Given that 52416 ÷ 312 = 168, what will be the quotient when 52.416 is divided by 0.0168?

Introduction / Context:
This question tests your understanding of how scaling both the dividend and the divisor of a division problem by the same factor affects the quotient. It also involves decimal placement and powers of ten, which are key in simplifying decimal division.

Given Data / Assumptions:

• Known quotient: 52416 ÷ 312 = 168.
• We must find the value of 52.416 ÷ 0.0168.
• All values are exact, and we assume standard decimal arithmetic.

Concept / Approach:
Dividing two numbers does not change if we multiply both numbers by the same non zero factor. In decimal division, we often choose a power of ten to remove decimals. Here we can express 52.416 and 0.0168 in a way that relates them to 52416 and 312 by an appropriate scaling factor.

Step-by-Step Solution:
Step 1: Observe that 52.416 is 52416 divided by 1000. In other words, 52.416 = 52416 / 1000. Step 2: Observe that 0.0168 is 16.8 divided by 1000, and 16.8 equals 168/10, so 0.0168 = 168 / 10000. Step 3: Write the division: 52.416 ÷ 0.0168 = (52416 / 1000) ÷ (168 / 10000). Step 4: Rewrite the division as multiplication: (52416 / 1000) × (10000 / 168). Step 5: Simplify the powers of ten: (52416 / 1000) × (10000 / 168) = 52416 × (10 / 168). Step 6: Notice that 52416 ÷ 312 = 168 is given. Rearranging, 52416 = 168 × 312. Step 7: Replace 52416 with 168 × 312 in our expression and simplify; the calculation leads to a final quotient of 3120.

Verification / Alternative check:
We can directly compute 52.416 ÷ 0.0168 by scaling both numbers by 1000 to remove decimals in the divisor. Multiply numerator and denominator by 1000 to get 52416 ÷ 16.8. Multiply by 10 again to remove the remaining decimal in 16.8, obtaining 524160 ÷ 168. Now use the given relation 52416 ÷ 312 = 168 to see that 524160 ÷ 168 equals 3120. This confirms our result.

Why Other Options Are Wrong:
32.10 is too small and does not match the effect of dividing by a very small number like 0.0168.
321.0 is also too small; the quotient should increase significantly when dividing by a small decimal.
3.120 is far too small and results from a misunderstanding of decimal placement and scaling.

Common Pitfalls:
Common mistakes include dividing 52.416 by 16.8 directly without properly accounting for the shift in decimal places or thinking that both numbers should be scaled differently. Always remember that multiplying both dividend and divisor by the same non zero number leaves the quotient unchanged and can greatly simplify decimal division.

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