📝 Translate sentences to equations and solve (13 MCQs)

📖 Explanation: The student's translation and solution are both correct. The phrase "decreased by 5" means subtract 5 from $2x$, and "3 more than the number" means $x+3$. Solving $2x-5 = x+3$ gives $x=8$, and checking confirms. This question tests the ability to verify a translation and identify that no error exists, a key HOTS skill in error analysis.

📖 Explanation: The correct perimeter formula is $2w + 2l = 60$ with $l = 2w+4$, so $2w + 2(2w+4)=60$ simplifies to $6w+8=60$ → $w=8.67$. Student 1 used $2(w + 2w+4)$ which simplifies to $6w+8$ as well, actually both give the same $w=8.67$ if solved correctly. Wait, Student 1's equation $2(w+2w+4)=60$ → $2(3w+4)=60$ → $6w+8=60$ → $w=8.67$. So both are correct and equivalent. But option B says Student 2 gives 8.67, which is correct. However, Student 1 also gives 8.67 if solved correctly, but the option says 9.33 which is a solving error. So correct is B.

📖 Explanation: Consecutive odd integers differ by 2, so if the smallest is $n$, the next two are $n+2$ and $n+4$. The equation $n + (n+2) + (n+4) = 57$ gives $3n+6=57$ → $n=17$, so the integers are 17, 19, 21; largest is 21. Option A uses consecutive integers, not odd, a common misconception.

📖 Explanation: The flat fee is a one-time charge, not per mile, so the variable $m$ must multiply the per-mile rate only. First student's equation is correct: $30 + 0.20m = 50 + 0.10m$ → $0.10m = 20$ → $m=200$. Second student incorrectly multiplied the flat fee by $m$, a classic misconception. At 200 miles, both cost $70.

📖 Explanation: Setting the two equations equal gives the x-coordinate of intersection: $2x+3 = -x+9$ → $3x=6$ → $x=2$. Substituting into either gives $y=2(2)+3=7$. The intersection point (2,7) is the solution to the system, meaning both equations are satisfied. This connects algebraic solving to graphical interpretation, a key HOTS skill.

📖 Explanation: The phrase "subtracting the number from 20" translates to $20 - x$, not $x - 20$. The order matters in subtraction. First student's equation $3x+4 = 20-x$ is correct; solving gives $4x = 16$ → $x=4$. Second student reversed the order, a common error. This tests understanding of operational order in verbal phrases.

📖 Explanation: Perimeter formula is $2w + 2l = 38$. With $l = w+5$, we get $2w + 2(w+5) = 38$ → $4w+10=38$ → $w=7$, $l=12$, area $= 84$. First student used $w + l = 38$ which is half the perimeter, a common mistake. This question tests the ability to distinguish between perimeter and semi-perimeter and to compute area from correct dimensions.

📖 Explanation: 'Five less than twice a number' means start with $2x$ and subtract 5, i.e., $2x - 5$. The equation $2x-5=15$ gives $2x=20$ → $x=10$. The peer's equation $5-2x=15$ represents 'twice a number less than 5', which is a different phrase. This tests the ability to correctly translate subtraction order in verbal expressions, a frequent source of errors.

📖 Explanation: The equation $x + 1/x = 5/2$ is correct. Multiplying by $2x$ gives $2x^2 + 2 = 5x$ → $2x^2 -5x +2=0$ → factors to $(2x-1)(x-2)=0$ → $x=1/2$ or $x=2$. Both are valid because the reciprocal of $1/2$ is 2, and sum is $0.5+2=2.5$. Second student missed the $1/2$ solution due to not clearing the fraction completely or dividing incorrectly. This is an Olympiad-style problem testing algebraic manipulation and checking for extraneous roots.

📖 Explanation: Student A's expansion is straightforward and valid. Student B divides by $x-4$, which is allowed only if $x eq 4$. Since $x=4$ does not satisfy the original equation (3(0)=2(5) → 0=10 false), division is safe and the method works. However, it's a riskier approach and can lead to loss of solutions if the expression could be zero. This question tests understanding of the validity of algebraic operations and when division by a variable is permissible.

📖 Explanation: First line slope = (6-2)/(2-0) = 4/2=2, so $y=2x+2$. Second line slope = (4-8)/(4-0) = -4/4=-1, so $y=-x+8$. Setting equal: $2x+2 = -x+8$ → $3x=6$ → $x=2$, then $y=2(2)+2=6$. (2,6) satisfies both: first gives 6, second gives -2+8=6. So the student is correct. This tests the ability to derive equations from a graph and solve the system, connecting graphical and algebraic representations.

📖 Explanation: The phrase "divided by 3 and then 2 is added" means division operation happens first, then addition: $\frac{x}{3} + 2 = 10$. Solving gives $\frac{x}{3}=8$ → $x=24$. The second equation $\frac{x+2}{3}=10$ represents "add 2 to the number, then divide by 3", which is a different order of operations. This tests the ability to correctly sequence operations from verbal descriptions, a key HOTS application.

📖 Explanation: The student's step: $4(x-2) = 2x+6$ → adding 2 to both sides gives $4x = 2x+8$ is INCORRECT because $4(x-2)$ is $4x-8$, not $4x-2$. To isolate, you must first distribute: $4x-8 = 2x+6$ → $2x = 14$ → $x=7$. The error was treating $4(x-2)$ as $4x-2$, a common distributive property mistake. Correct solution is $x=7$. This tests error analysis in multi-step equation solving.