One-Step Equations - SSAT Middle Level Quantitative

Maria bought a notebook and spent $3 more than twice what she had planned to spend. If she actually spent $15, how much had she planned to spend?
Answer: $6 This is a classic "working backwards" word problem that tests your ability to translate English into algebra. When you see phrases like "more than twice," you need to carefully set up an equation based on what actually happened. Let's call the planned amount $p$ . The problem states Maria spent "$3 more than twice what she had planned," which translates to $2p + 3$ . Since she actually spent $15, we can write: $2p + 3 = 15$ Solving for $p$ : $2p + 3 = 15$ $2p = 12$ $p = 6$ So Maria had planned to spend $6, making (A) correct. Let's check why the other answers don't work. If you choose (B) $9, then twice that plus $3 would be $2(9) + 3 = 21$ , not $15. For (C) $12, you get $2(12) + 3 = 27$ , which is way too high. Answer (D) $18 gives you $2(18) + 3 = 39$ , which is even further off. The most common mistake here is confusing the relationship. Some students might think "she spent $3 more" means you add $3 to the actual amount spent ($15 + $3 = $18), leading them to pick (D). Others might subtract $3 from $15 to get $12, choosing (C). Strategy tip: Always define your variable as what the question asks for (here, "planned to spend"), then translate the word problem into an equation step by step. Don't try to shortcut the algebra—write it out clearly to avoid mix-ups with the relationships.
Six identical water bottles cost $42$ ; solve $6x = 42$ .
Answer: 7 This question tests middle-level algebra skills: solving one-step equations with a variable. Understanding one-step equations involves performing inverse operations to isolate the variable. For example, if 6x = 12, divide both sides by 6 to solve for x. In this specific question, the equation 6x = 42 requires dividing both sides by 6, resulting in x = 7. Choice C is correct because it accurately reflects the solution to the equation. Choice A is incorrect as it results from multiplying instead of dividing, a common mistake in costs. To help students: Encourage practicing inverse operations, using bottle purchases for visualization, and checking solutions by substituting back into the original equation. Watch for: misunderstanding inverse operations, and ensure clarity in distinguishing between multiplication and division.
If $\frac{k + 6}{2} = 9$ , what is the value of $k$ ?
Answer: 12 This problem tests your ability to solve one-step equations involving fractions. When you see an equation with a variable in a fraction, your goal is to isolate the variable by undoing the operations in reverse order. Starting with $\frac{k + 6}{2} = 9$ , you need to eliminate the fraction first. Since $(k + 6)$ is divided by 2, multiply both sides by 2 to cancel out the denominator: $k + 6 = 18$ . Now subtract 6 from both sides to isolate $k$ : $k = 12$ . Let's verify: $\frac{12 + 6}{2} = \frac{18}{2} = 9$ ✓ Looking at the wrong answers: Choice (A) 3 likely comes from incorrectly solving $k + 6 = 9$ without first multiplying by 2 to clear the fraction. Choice (C) 15 might result from adding 6 instead of subtracting it in the final step, giving $k = 18 - 6 + 3 = 15$ through confused arithmetic. Choice (D) 18 comes from stopping too early—this is the value of $k + 6$ , not $k$ itself. The key strategy for fraction equations is to clear denominators first by multiplying both sides by the denominator, then solve the resulting simpler equation. Always check your answer by substituting back into the original equation. This verification step catches most algebraic mistakes and builds confidence in your solution.
If $4(p + 3) = 28$ , what is the value of $p$ ?
Answer: 4 This is a linear equation that you solve by isolating the variable $p$ using inverse operations. When you see an equation like this, work systematically to "undo" each operation in reverse order. Starting with $4(p + 3) = 28$ , first divide both sides by 4 to eliminate the coefficient: $p + 3 = 7$ . Then subtract 3 from both sides to isolate $p$ : $p = 4$ . You can verify this by substituting back: $4(4 + 3) = 4(7) = 28$ ✓ Looking at the wrong answers: Choice B (7) is what you get if you stop after the first step and forget to subtract 3 from $p + 3 = 7$ . This is a common error when students see that 7 appears in their work and mistakenly think it's the final answer. Choice C (10) results from adding 3 instead of subtracting it in the final step, giving $p = 7 + 3 = 10$ . Choice D (16) comes from incorrectly distributing or making an arithmetic error, possibly thinking $28 - 4 \times 3 = 16$ . Remember to work through multi-step equations methodically: eliminate multiplication/division first, then handle addition/subtraction. Always check your answer by substituting it back into the original equation. This verification step catches most algebraic mistakes and builds confidence in your solution.
Four identical bracelets cost $32$ ; solve $4x = 32$ .
Answer: 8 This question tests middle-level algebra skills: solving one-step equations with a variable. Understanding one-step equations involves performing inverse operations to isolate the variable. For example, if 4x = 16, divide both sides by 4 to solve for x. In this specific question, the equation 4x = 32 requires dividing both sides by 4, resulting in x = 8. Choice C is correct because it accurately reflects the solution to the equation. Choice A is incorrect as it results from multiplying instead of dividing, a common mistake with even numbers. To help students: Encourage practicing inverse operations, using group purchases for visualization, and checking solutions by substituting back into the original equation. Watch for: misunderstanding inverse operations, and ensure clarity in distinguishing between multiplication and division.
A runner jogs 72 miles in eight hours; solve $8x = 72$ .
Answer: 9 This question tests middle-level algebra skills: solving one-step equations with a variable. Understanding one-step equations involves performing inverse operations to isolate the variable. For example, if 8x = 40, divide both sides by 8 to solve for x. In this specific question, the equation 8x = 72 requires dividing both sides by 8, resulting in x = 9. Choice B is correct because it accurately reflects the solution to the equation. Choice C is incorrect as it results from multiplying instead of dividing, a common mistake when students invert operations. To help students: Encourage practicing inverse operations, using distance and time for visualization, and checking solutions by substituting back into the original equation. Watch for: misunderstanding inverse operations, and ensure clarity in distinguishing between multiplication and division.
Seven comic books cost $49$ ; solve $7x = 49$ .
Answer: 7 This question tests middle-level algebra skills: solving one-step equations with a variable. Understanding one-step equations involves performing inverse operations to isolate the variable. For example, if 7x = 21, divide both sides by 7 to solve for x. In this specific question, the equation 7x = 49 requires dividing both sides by 7, resulting in x = 7. Choice C is correct because it accurately reflects the solution to the equation. Choice B is incorrect as it results from multiplying instead of dividing, a common mistake in equal costs. To help students: Encourage practicing inverse operations, using collection items for visualization, and checking solutions by substituting back into the original equation. Watch for: misunderstanding inverse operations, and ensure clarity in distinguishing between multiplication and division.
A car travels 150 miles in five hours; solve $5x = 150$ .
Answer: 30 This question tests middle-level algebra skills: solving one-step equations with a variable. Understanding one-step equations involves performing inverse operations to isolate the variable. For example, if 5x = 25, divide both sides by 5 to solve for x. In this specific question, the equation 5x = 150 requires dividing both sides by 5, resulting in x = 30. Choice C is correct because it accurately reflects the solution to the equation. Choice B is incorrect as it results from adding instead of dividing, a common mistake in travel problems. To help students: Encourage practicing inverse operations, using speed calculations for visualization, and checking solutions by substituting back into the original equation. Watch for: misunderstanding inverse operations, and ensure clarity in distinguishing between multiplication and division.
A recipe uses 18 cups for nine servings; solve $9x = 18$ .
Answer: 2 This question tests middle-level algebra skills: solving one-step equations with a variable. Understanding one-step equations involves performing inverse operations to isolate the variable. For example, if 9x = 9, divide both sides by 9 to solve for x. In this specific question, the equation 9x = 18 requires dividing both sides by 9, resulting in x = 2. Choice A is correct because it accurately reflects the solution to the equation. Choice B is incorrect as it results from multiplying instead of dividing, a common mistake in recipes. To help students: Encourage practicing inverse operations, using serving sizes for visualization, and checking solutions by substituting back into the original equation. Watch for: misunderstanding inverse operations, and ensure clarity in distinguishing between multiplication and division.
A recipe calls for 4 cups of flour to make 12 muffins. If Lisa wants to make 18 muffins, how many cups of flour does she need?
Answer: 6 cups This is a proportion problem where you need to find how much of one ingredient you need when scaling a recipe up or down. When you see questions about recipes, rates, or "per unit" relationships, think about setting up proportions to maintain the same ratio. The original recipe uses 4 cups of flour for 12 muffins. To find how much flour Lisa needs for 18 muffins, set up a proportion: $\frac{4 \text{ cups}}{12 \text{ muffins}} = \frac{x \text{ cups}}{18 \text{ muffins}}$ Cross multiply: $4 \times 18 = 12 \times x$ , which gives you $72 = 12x$ . Dividing both sides by 12: $x = 6$ cups. You can also think of this as finding the rate first: 4 cups ÷ 12 muffins = $\frac{1}{3}$ cup per muffin. Then multiply: $\frac{1}{3} \times 18 = 6$ cups. Looking at the wrong answers: B) 8 cups comes from incorrectly adding 4 cups to the original amount without considering the actual ratio needed. C) 14 cups might result from adding 18 - 12 = 6 to the original 4 cups plus some calculation error. D) 22 cups is far too large and suggests multiplying instead of using proportional reasoning. Study tip: For proportion problems on the SSAT, always check if your answer makes sense. Since Lisa is making 1.5 times as many muffins (18 ÷ 12 = 1.5), she should need 1.5 times as much flour: 4 × 1.5 = 6 cups. This "scaling factor" approach often provides a quick way to verify your answer.
A store sells five notebooks for $25$ ; solve $5x = 25$ .
Answer: 5 This question tests middle-level algebra skills: solving one-step equations with a variable. Understanding one-step equations involves performing inverse operations to isolate the variable. For example, if 5x = 10, divide both sides by 5 to solve for x. In this specific question, the equation 5x = 25 requires dividing both sides by 5, resulting in x = 5. Choice C is correct because it accurately reflects the solution to the equation. Choice D is incorrect as it results from multiplying instead of dividing, a common mistake in cost problems. To help students: Encourage practicing inverse operations, using shopping contexts for visualization, and checking solutions by substituting back into the original equation. Watch for: misunderstanding inverse operations, and ensure clarity in distinguishing between multiplication and division.
A car goes 180 miles in six hours; solve $6x = 180$ .
Answer: 30 This question tests middle-level algebra skills: solving one-step equations with a variable. Understanding one-step equations involves performing inverse operations to isolate the variable. For example, if 6x = 30, divide both sides by 6 to solve for x. In this specific question, the equation 6x = 180 requires dividing both sides by 6, resulting in x = 30. Choice A is correct because it accurately reflects the solution to the equation. Choice B is incorrect as it results from adding instead of dividing, a common mistake when students misapply operations. To help students: Encourage practicing inverse operations, using real-world speed contexts for visualization, and checking solutions by substituting back into the original equation. Watch for: misunderstanding inverse operations, and ensure clarity in distinguishing between multiplication and division.
Three movie tickets cost $27$ ; solve $3x = 27$ .
Answer: 9 This question tests middle-level algebra skills: solving one-step equations with a variable. Understanding one-step equations involves performing inverse operations to isolate the variable. For example, if 3x = 12, divide both sides by 3 to solve for x. In this specific question, the equation 3x = 27 requires dividing both sides by 3, resulting in x = 9. Choice C is correct because it accurately reflects the solution to the equation. Choice A is incorrect as it results from multiplying instead of dividing, a common mistake when students confuse operations. To help students: Encourage practicing inverse operations, using multiplication tables for visualization, and checking solutions by substituting back into the original equation. Watch for: misunderstanding inverse operations, and ensure clarity in distinguishing between multiplication and division.
A team earns $84$ from $x$ tickets at $7$ each: $7x = 84$ .
Answer: 12 This question tests middle-level algebra skills: solving one-step equations with a variable. Understanding one-step equations involves performing inverse operations to isolate the variable. For example, if 7x = 14, divide both sides by 7 to solve for x. In this specific question, the equation 7x = 84 requires dividing both sides by 7, resulting in x = 12. Choice C is correct because it accurately reflects the solution to the equation. Choice B is incorrect as it results from multiplying instead of dividing, a common mistake in team earnings. To help students: Encourage practicing inverse operations, using ticket earnings for visualization, and checking solutions by substituting back into the original equation. Watch for: misunderstanding inverse operations, and ensure clarity in distinguishing between multiplication and division.
A store sells books for $8 each. After buying some books, Jake has $17 left from his original $65. How many books did Jake buy?
Answer: 6 books This is a classic "working backwards" word problem that tests your ability to set up and solve equations from real-world scenarios. When you see problems involving money spent and money remaining, think about the relationship: starting amount - amount spent = amount left. Let's work through this step by step. Jake started with $65 and ended with $17, so he spent $65 - $17 = $48 on books. Since each book costs $8, you divide the total spent by the price per book: $48 ÷ $8 = 6 books. You can verify this: 6 books × $8 each = $48 spent, and $65 - $48 = $17 remaining. ✓ Now let's examine why each answer choice works or fails: A) 6 books is correct, as we just calculated. B) 8 books would cost $8 × 8 = $64, leaving Jake with $65 - $64 = $1, not $17. C) 48 books would cost $8 × 48 = $384, which exceeds Jake's original $65 entirely. D) 82 books would cost $8 × 82 = $656, also far beyond Jake's budget. Notice how choices C and D represent common mistakes: C uses the dollar amount spent ($48) as the number of books, while D combines the starting amount and remaining amount (65 + 17 = 82). Study tip: For money problems, always identify what you're looking for first, then set up the equation. Remember the basic relationship: initial amount - spent = remaining. This framework works for any "purchase and change" scenario you'll encounter.
Three times a number decreased by 5 equals 19. What is the number?
Answer: 8 When you see "three times a number decreased by 5 equals 19," you're dealing with translating words into algebra. This type of word problem tests your ability to set up and solve equations systematically. Let's call the unknown number $x$ . Breaking down the phrase: "three times a number" means $3x$ , "decreased by 5" means subtract 5, so we get $3x - 5$ . Since this "equals 19," our equation is $3x - 5 = 19$ . To solve, add 5 to both sides: $3x - 5 + 5 = 19 + 5$ , which gives us $3x = 24$ . Dividing both sides by 3: $x = 8$ . Let's verify: three times 8 is 24, decreased by 5 gives us 19. ✓ Looking at the wrong answers: Choice A (6) comes from incorrectly setting up the equation as $3x + 5 = 19$ instead of $3x - 5 = 19$ . This gives $3x = 14$ , so $x = 4.67$ , but if you round or make calculation errors, you might pick 6. Choice C (14) results from forgetting to divide by 3 after finding $3x = 24$ . Choice D (24) happens when you solve for $3x$ but forget that the question asks for $x$ , not $3x$ . The key strategy here is to translate the words methodically into mathematical symbols, then solve step by step. Always check your answer by substituting it back into the original word problem to ensure it makes sense.
Tom's age in 6 years will be 21. How old is Tom now?
Answer: 15 years old Age problems involve working backward from future information to find present values. When you see a question telling you someone's age at a future time, you need to subtract the number of years to find their current age. Let's set up the problem systematically. If Tom's age in 6 years will be 21, we can write this as: Tom's current age + 6 = 21. To find Tom's current age, we subtract 6 from both sides: Tom's current age = 21 - 6 = 15 years old. We can verify this: if Tom is currently 15, then in 6 years he'll be 15 + 6 = 21 years old, which matches the given information. Looking at the wrong answers: Choice B (16 years old) is close but represents adding one extra year — if Tom were 16 now, he'd be 22 in 6 years, not 21. Choice C (21 years old) is the trap of using his future age as his current age; if he's 21 now, he'd be 27 in 6 years. Choice D (27 years old) comes from mistakenly adding 6 to 21 instead of subtracting; this would make Tom 33 years old in 6 years. For age problems, always identify what time period you're given information about, then work backward or forward as needed. When the problem states "age in X years will be Y," use the formula: current age = Y - X. Double-check by adding the years back to your answer.
Sarah has some stickers. After giving away 15 stickers, she has 42 stickers left. How many stickers did Sarah start with?
Answer: 57 stickers This is a classic "working backwards" word problem that tests your ability to set up and solve a simple equation. When you see phrases like "after giving away" or "has left," you need to think about the relationship between the starting amount, what changed, and the final result. Let's work through this step by step. Sarah started with some unknown number of stickers, gave away 15, and ended up with 42. If we call her starting amount $x$ , then: $x - 15 = 42$ . To find $x$ , we add 15 to both sides: $x = 42 + 15 = 57$ stickers. You can verify this makes sense: if Sarah started with 57 stickers and gave away 15, she would have $57 - 15 = 42$ stickers remaining, which matches the problem. Looking at the wrong answers: Choice A (27) represents the common error of subtracting instead of adding—students might think $42 - 15 = 27$ , but this doesn't answer what we're looking for. Choice C (630) is the result of multiplying $42 \times 15$ , which doesn't make logical sense in this context. Choice D (84) comes from adding $42 + 42$ , perhaps from misreading the problem or confusing the numbers involved. Strategy tip: For "working backwards" problems, always check your answer by plugging it back into the original situation. This verification step catches most calculation errors and helps confirm you set up the problem correctly. The key phrase "after giving away... left" should immediately signal addition to reverse the subtraction that occurred.
A taxi fare costs $3.50 plus $0.75 for each mile traveled. If the total fare was $11.75, how many miles were traveled?
Answer: 11 miles When you encounter a taxi fare problem like this, you're dealing with a linear equation where there's a fixed cost plus a variable cost per unit. The key is to set up an equation that separates the base fare from the per-mile charge. Let's call the number of miles traveled $x$ . The total fare equation is: $3.50 + 0.75x = 11.75$ To solve for $x$ , first subtract the base fare from both sides: $0.75x = 11.75 - 3.50 = 8.25$ Then divide by the per-mile rate: $x = \frac{8.25}{0.75} = 11$ miles This confirms that choice B (11 miles) is correct. Let's examine why the other options are wrong. Choice A (8.8 miles) would give you a total fare of $3.50 + 0.75(8.8) = 10.10$ , which is too low. Choice C (15.67 miles) would cost $3.50 + 0.75(15.67) = 15.25$ , which exceeds the actual fare. Choice D (20.33 miles) would result in $3.50 + 0.75(20.33) = 18.75$ , which is way too high. These incorrect answers likely come from common mistakes: mixing up which values to add or subtract, or incorrectly setting up the equation by forgetting about the base fare. Strategy tip: For any "fixed cost plus variable cost" problem, always isolate the variable portion first by subtracting the fixed cost from the total, then divide by the rate. Double-check your answer by plugging it back into the original equation.
If $-3x + 7 = -8$ , what is the value of $x$ ?
Answer: 5 This is a linear equation that requires you to isolate the variable $x$ using inverse operations. When you see an equation like $-3x + 7 = -8$ , your goal is to get $x$ by itself on one side. Start by subtracting 7 from both sides to eliminate the constant term: $-3x + 7 - 7 = -8 - 7$ , which simplifies to $-3x = -15$ . Next, divide both sides by $-3$ to isolate $x$ : $\frac{-3x}{-3} = \frac{-15}{-3}$ . This gives you $x = 5$ . You can verify this by substituting back: $-3(5) + 7 = -15 + 7 = -8$ ✓ Looking at the wrong answers: Choice (A) gives $-5$ , which occurs if you forget that dividing two negative numbers yields a positive result—a common sign error. Choice (B) gives $\frac{1}{3}$ , which suggests confusion about the order of operations or possibly dividing $-3$ by $-15$ instead of $-15$ by $-3$ . Choice (D) gives $15$ , which happens if you correctly get $-3x = -15$ but then incorrectly think $x = 15$ by dropping the negative signs entirely. The correct answer is (C) $5$ . Study tip: Always work systematically with linear equations—handle addition/subtraction first, then multiplication/division. Pay special attention to negative signs, and always substitute your answer back into the original equation to check your work. This catches most algebraic mistakes before you submit your answer.

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