P$, into two different accounts. Three-fifths of the principal is deposited into Account A, which earns simple interest at an annual rate of $r_A$. The remainder of the principal is deposited into Account B, which earns simple interest at an annual rate of $r_B$. Which expression represents the total interest earned from both accounts after $t$ years?
    Answer: $P \cdot t \cdot (\frac{3}{5}r_A + \frac{2}{5}r_B)$ The formula for simple interest is $I = Prt$. First, consider Account A. The principal in this account is $\frac{3}{5}P$. The interest earned is $I_A = (\frac{3}{5}P) \cdot r_A \cdot t$. The remainder of the principal is $P - \frac{3}{5}P = \frac{2}{5}P$, which is deposited into Account B. The interest earned from Account B is $I_B = (\frac{2}{5}P) \cdot r_B \cdot t$. The total interest is the sum $I_A + I_B = (\frac{3}{5}P)r_A t + (\frac{2}{5}P)r_B t$. We can factor out the common terms $P$ and $t$ from this expression: $P \cdot t \cdot (\frac{3}{5}r_A + \frac{2}{5}r_B)$.

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