Without calculating the exact product, estimate the value of (?) in the approximation equation: 21.003 × 39.998 − 209.91 = 126 × ? Choose the closest approximate integer value.

Introduction / Context:
This problem checks approximation skills using smart rounding. The numbers are very close to easy values (21, 40, and 210), so you can estimate the left-hand side quickly and then divide by 126 to get the approximate value of (?).

Given Data / Assumptions:

• Expression: 21.003 × 39.998 − 209.91 • This equals: 126 × ? • Exact computation is not required; choose closest integer

Concept / Approach:
Round to nearby convenient numbers: 21.003 ≈ 21, 39.998 ≈ 40, 209.91 ≈ 210. Compute the rough result, then refine slightly if desired, and divide by 126. Since 126 × 5 = 630, we expect the answer to be near 5 if the left side is near 630.

Step-by-Step Solution:
1) Round the product: 21.003 × 39.998 ≈ 21 × 40 = 840 2) Round the subtraction term: 209.91 ≈ 210 3) Estimate left-hand side: 840 − 210 = 630 4) Now solve 126 × ? ≈ 630: ? ≈ 630/126 = 5 5) Quick sanity: 126 × 5 = 630 exactly, so 5 is the best approximate integer.

Verification / Alternative check:
If you refine slightly: 21.003 × 39.998 is extremely close to 21 × 40, and 209.91 is extremely close to 210, so the left side stays extremely close to 630. Therefore the quotient stays extremely close to 5, not 4 or 6.

Why Other Options Are Wrong:
• 4 would imply left side near 504, too small. • 6 would imply left side near 756, too large. • 3 or 2 are far too small compared to the estimated 630.

Common Pitfalls:
• Multiplying precisely (wastes time) instead of rounding first. • Dividing by 126 incorrectly or confusing it with 125.

Final Answer:
5