It is known that 52,416 divided by 312 gives a quotient of 168. Using this information, what will be the quotient when 52.416 is divided by 0.0168?

Introduction:
This problem tests your understanding of how scaling both the dividend and the divisor by powers of 10 affects the quotient. Instead of directly performing a messy decimal division, you can use the given exact division involving whole numbers to find the new quotient efficiently.

Given Data / Assumptions:

• Given: 52,416 ÷ 312 = 168.
• We must find: 52.416 ÷ 0.0168.
• The relationship between 52,416 and 52.416, and between 312 and 0.0168, involves powers of 10.

Concept / Approach:
When both the numerator (dividend) and the denominator (divisor) of a division are multiplied or divided by the same nonzero number, the quotient does not change. By expressing 52.416 and 0.0168 in terms of 52,416 and 168/10,000, we can rewrite the division to leverage the known result 52,416 ÷ 312 = 168.

Step-by-Step Solution:
Step 1: Express 52.416 as a fraction of 52,416: 52.416 = 52,416 / 1000.Step 2: Express 0.0168 as a fraction: 0.0168 = 168 / 10,000.Step 3: Write the required division: 52.416 ÷ 0.0168 = (52,416 / 1000) ÷ (168 / 10,000).Step 4: Dividing by a fraction is the same as multiplying by its reciprocal: (52,416 / 1000) * (10,000 / 168).Step 5: Simplify: (52,416 * 10,000) / (1000 * 168) = (52,416 * 10) / 168.Step 6: So the result is 52,4160 / 168.Step 7: Note that 52,416 ÷ 312 = 168 is given. We can see that 52,4160 = 10 * 52,416.Step 8: Compute 52,4160 / 168 = 10 * (52,416 / 168).Step 9: Since 52,416 / 312 = 168, and 168 * 312 = 52,416, we can verify that 52,416 / 168 = 312.Step 10: Therefore, 52,4160 / 168 = 10 * 312 = 3120.Thus, the desired quotient is 3120.

Verification / Alternative check:
We can quickly approximate: 52.416 ÷ 0.0168 is like 52 ÷ 0.017, which is roughly 3,000. The exact answer 3120 is close to this estimate and matches the detailed calculation.

Why Other Options Are Wrong:
Values such as 32.10, 321.0, 3.120, and 0.312 are far too small compared to the approximate ratio of 52 to 0.017. They arise from misplacing decimal points or dividing incorrectly by powers of 10. The only value consistent with the scaling and known division fact is 3120.

Common Pitfalls:
Many students attempt to do the decimal division directly, leading to errors with decimal placement. Others may incorrectly cancel zeros without keeping track of the effect on both numerator and denominator. Using the relationship between the given whole number division and the target division simplifies the problem and avoids such mistakes.

Final Answer:
The quotient when 52.416 is divided by 0.0168 is 3120.