Question

A student was asked to divide a number by 6 and add 12 to the quotient. He however, first added 12 to the number and then divided it by 6, getting 112 as the answer. The correct answer should have been?

• A. 124
• B. 172
• C. 118
• D. 122

Hint:

When a problem involves misinterpretation of operations, carefully set up both the intended and the mistaken operations. Always verify with the final options to avoid errors.

Answer 1

The Correct Option is D

Step 1: Express condition mathematically.
Let the number be \(N\). Correct operation: \(\frac{N}{6} + 12\). Mistaken operation: \(\frac{N+12}{6} = 112\). 

Step 2: Solve for \(N\).
\[ \frac{N+12}{6} = 112 \quad \Rightarrow \quad N+12 = 672 \quad \Rightarrow \quad N=660. \] 

Step 3: Compute correct answer.
\[ \frac{N}{6} + 12 = \frac{660}{6}+12 = 110+12=122. \] \[ \boxed{122} \]

Answer 2

A quicker way avoids finding the number at all: compare the correct process and the mistaken process directly to see how far apart their results must be.

1. 124: Checking against \(112+10=122\), this option is 2 more than the required value, so it does not match.
2. 172: This is 50 more than 122, far from the required difference, so it is incorrect.
3. 118: This is 4 less than 122, so it does not satisfy the relationship either.
4. 122: This exactly equals \(112+10\), matching the required relationship between the correct and mistaken results.

The correct process is \(\dfrac{N}{6}+12\), while the mistaken process is \(\dfrac{N+12}{6}=\dfrac{N}{6}+2\). Subtracting, (correct) \(-\) (mistaken) \(=12-2=10\), so the correct answer is always exactly 10 more than the mistaken one, regardless of \(N\): \(112+10=122\).

So the correct answer is 122.