In a division sum, the divisor is 12 times the quotient and 5 times the remainder. If the remainder is 36, then the dividend is:

Question

Accepted Answer

The correct option is 2:
The correct option is 2: $$Detailed explanation of why it is correct — include conceptual, or contextual depth.$$ The incorrect option is 1: $$Detailed explanation of why it is incorrect — include conceptual, or contextual depth.$$ The incorrect option is 3: $$Detailed explanation of why it is incorrect — include conceptual, or contextual depth.$$ The incorrect option is 4: $$Detailed explanation of why it is incorrect — include conceptual, or contextual depth.$$

The correct option is 2: 2736

This problem requires us to recall and apply the fundamental relationship in division, known as the Division Algorithm:

$$\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$$

We are given the remainder and the proportional relationships governing the divisor and the quotient. We can find the unknown components (divisor and quotient) in a step-by-step manner and then use the Division Algorithm to calculate the dividend.

**Step 1: Determine the Divisor**

The question states that the divisor is 5 times the remainder.
We are given that the Remainder = 36.
Therefore, Divisor = $5 \times \text{Remainder}$
Divisor = $5 \times 36$
Divisor = 180.

**Step 2: Determine the Quotient**

The question also states that the divisor is 12 times the quotient.
We know the Divisor = 180.
So, $\text{Divisor} = 12 \times \text{Quotient}$
$180 = 12 \times \text{Quotient}$
$\text{Quotient} = \frac{180}{12}$
$\text{Quotient} = 15$.

**Step 3: Calculate the Dividend**

Now we substitute the known values into the Division Algorithm:
$$\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$$
$$\text{Dividend} = (180 \times 15) + 36$$

First, multiply the Divisor and the Quotient:
$180 \times 15 = 2700$

Next, add the Remainder:
$\text{Dividend} = 2700 + 36$
$\text{Dividend} = 2736$

Therefore, the dividend is 2736. This mathematical sequence confirms that Option 2 is the correct answer.

The incorrect option is 1: 2706
This result (2706) would be obtained if a student correctly calculated the product of the divisor and quotient (2700) but then incorrectly added 6 instead of the correct remainder of 36. Alternatively, this number results from an error in calculating the divisor or quotient, or if a different, incorrect remainder was assumed. This option is too low because it misses the full value of the remainder (36) in the final calculation.

The incorrect option is 3: 2796
This result (2796) is derived if a significant multiplication error occurred while calculating the dividend. Specifically, if the student mistakenly calculated the product of the divisor (180) and quotient (15) as 2760 instead of 2700, and then added the remainder (36), they would get $2760 + 36 = 2796$. Another possibility is if the quotient was mistakenly calculated as 15.6, leading to an incorrect product. Since our calculated product is 2700, this option involves a considerable arithmetic error.

The incorrect option is 4: 2766
This result (2766) is mathematically plausible if a student correctly calculated the quotient (15) and divisor (180), but then made an error in the product calculation, perhaps computing $180 \times 15$ as 2730, and then correctly adding the remainder: $2730 + 36 = 2766$. More likely, this results from an error where the remainder was added twice, or if the initial calculations for the divisor or quotient were slightly off, leading to a higher dividend than correct.