PSAT Math Quiz: Equations With One Variable

What this quiz covers

This quiz focuses on Equations With One Variable, giving you a quick way to practice the rules, question types, and explanations that matter most for PSAT Math.

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.

Question 1

1. $4$ (correct answer)
2. $6$
3. $28$
4. $\dfrac{46}{7}$

Explanation: The problem asks to find x from the phone plan equation $18 + 7x = 46$, where 18 is the base charge and $7x$ represents 7 gigabytes at x dollars each. Subtract 18 from both sides: $7x = 46 - 18 = 28$. Divide both sides by 7: $x = 28 \div 7 = 4$. A common mistake would be dividing the wrong terms or making arithmetic errors in subtraction. Always isolate the variable term first, then divide by its coefficient.

Question 2

1. $8$
2. $10$
3. $12$ (correct answer)
4. $18$

Explanation: This problem asks us to find the monthly fee $x$ when the total cost after 6 months equals $18 + 6x = 3x + 54$. To solve, we first subtract $3x$ from both sides: $18 + 6x - 3x = 54$, which simplifies to $18 + 3x = 54$. Next, we subtract 18 from both sides: $3x = 54 - 18 = 36$. Finally, dividing both sides by 3 gives us $x = 12$. A common error is incorrectly combining the $x$ terms or making arithmetic mistakes when subtracting. When solving equations with variables on both sides, always collect like terms on one side first.

Question 3

1. 300
2. 400
3. 450
4. 500 (correct answer)

Explanation: This problem tests your ability to set up and solve profit equations. When you see questions about company costs, revenues, and profits, remember that profit equals revenue minus costs. First, let's establish the key relationships. The cost function is $C(x) = 15x + 500$, where $x$ is the number of units produced. Since each unit sells for $25, the revenue function is $$R(x) = 25x$ $. Profit is calculated as$ $P(x) = R(x) - C(x) = 25x - (15x + 500) = 10x - 500$ $. To find when profit equals $4,500, set up the equation: $10x - 500 = 4500$. Adding 500 to both sides gives $10x = 5000$, so $x = 500$. This confirms answer choice D is correct. Let's examine why the other answers are wrong. Choice A (300 units) would yield a profit of $10(300) - 500 = 2500$, which is $2,000 short of the target. Choice B (400 units) gives$ $10(400) - 500 = 3500$ $, still$1,000 below the goal. Choice C (450 units) results in $10(450) - 500 = 4000$, which is $500 less than needed. Each incorrect answer represents a common calculation error: either mistakes in setting up the profit equation or arithmetic errors when solving. Study tip: Always write out the profit equation explicitly as Revenue - Costs before substituting numbers. This prevents confusion about which values represent income versus expenses, and double-check your arithmetic by substituting your answer back into the original profit equation.

Question 4

1. 8 cm
2. 12 cm (correct answer)
3. 20 cm
4. 40 cm

Explanation: This problem tests your ability to set up equations using perimeter formulas and solve for unknown variables. When you see problems involving equal perimeters of different shapes, focus on writing expressions for each perimeter and setting them equal. Let's define our variables. The rectangle has width $w$ and length $w + 4$. The pentagon has side length $w$ (equal to the rectangle's width). The perimeter of a rectangle is $2(\text{length} + \text{width})$, so our rectangle's perimeter is $2(w + 4 + w) = 2(2w + 4) = 4w + 8$. The perimeter of a regular pentagon is $5 \times \text{side length}$, so our pentagon's perimeter is $5w$. Since the perimeters are equal: $4w + 8 = 5w$ Solving for $w$: $8 = w$ Therefore, the rectangle's length is $w + 4 = 8 + 4 = 12$ cm. Looking at the wrong answers: Choice A (8 cm) gives you the width, not the length—a common mistake when solving multi-step problems. Choice C (20 cm) results from incorrectly setting up the pentagon's perimeter as $4w$ instead of $5w$, leading to $w = 12$ and length = 16, but this doesn't match any calculation. Choice D (40 cm) comes from computational errors in the equation setup. The correct answer is B. Strategy tip: In perimeter problems with multiple shapes, always write out the perimeter formula for each shape explicitly before setting up your equation. Double-check that you're solving for the right quantity—here, length, not width.

Question 5

1. $h = \frac{A}{2\pi r} - r$ (correct answer)
2. $h = \frac{A}{2\pi r} - r^2$
3. $h = \frac{A - 2\pi r^2}{2\pi}$
4. $h = A - r$

Explanation: When you encounter a formula that needs to be solved for a different variable, you're working with algebraic manipulation—a core skill tested throughout the PSAT Math section. To solve for $h$ in the surface area formula $A = 2\pi r^2 + 2\pi rh$, you need to isolate $h$ on one side. Start by subtracting $2\pi r^2$ from both sides: $A - 2\pi r^2 = 2\pi rh$. Then divide both sides by $2\pi r$ to get: $h = \frac{A - 2\pi r^2}{2\pi r}$. You can simplify this by splitting the fraction: $h = \frac{A}{2\pi r} - \frac{2\pi r^2}{2\pi r} = \frac{A}{2\pi r} - r$. This confirms that choice A is correct. Choice B contains a critical error: it shows $h = \frac{A}{2\pi r} - r^2$, which incorrectly simplifies $\frac{2\pi r^2}{2\pi r}$. When you divide $2\pi r^2$ by $2\pi r$, you get $r$, not $r^2$. Choice C gives $h = \frac{A - 2\pi r^2}{2\pi}$, which represents the intermediate step before dividing by $r$. This student forgot to complete the division by the full denominator $2\pi r$. Choice D shows $h = A - r$, which completely ignores the coefficients and demonstrates a fundamental misunderstanding of algebraic manipulation. Remember: when solving literal equations, perform the same operations you would with numbers, and always check your algebra by substituting back into the original equation.

Question 6

1. $4$
2. $5$
3. $6$ (correct answer)
4. $7$

Explanation: The problem describes a water tank draining equation 120 - 9x = 66, where 120 is initial volume, 9x is amount drained in 9 minutes at x liters per minute, and 66 is final volume. Subtract 120 from 66: -9x = 66 - 120 = -54. Divide by -9: x = 6. A common mistake would be incorrect subtraction or sign handling. When solving real-world problems, ensure the equation makes physical sense before solving.

Question 7

1. $12$
2. $13$
3. $14$
4. $15$ (correct answer)

Explanation: The problem describes a class collection equation 30 + 10x = 180, where 30 is a fixed fee and 10x represents 10 students at x dollars each. Subtract 30 from both sides: 10x = 150, then divide by 10: x = 15. A common mistake would be dividing by the wrong coefficient or making subtraction errors. When solving real-world problems, identify the fixed and variable components clearly before isolating the variable.

Question 8

1. $8$ (correct answer)
2. $9$
3. $7$
4. $\dfrac{21.5}{2.25}$

Explanation: The problem asks to solve the taxi fare equation 3.50 + 2.25x = 21.50 for x, where x represents miles traveled. Subtract 3.50 from both sides: 2.25x = 18. Divide both sides by 2.25: x = 18 ÷ 2.25 = 8. A common error would be incorrect decimal division or forgetting to subtract the base fare first. When solving real-world linear equations, identify the fixed cost and variable cost components before isolating the variable.

Question 9

1. $30$
2. $35$
3. $40$ (correct answer)
4. $120$

Explanation: The problem describes a contractor's billing equation $50 + 3x = 170$, where 50 is the visit fee and $3x$ represents 3 hours at x dollars per hour. Subtract 50 from both sides: $3x = 120$, then divide by 3: $x = 40$. A common mistake would be dividing by the wrong coefficient or making subtraction errors. When solving real-world linear equations, clearly identify the fixed and variable components before isolating the variable.

Question 10

1. $\dfrac{7}{24}$ (correct answer)
2. $\dfrac{1}{8}$
3. $\dfrac{7}{6}$
4. $\dfrac{7}{12}$

Explanation: The problem asks to solve 4x - 1/3 = 5/6 for x. Add 1/3 to both sides: 4x = 5/6 + 1/3. Convert to common denominator: 4x = 5/6 + 2/6 = 7/6. Divide by 4: x = 7/6 ÷ 4 = 7/24. A common error is incorrect fraction arithmetic when adding or dividing fractions. When combining fractions, always find a common denominator first, then divide carefully.

Question 11

1. $-13$ (correct answer)
2. $13$
3. $-7$
4. $7$

Explanation: The problem asks to solve $2x + 5 = \dfrac{3}{2}(x - 1)$ for x. First, multiply both sides by 2 to clear the fraction: $4x + 10 = 3(x - 1) = 3x - 3$. Subtract 3x from both sides: $x + 10 = -3$, then subtract 10: $x = -13$. A common mistake is not properly clearing fractions or making sign errors during distribution. When fractions appear in equations, multiply by the denominator to eliminate them before proceeding.

Question 12

1. $$$22.50
2. $$$30.00 (correct answer)
3. $$$37.50
4. $$$45.00

Explanation: When you encounter markup and discount problems, you're working with percentage changes applied in sequence. The key is to track how the price changes step by step, then work backwards from the final price. Let's trace through the price changes. The store owner starts with cost $c$, then marks it up 50%. This gives a selling price of $c + 0.50c = 1.50c$. Next comes a 20% discount off this selling price: $1.50c - 0.20(1.50c) = 1.50c(1 - 0.20) = 1.50c \times 0.80 = 1.20c$. Since the final price is $36, we have $$1.20c = 36$ $, so$ $c = 36 ÷ 1.20 = 30$ $. The original cost was $30. Looking at the wrong answers: Choice A ($22.50) represents a common error where students might subtract both percentages from 100$37.50) could result from confusing which percentage to apply when, perhaps treating the discount as applying to the original cost. Choice D ($45.00) might come from incorrectly adding the percentages or applying them in the wrong direction. The correct answer is B. Study tip: For sequential percentage problems, convert each percentage to a multiplier (50% markup means multiply by 1.50, 20% discount means multiply by 0.80), then combine them before solving. This prevents calculation errors and makes the algebra cleaner.

Question 13

1. $2$
2. $3$ (correct answer)
3. $5$
4. $15$

Explanation: The problem describes a streaming service billing equation 9 + 5x = 24, where 9 is the base fee and 5x represents 5 movies at x dollars each. Subtract 9 from both sides: 5x = 15, then divide by 5: x = 3. A common mistake would be dividing by the wrong coefficient or making arithmetic errors in subtraction. When solving real-world problems, identify the fixed and variable costs before isolating the variable.

Question 14

1. $16$
2. $20$
3. $24$ (correct answer)
4. $30$

Explanation: We need to solve $12 + 0.75x = 0.5x + 18$ for the number of gigabytes $x$. Subtracting $0.5x$ from both sides: $12 + 0.75x - 0.5x = 18$, which simplifies to $12 + 0.25x = 18$. Subtracting 12 from both sides: $0.25x = 6$. Dividing by 0.25: $x = 6 ÷ 0.25 = 24$. A common error is making arithmetic mistakes with decimals or incorrectly combining the $x$ terms. When working with decimal coefficients, consider converting to fractions to avoid calculation errors.

Question 15

1. $17$ (correct answer)
2. $-17$
3. $13$
4. $11$

Explanation: The problem asks to solve (2x - 1)/3 = (x + 5)/2 for x. Cross-multiply to get 2(2x - 1) = 3(x + 5), which gives 4x - 2 = 3x + 15. Subtract 3x from both sides: x - 2 = 15, then add 2: x = 17. A common error is making mistakes during cross-multiplication or not properly distributing. When solving rational equations, cross-multiplication is often the most efficient method to eliminate fractions.

Question 16

1. $33$ (correct answer)
2. $57$
3. $45$
4. $12$

Explanation: The problem describes a lab mixture equation x + 12 = 45, where x is the initial amount and 12 mL is added to get 45 mL total. Subtract 12 from both sides: x = 45 - 12 = 33. A common mistake would be adding 12 instead of subtracting or making arithmetic errors. This is a simple one-step equation requiring only subtraction to isolate the variable.

Question 17

1. $\dfrac{11}{10}$
2. $\dfrac{23}{10}$ (correct answer)
3. $\dfrac{11}{5}$
4. $\dfrac{23}{5}$

Explanation: The problem asks to solve 7x - 3 = 2x + 17/2 for x. Subtract 2x from both sides: 5x - 3 = 17/2, then add 3: 5x = 17/2 + 6/2 = 23/2. Finally, divide by 5: x = 23/10. A common error is incorrect fraction arithmetic when combining terms. When adding fractions to whole numbers, convert the whole number to a fraction with the same denominator first.

Question 18

1. $-\frac{1}{3}$
2. 1
3. 5 (correct answer)
4. 25

Explanation: When you encounter an equation with fractions, your goal is to clear the denominators and solve for the variable systematically. To solve $\frac{x+1}{2} + \frac{x+2}{3} = \frac{16}{3}$, find a common denominator for the left side. The LCD of 2 and 3 is 6, so multiply each term appropriately: $\frac{3(x+1)}{6} + \frac{2(x+2)}{6} = \frac{16}{3}$ This simplifies to: $\frac{3x+3+2x+4}{6} = \frac{16}{3}$ $\frac{5x+7}{6} = \frac{16}{3}$ Cross-multiply to eliminate fractions: $3(5x+7) = 6 \cdot 16$ $15x + 21 = 96$ $15x = 75$ $x = 5$ You can verify: $\frac{5+1}{2} + \frac{5+2}{3} = \frac{6}{2} + \frac{7}{3} = 3 + \frac{7}{3} = \frac{16}{3}$ ✓ Choice A ($-\frac{1}{3}$) likely results from sign errors during fraction manipulation. Choice B (1) might come from incorrectly combining like terms or making arithmetic mistakes when finding the common denominator. Choice D (25) could result from errors in cross-multiplication, such as multiplying $15x = 75$ incorrectly or confusing the final division step. Strategy tip: When solving equations with multiple fractions, always find a common denominator first, then cross-multiply to eliminate fractions entirely. This reduces the chance of arithmetic errors and makes the algebra more straightforward. Always substitute your answer back into the original equation to verify it works.

Question 19

1. 3
2. $\frac{16}{7}$
3. $\frac{16}{3}$
4. $\frac{12}{7}$ (correct answer)

Explanation: This equation requires careful algebraic manipulation, particularly when dealing with nested parentheses. The key is to work systematically from the inside out, then combine like terms to isolate the variable. Start with the left side: $3(x - 2(x + 1))$. First, distribute the $-2$ inside the inner parentheses: $x - 2(x + 1) = x - 2x - 2 = -x - 2$. Now distribute the $3$: $3(-x - 2) = -3x - 6$. For the right side: $4(x - 5) + 2 = 4x - 20 + 2 = 4x - 18$. The equation becomes: $-3x - 6 = 4x - 18$ Collect like terms by adding $3x$ to both sides: $-6 = 7x - 18$ Add $18$ to both sides: $12 = 7x$ Therefore: $x = \frac{12}{7}$ Choice A (3) likely comes from making errors in the distribution process and getting a simpler equation. Choice B ($\frac{16}{7}$) probably results from a sign error when combining the constant terms—getting $16 = 7x$ instead of $12 = 7x$. Choice C ($\frac{16}{3}$) suggests confusion in both the coefficient of $x$ and the constant term, possibly mixing up the $7$ and $3$ from the distribution steps. When solving multi-step equations with nested parentheses, always distribute from the innermost parentheses outward, then carefully track positive and negative signs as you combine like terms. Double-check by substituting your answer back into the original equation.

Question 20

1. -2
2. 0
3. 2 (correct answer)
4. 4

Explanation: When you encounter an equation with a parameter and are asked when it has "no solution," you're dealing with a situation where the equation becomes contradictory—like $3 = 5$. Let's simplify both sides of $5(x - 2) - x = 2(ax - 3)$ and see what happens. On the left side: $5(x - 2) - x = 5x - 10 - x = 4x - 10$. On the right side: $2(ax - 3) = 2ax - 6$. So our equation becomes $4x - 10 = 2ax - 6$. Rearranging to get all terms on one side: $4x - 2ax = -6 + 10$, which gives us $(4 - 2a)x = 4$. For this equation to have a unique solution, we'd divide both sides by $(4 - 2a)$. But if $4 - 2a = 0$, we can't divide, and we get $0 \cdot x = 4$, or $0 = 4$—which is impossible. This happens when $4 - 2a = 0$, so $a = 2$. Looking at the wrong answers: Choice A (-2) would give us $8x = 4$, which has solution $x = \frac{1}{2}$. Choice B (0) would give us $4x = 4$, so $x = 1$. Choice D (4) would give us $-4x = 4$, so $x = -1$. All of these produce valid solutions. Only choice C creates the contradiction that results in no solution. Strategy tip: When asked about "no solution" scenarios, look for cases where you end up with a false statement like $0 = \text{nonzero number}$ after simplifying.