Reverse division with conditions — In a division, the divisor equals 12 times the quotient and 5 times the remainder. If the remainder is 48, find the dividend using the division identity.

Introduction / Context:Division problems with constraints on the divisor, quotient, and remainder are classic applications of the division identity: dividend = divisor * quotient + remainder. Here, you are given relationships between divisor, quotient, and remainder and must compute the dividend directly and cleanly.

Given Data / Assumptions:

• Let divisor = d, quotient = q, remainder = r.
• Given: d = 12 * q and d = 5 * r.
• Given remainder r = 48. Standard condition: 0 ≤ r < d.

Concept / Approach:Use the given relationships to resolve d and q. From d = 5 * r with r = 48, get d. Then from d = 12 * q, obtain q. Finally, use dividend = d * q + r to compute the desired number. This sequence avoids guesswork and stays faithful to the constraints.

Step-by-Step Solution:From d = 5 * r and r = 48, compute d = 5 * 48 = 240.From d = 12 * q, compute q = d / 12 = 240 / 12 = 20.Apply the division identity: dividend = d * q + r = 240 * 20 + 48 = 4800 + 48 = 4848.Thus, the dividend is 4848.

Verification / Alternative check:Divide 4848 by 240: 4848 = 240 * 20 + 48, so quotient 20 and remainder 48. The relationships d = 12 * q (240 = 12 * 20) and d = 5 * r (240 = 5 * 48) hold perfectly.

Why Other Options Are Wrong:

• 240 / 576 / 4800 / 5280: These do not satisfy dividend = d * q + r under the given constraints or correspond to intermediate values (divisor/product) rather than the actual dividend.

Common Pitfalls:Confusing which variable is 12 times which; forgetting to add the remainder after the product; violating the condition r < d.

Final Answer:4848