Middle School Math Quiz: Problem Solving Perseverance

7 questions · exam conditions

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Problem Solving PerseveranceQuestion 1 of 7

Marcus is planning a school fundraiser selling two types of items: keychains for $3 each and bookmarks for $2 each. He needs to raise at least $240 and wants to sell exactly 100 items total. If he sells $ $k$ $ keychains, which inequality represents the constraint for reaching his fundraising goal, and what is the minimum number of keychains he must sell?

3k + 2(100-k) \geq 240

; he must sell at least 40 keychains

3k + 2(100-k) \geq 240

; he must sell at least 50 keychains

2k + 3(100-k) \geq 240

; he must sell at least 40 keychains

3k + 200 \geq 240

; he must sell at least 60 keychains

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Middle School Math Quiz

Middle School Math Quiz: Problem Solving Perseverance

Practice Problem Solving Perseverance in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Problem Solving Perseverance, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School 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.

All questions

Question 1

$3k + 2(100-k) \geq 240$ ; he must sell at least 40 keychains (correct answer)
$3k + 2(100-k) \geq 240$ ; he must sell at least 50 keychains
$2k + 3(100-k) \geq 240$ ; he must sell at least 40 keychains
$3k + 200 \geq 240$ ; he must sell at least 60 keychains

Explanation: The inequality

3k + 2(100-k) \geq 240

correctly represents the constraint where Marcus sells k keychains at $3 each and (100-k) bookmarks at $2 each. Solving: $

3k + 200 - 2k \geq 240

, so

k \geq 40

$. Choice B uses the correct inequality but miscalculates the solution. Choice C switches the prices. Choice D incorrectly assumes all non-keychains cost $0.

Question 2

Tom is working on this problem: "The sum of three consecutive integers is 48. Find the integers." He sets up the equation $x + (x+1) + (x+2) = 48$ and solves to get $x = 15$ . When he checks his work by substituting back, he gets $15 + 16 + 17 = 48$ , which is correct. However, he notices that if the problem had asked for consecutive even integers instead, his method would need adjustment. What should his equation be for consecutive even integers with the same sum?

$2x + (2x+1) + (2x+2) = 48$ because consecutive even integers differ by 1
$x + (x+2) + (x+4) = 48$ because consecutive even integers differ by 2 (correct answer)
$2x + (2x+2) + (2x+4) = 48$ because even integers are multiples of 2
$x + (x+1) + (x+2) = 48$ because the sum is the same regardless of the type of integers

Explanation: Consecutive even integers differ by 2, so if the first is x, the next two are (x+2) and (x+4). Choice A incorrectly suggests even integers differ by 1. Choice C uses an unnecessarily complicated representation - while 2x, (2x+2), (2x+4) are consecutive even integers, the simpler x, (x+2), (x+4) form works better where x represents the first even integer directly. Choice D incorrectly assumes the same equation works for different types of consecutive integers.

Question 3

Sarah is solving a multi-step equation and gets stuck. She writes: " $2(x+3) - 5 = 3x - 7$ . I distributed and got $2x + 6 - 5 = 3x - 7$ , then $2x + 1 = 3x - 7$ . Now I have $2x - 3x = -7 - 1$ , so $-x = -8$ , which means $x = 8$ ." When she checks her answer, it doesn't work. What should she do next to fix her error?

Re-examine her distribution step because $2(x+3) = 2x + 6$ might be wrong
Re-examine her subtraction step because she wrote $-7 - 1 = -8$ instead of $-7 - 1 = -6$
Re-examine her variable collection because she should have $-x = -6$ instead of $-x = -8$
Re-examine her sign handling because when moving terms she should have $2x - 3x = -7 + 1$ (correct answer)

Explanation: Sarah's error occurs when collecting terms. From

2x + 1 = 3x - 7

, subtracting

2x

and adding

7

to both sides gives

1 + 7 = 3x - 2x

, so

8 = x

. Her mistake was writing

2x - 3x = -7 - 1

when it should be

2x - 3x = -7 + 1

(since she's moving +1 to the right side). Choice A is wrong because her distribution is correct. Choice B incorrectly identifies the arithmetic error. Choice C identifies a consequence but not the root cause.

Question 4

Carmen is solving the system: $\begin{cases} 2x + 3y = 12 \\ 4x + 6y = 20 \end{cases}$ . She multiplies the first equation by -2 to get $-4x - 6y = -24$ , then adds it to the second equation and gets $0 = -4$ . She concludes the system has no solution. Her friend David says she made an error because the second equation should be $4x + 6y = 24$ to be consistent. Who is correct and why?

Carmen is correct; the system as written is inconsistent and has no solution regardless of David's suggested change (correct answer)
David is correct; Carmen made an arithmetic error and the system actually has infinitely many solutions
Both are partially correct; Carmen's method is right but David correctly identifies that the original system contains a typo
Neither is correct; the system has exactly one solution that Carmen missed due to calculation errors

Explanation: Carmen's analysis is mathematically sound. The system as given is inconsistent because the left sides of the equations are proportional (the second equation's left side is exactly twice the first), but the right sides are not proportional (24 ≠ 2×12 = 20). This creates a contradiction, so there is no solution. David's suggestion would create a system with infinitely many solutions, but that doesn't mean the original system was wrong - it could intentionally be inconsistent. Choice B misunderstands the mathematics. Choice C assumes there's a typo without justification. Choice D is mathematically impossible given the structure.

Question 5

Jake is solving the inequality $-3(x - 4) < 2x + 8$ . He distributes to get $-3x + 12 < 2x + 8$ , then subtracts $2x$ from both sides: $-5x + 12 < 8$ . Next, he subtracts 12 from both sides: $-5x < -4$ . At this point, he needs to divide by $-5$ . What must he remember to do, and what will his final answer be?

Flip the inequality sign when dividing by a negative; final answer: $x > \frac{4}{5}$ (correct answer)
Keep the inequality sign the same when dividing by a negative; final answer: $x < \frac{4}{5}$
Flip the inequality sign when dividing by a negative; final answer: $x < \frac{4}{5}$
Change the problem to an equation since dividing by negatives in inequalities is undefined; solve $-5x = -4$

Explanation: When dividing both sides of an inequality by a negative number, the inequality sign must be flipped. From

-5x < -4

, dividing by

-5

gives

x > \frac{4}{5}

. Choice B forgets to flip the sign. Choice C flips the sign correctly but gets the wrong final inequality direction. Choice D is completely incorrect - division by negative numbers is perfectly valid in inequalities, just requires flipping the sign.

Question 6

Maria is planning a rectangular vegetable garden where the length is 4 feet more than twice the width. She has 60 feet of fencing to enclose the garden. After setting up her equation $2w + 2(2w + 4) = 60$ where $w$ is the width, she solves and gets $w = 8.67$ feet. However, she realizes this creates practical problems for her garden layout. What should she consider doing next to make this more practical?

Round to $w = 9$ feet and accept that she'll use slightly more than 60 feet of fencing
Round to $w = 8$ feet and use the leftover fencing for a separate small garden bed (correct answer)
Keep the exact decimal value since precision is important for maximizing garden area
Recalculate assuming she can buy additional fencing to make the width exactly 9 feet

Explanation: With

w = 8

feet, the length becomes

2(8) + 4 = 20

feet, and the perimeter is

2(8) + 2(20) = 56

feet, leaving 4 feet of fencing unused - a practical solution. Choice A would require 62 feet of fencing (more than available). Choice C ignores the practical constraint that fractional feet are difficult to measure and work with in construction. Choice D unnecessarily complicates the problem by changing the constraint rather than working within it.

Question 7

A rectangular garden has length $(2x + 5)$ feet and width $(x - 3)$ feet. The gardener wants the area to be exactly 84 square feet. After setting up the equation $(2x + 5)(x - 3) = 84$ and expanding to get $2x^2 - x - 15 = 84$ , what should be the next strategic step to solve this problem efficiently?

Move 84 to the left side to get $2x^2 - x - 99 = 0$ and use the quadratic formula immediately
Factor the left side as $(2x - 5)(x + 3) = 84$ and set each factor equal to 84
Move 84 to the left side to get $2x^2 - x - 99 = 0$ and look for rational roots using the rational root theorem (correct answer)
Substitute test values like $x = 1, 2, 3$ into the original equation to find the solution by trial

Explanation: After getting

2x^2 - x - 99 = 0

, the rational root theorem is most efficient since potential rational roots are limited to factors of 99 divided by factors of 2. Testing shows

x = 9

works. Choice A uses the quadratic formula unnecessarily when simpler methods exist. Choice B incorrectly factors (the correct factorization of

2x^2 - x - 15

(2x - 5)(x + 3)

, but this doesn't help with the +84). Choice D is inefficient trial-and-error without systematic approach.