Without computing every digit exactly, approximate the value of (111.93 × 5) ÷ 14.02 and solve the equation (111.93 × 5) ÷ 14.02 = 11.002 + ?. Which option is closest to the value of the question mark?

Introduction / Context:
This question tests estimation of a compound arithmetic expression and solving for an unknown in a simple equation. It is typical in aptitude exams where you need to quickly approximate products and quotients and then isolate the unknown term without relying on long exact calculations.

Given Data / Assumptions:

• The expression is (111.93 × 5) ÷ 14.02.
• The equation given is (111.93 × 5) ÷ 14.02 = 11.002 + ?.
• We must approximate the value of ?.

Concept / Approach:
First, approximate the product 111.93 × 5 and the divisor 14.02 using nearby convenient numbers. Then evaluate the approximate quotient. After finding this approximate value, subtract 11.002 to isolate ?. Finally, choose the option that most closely matches the result. Estimation is enough because the options are spaced far apart.

Step-by-Step Solution:
Approximate 111.93 as 112 and 14.02 as 14 for simpler arithmetic.Compute the approximate product: 112 × 5 = 560.Compute the approximate quotient: 560 ÷ 14 = 40.Now we have roughly 40 ≈ 11.002 + ?, so ? is approximately 40 − 11.002 ≈ 29.Thus, the value of ? is closest to 29 among the given options.

Verification / Alternative check:
A more precise calculation gives (111.93 × 5) ÷ 14.02 approximately equal to 39.92. Subtracting 11.002 from 39.92 yields around 28.92, which rounds to 29. This more accurate computation supports the initial estimate and confirms that 29 is the nearest correct option.

Why Other Options Are Wrong:

• 11 and 10 are far too small. If ? were of that size, the right side 11.002 + ? would be near 21, not near 40.
• 34 is larger than 29 and would make the right side about 45, which is noticeably bigger than the approximate left side.
• 0 would make the right side just above 11, which is clearly too small compared with the quotient.

Common Pitfalls:

• Not approximating carefully and making the product or divisor too rough, causing a poor estimate.
• Forgetting to subtract 11.002 after computing the quotient, leading to a wrong match with the options.
• Misjudging which integer is closest when comparing approximate decimal values to the provided options.

Final Answer:
29