One- and Two-Step Equations - ISEE Middle Level Quantitative Reasoning

The sum of three consecutive odd integers is 87. What is the largest of these three integers?
Answer: 31 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.
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
Answer: The two quantities are equal. 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).
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?
Answer: 16 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.
Jamal owes his friend $60. He pays her $15 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?
Answer: 10 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 $5 per week. The number of $5 payments needed is $\$45 \div \$5 = 9$ weeks. The total time is the first week (for the $15 payment) plus the 9 subsequent weeks (for the $5 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 ( $60 \div 5 = 12$ ).
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?
Answer: 3 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 phone company charges $0.20 for the first minute of a call and $0.08 for each additional minute. If a call costs $1.24, what was the total duration of the call in minutes?
Answer: 14 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.
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
Answer: The two quantities are equal. 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.
If 12 is subtracted from 4 times a number, the result is 20. What is the number?
Answer: 8 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 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?
Answer: 12 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: $(50+20) \div 2.5 = 28$ .
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?
Answer: 30 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?
Answer: 17 inches 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 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?
Answer: 15 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 ( $(240+60) \div 12 = 25$ ).
Which of the following situations is best modeled by the equation $25 - 2x = 9$ ?
Answer: Mark has $25. He buys 2 books that each cost $x$ dollars and has $9 in change remaining. 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.
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?
Answer: 18th 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?
Answer: 8 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.
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, $c$ ?
Answer: $240 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 result of dividing a number $p$ by -6 is 8. What is the result of dividing $p$ by 4?
Answer: -12 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.
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 $k - 2$ ?
Answer: 22 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 $k-2=94$ .
Given the equation $y = 2x + 5$ . Column A The value of $x$ when $y = 1$ Column B The value of $y$ when $x = -1$
Answer: The quantity in Column B is greater. 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.

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