A sequence of numbers is generated by the rule (3n - 5), where (n) is the position of the number in the sequence. Which term in the sequence has a value of 49?
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ISEE Middle Level Quantitative Reasoning Quiz
Practice One And Two Step Equations in ISEE Middle Level Quantitative Reasoning with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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A sequence of numbers is generated by the rule (3n - 5), where (n) is the position of the number in the sequence. Which term in the sequence has a value of 49?
This quiz focuses on One And Two Step Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Quantitative Reasoning.
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.
A sequence of numbers is generated by the rule (3n - 5), where (n) is the position of the number in the sequence. Which term in the sequence has a value of 49?
Explanation: We are looking for the value of (n) when the term's value is 49. Set up the equation: (3n - 5 = 49). Add 5 to both sides: (3n = 54). Divide by 3: (n = 18). So, the 18th term in the sequence has a value of 49. Distractor C is the intermediate value of (3n). Distractor D is the value of the 49th term ((3(49) - 5 = 147 - 5 = 142)). Distractor A is a result of an estimation or calculation error.
The perimeter of a rectangle is 54 cm. The length of the rectangle is 3 cm more than twice its width. What is the width of the rectangle in centimeters?
Explanation: Let (w) be the width. The length (l) is (2w + 3). The formula for the perimeter of a rectangle is (P = 2(l + w)). Substitute the given values and expressions: (54 = 2((2w + 3) + w)). Simplify the expression inside the parentheses: (54 = 2(3w + 3)). Distribute the 2: (54 = 6w + 6). Subtract 6 from both sides: (48 = 6w). Divide by 6: (w = 8). The width is 8 cm. Distractor B (17) comes from using (l + w = 54) instead of (2(l + w) = 54). Distractor C (19) is the length. Distractor D (25.5) comes from solving (2w + 3 = 54) directly.
When a number (k) is divided by 4 and then 9 is added to the result, the answer is 15. What is the value of (k - 2)?
Explanation: First, set up the equation to find (k): (\frac{k}{4} + 9 = 15). Subtract 9 from both sides: (\frac{k}{4} = 6). Multiply both sides by 4: (k = 24). The question asks for the value of (k - 2), not (k). So, substitute the value of (k): (24 - 2 = 22). Distractor B is the value of (k). Distractor C results from misinterpreting the order of operations as (\frac{k+9}{4}=15), which gives (k+9=60) and (k=51), so (k-2=49). Distractor D results from adding 9 instead of subtracting: (\frac{k}{4}=15+9=24), so (k=96) and (k-2=94).
The sum of three consecutive odd integers is 87. What is the largest of these three integers?
Explanation: Let the three consecutive odd integers be (n), (n+2), and (n+4). Their sum is (n + (n+2) + (n+4) = 87). Combining like terms gives (3n + 6 = 87). Subtracting 6 from both sides gives (3n = 81). Dividing by 3 gives (n = 27). This is the smallest of the three integers. The integers are 27, 29, and 31. The largest is 31. Distractor A is the smallest integer. Distractor B is the middle integer (which is also the average, (87 \div 3 = 29)). Distractor C is incorrect because the integer must be odd.
The result of dividing a number (p) by -6 is 8. What is the result of dividing (p) by 4?
Explanation: First, find the value of (p). The problem states (\frac{p}{-6} = 8). To solve for (p), multiply both sides by -6: (p = 8 \times (-6) = -48). Next, divide (p) by 4: (\frac{-48}{4} = -12). The final result is -12. Distractor C is the value of (p). Distractor A is the result of a sign error when finding (p) (if (p=48)) or when performing the final division. Distractor D does not follow from a likely calculation error.
A store owner buys a tablet for a certain cost, (c). To set the selling price, she marks up the cost by 25%. If the final selling price is $300, what was the original cost, (c)?
Explanation: The selling price is the original cost plus a 25% markup. This can be written as (c + 0.25c = 300), or (1.25c = 300). To find the original cost (c), divide the selling price by 1.25: (c = 300 \div 1.25 = 240). The original cost was $240. Distractor A is a common error where one finds 25% of the selling price ((0.25 \times 300 = 75)) and subtracts it from the selling price ((300 - 75 = 225)). Distractor D is another common error where one finds 25% of the cost and adds it to the selling price.
The formula to convert temperature from Celsius (C) to Fahrenheit (F) is (F = \frac{9}{5}C + 32). If the temperature is 86°F, what is the temperature in degrees Celsius?
Explanation: Substitute 86 for F in the formula: (86 = \frac{9}{5}C + 32). First, subtract 32 from both sides: (86 - 32 = \frac{9}{5}C), which simplifies to (54 = \frac{9}{5}C). To solve for C, multiply both sides by the reciprocal of (\frac{9}{5}), which is (\frac{5}{9}): (C = 54 \times \frac{5}{9}). This can be calculated as ((54 \div 9) \times 5 = 6 \times 5 = 30). The temperature is 30°C. Distractor C is the intermediate result before multiplying by the reciprocal. Distractor D is the result of multiplying 54 by (\frac{9}{5}) instead of (\frac{5}{9}).
An isosceles triangle has a perimeter of 39 inches. The base is 6 inches longer than each of the two equal sides. What is the length of the base?
Explanation: Let (s) be the length of each of the two equal sides. The base (b) is (s + 6). The perimeter is the sum of the three sides: (s + s + b = 39). Substitute the expression for (b): (s + s + (s + 6) = 39). Combine like terms: (3s + 6 = 39). Subtract 6 from both sides: (3s = 33). Divide by 3: (s = 11). This is the length of an equal side. The question asks for the length of the base, which is (s + 6 = 11 + 6 = 17) inches. Distractor A is the length of one of the equal sides. Distractors B and D are calculation errors.
A number (n) is decreased by 6, and the result is multiplied by -3. The final result is 24.
Column A (n)
Column B -2
Explanation: First, set up the equation based on the information given: (-3(n - 6) = 24). To solve for (n), first divide both sides by -3: (n - 6 = 24 \div (-3)), which simplifies to (n - 6 = -8). Then, add 6 to both sides: (n = -8 + 6), which gives (n = -2). Therefore, the value in Column A is -2. The value in Column B is also -2. The two quantities are equal. A common mistake is a sign error in distribution ((-3n - 18 = 24 \implies -3n = 42 \implies n = -14), leading to B) or in solving ((n - 6 = -8 \implies n = -14), leading to B).
A phone company charges 0.20forthefirstminuteofacalland0.08 for each additional minute. If a call costs $1.24, what was the total duration of the call in minutes?
Explanation: Let (m) be the total number of minutes. The cost is for the first minute plus the additional minutes. The cost of the additional minutes is (1.24−0.20 = 1.04\). The number of additional minutes is the cost of additional minutes divided by the rate: \(1.04 \div 0.08 = 13\). These are the additional minutes. The total duration is the first minute plus the additional minutes: \(1 + 13 = 14\) minutes. Distractor B is the number of additional minutes, not the total duration. Distractor A is an estimation error. Distractor D is the result of dividing the total cost by the additional minute rate \(1.24 / 0.08 = 15.5), which ignores the special rate for the first minute.
Which of the following situations is best modeled by the equation (25 - 2x = 9)?
Explanation: The equation (25 - 2x = 9) represents a starting amount of 25, from which an amount of (2x) is subtracted, leaving a result of 9. Choice C matches this perfectly: Mark starts with 25, the cost of the two books is \(2x\), and his remaining change is 9. Choice A is modeled by (9 + 2x = 25). Choice B is not modeled correctly by any simple equation provided. Choice D is modeled by (25 - x = 9) if she gave away x cookies total, or (25-2x=9) if she gave x cookies to each of two friends. The phrasing in D is ambiguous, while C is precise and correct.
Jamal owes his friend 60.Hepaysher15 at the end of the first week, and then $5 each week after that until the debt is fully paid. How many weeks in total will it take him to pay off the entire debt?
Explanation: After the first week's payment of 15, Jamal has \(60 - 15=45) left to pay. He pays this remaining amount at a rate of 5perweek.Thenumberof5 payments needed is (45÷5 = 9) weeks. The total time is the first week (for the 15payment)plusthe9subsequentweeks(forthe5 payments). So, (1 + 9 = 10) weeks in total. Distractor A is the number of weeks after the first payment. Distractor D is from dividing the total debt by the smaller payment amount ((60 \div 5 = 12)).
A pizza is cut into 8 equal slices. After Maya and her friends eat some slices, (\frac{3}{8}) of the pizza remains. Let (n) be the number of slices that were eaten.
Column A (n)
Column B 5
Explanation: If (\frac{3}{8}) of the pizza remains, then the fraction of the pizza that was eaten is (1 - \frac{3}{8} = \frac{5}{8}). Since the pizza had 8 slices, the number of slices eaten, (n), is (\frac{5}{8} \times 8 = 5). So, the value in Column A is 5. The value in Column B is also 5. Therefore, the two quantities are equal. A common mistake is to assume (n=3), which would make Column B greater.
Ana is twice as old as her brother, Sam. In 4 years, the sum of their ages will be 32. How old is Ana now?
Explanation: Let Sam's current age be (S). Ana's current age is (2S). In 4 years, Sam will be (S+4) and Ana will be (2S+4). The sum of their ages in 4 years will be ((S+4) + (2S+4) = 32). Combine like terms: (3S + 8 = 32). Subtract 8 from both sides: (3S = 24). Divide by 3: (S = 8). This is Sam's current age. The question asks for Ana's current age, which is (2S = 2(8) = 16). Distractor A is Sam's current age. Distractor B is Sam's age in 4 years. Distractor D is Ana's age in 4 years.
Leo is thinking of a number. If he multiplies his number by 4 and then adds 9, he gets the same result as when he multiplies his number by 2 and adds 15. What is Leo's number?
Explanation: Let the number be (x). The problem can be translated into the equation (4x + 9 = 2x + 15). To solve for (x), first gather the variable terms on one side by subtracting (2x) from both sides: (2x + 9 = 15). Next, isolate the variable term by subtracting 9 from both sides: (2x = 6). Finally, divide by 2: (x = 3). Distractor C comes from an error where the constants are added instead of subtracted: (2x = 15+9=24), so (x=12). Distractor D is that value. Distractor B is a guess. The error for D is (4x-2x = 15+9 \implies 2x=24 \implies x=12). The error for C (6) is (4x+2x = 15+9 \implies 6x=24 \implies x=4). No, wait. Error for D: (4x-2x = 15+9 \implies 2x=24 \implies x=12). Error for B: (4x+2x=15+9 \implies 6x=24 \implies x=4). Error for C: (4x+2x=15-9 \implies 6x=6 \implies x=1). The provided distractors seem to have a mixup. Let's re-calculate: (4x+9=2x+15 \implies 2x=6 \implies x=3). Correct is A. Distractor D (12) from (2x = 15+9). Let's get C=6. Maybe (2x=15-9=6) but they forget to divide by 2, so x=6. That's plausible. So A is correct. B is a guess. C is from stopping at an intermediate step. D is from a sign error.
A baker starts with a full 50-pound bag of flour. He uses (2\frac{1}{2}) pounds of flour for each cake he bakes. At the end of the day, he has 20 pounds of flour left. How many cakes did he bake?
Explanation: First, determine how much flour was used. The baker started with 50 pounds and had 20 pounds left, so he used (50 - 20 = 30) pounds of flour. Each cake requires (2\frac{1}{2}) or 2.5 pounds of flour. To find the number of cakes, divide the total flour used by the amount per cake: (30 \div 2.5 = 12). The baker made 12 cakes. Distractor A is from dividing 20 by 2.5. Distractor C is from dividing 50 by 2.5, ignoring the remaining flour. Distractor D is from adding the remaining flour instead of subtracting: ((50+20) \div 2.5 = 28).
Given the equation (y = 2x + 5).
Column A The value of (x) when (y = 1)
Column B The value of (y) when (x = -1)
Explanation: For Column A, substitute (y=1) into the equation: (1 = 2x + 5). Subtract 5 from both sides: (-4 = 2x). Divide by 2: (x = -2). So, the quantity in Column A is -2. For Column B, substitute (x=-1) into the equation: (y = 2(-1) + 5). Simplify: (y = -2 + 5), so (y = 3). The quantity in Column B is 3. Since (3 > -2), the quantity in Column B is greater.
If 12 is subtracted from 4 times a number, the result is 20. What is the number?
Explanation: Let the number be (n). The phrase '4 times a number' is (4n). '12 is subtracted from' this means (4n - 12). The result is 20, so the equation is (4n - 12 = 20). Add 12 to both sides: (4n = 32). Divide by 4: (n = 8). Distractor B results from subtracting 12 from 20 instead of adding: (4n = 20 - 12 = 8 \implies n=2). Distractor A results from a sign error: (4n = 12-20 = -8 \implies n=-2). Distractor D results from misreading the problem as 'a number minus 12, times 4': (4(n-12)=20 \implies n-12=5 \implies n=17).
A submarine is at a depth of 240 meters below sea level. It begins to ascend at a constant rate of 12 meters per minute. How many minutes will it take for the submarine to reach a depth of 60 meters below sea level?
Explanation: Let the initial depth be -240 meters and the final depth be -60 meters. The rate of change is +12 meters per minute. The equation is (-240 + 12m = -60), where (m) is the number of minutes. To solve for (m), add 240 to both sides: (12m = -60 + 240), which is (12m = 180). Then, divide by 12: (m = 180 \div 12 = 15). It will take 15 minutes. Distractor C comes from dividing the initial depth by the rate ((240 \div 12 = 20)). Distractor A comes from dividing the final depth by the rate ((60 \div 12 = 5)). Distractor D comes from adding the absolute values of the depths and dividing by the rate (((240+60) \div 12 = 25)).