When 52.416 is divided by 0.0168, what is the resulting quotient, given that 52,416 ÷ 312 = 168?

Introduction / Context:
This problem combines decimal division with a useful shortcut based on scaling by powers of 10. You are given one division fact (52416 ÷ 312 = 168) and asked to compute a related quotient involving decimals. Recognising how to relate the two problems is the key skill being tested.

Given Data / Assumptions:

• Given: 52,416 ÷ 312 = 168.
• Required: 52.416 ÷ 0.0168.
• All operations are standard real-number divisions.

Concept / Approach:
We can eliminate decimals by multiplying both the dividend and the divisor by the same power of 10. This does not change the value of the quotient. Once the decimals are removed, we may be able to relate the resulting division to a simple integer division that is easy to compute.

Step-by-Step Solution:
Step 1: Consider the target expression: 52.416 ÷ 0.0168. Step 2: Multiply both dividend and divisor by 1000 to remove three decimal places: (52.416 × 1000) ÷ (0.0168 × 1000). Step 3: This becomes 52416 ÷ 16.8. Step 4: Multiply both numbers by 10 to remove the remaining decimal: (52416 × 10) ÷ (16.8 × 10) = 524160 ÷ 168. Step 5: Now evaluate 524160 ÷ 168. Notice that 524160 = 3120 × 168. Step 6: Therefore, 524160 ÷ 168 = 3120. Step 7: Hence, 52.416 ÷ 0.0168 = 3120.

Verification / Alternative check:
Use the given relationship. Since 52416 ÷ 312 = 168, we can try to express 52.416 ÷ 0.0168 in a way related to 168. Recognise that dividing by 0.0168 is the same as multiplying by its reciprocal. Adjusting by factors of 10 eventually leads to the same result 3120. A calculator check will also confirm that 52.416 ÷ 0.0168 = 3120 exactly.

Why Other Options Are Wrong:
3.12 and 312 are much smaller than the true quotient, resulting from moving the decimal point incorrectly or not accounting for all the powers of 10.
31200 is ten times larger than the correct result, likely from overcompensating when clearing decimals.
None of these is incorrect because 3120 is a valid and exact answer.

Common Pitfalls:
The most common errors arise from mishandling decimal places. If you do not multiply both numerator and denominator by the same power of 10, you accidentally change the quotient. Another pitfall is stopping halfway, for example after removing only some of the decimal places, which makes the division more complicated than necessary.

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