Question

Which of the following would be the most appropriate setup for estimating the value 3.6 × 4.85 for questions in which answer choices differ by a small margin? (A) 3.5 × 5 (B) 3.5 × 4.5 (C) 4 × 4 (D) 4 × 5

Short answer

Option (A) 3.5 × 5 is the most appropriate setup.

Step-by-step solution

- Identify the Numbers to be Estimated

Identify the numbers in the problem: 3.6 and 4.85. We need to round these numbers appropriately to estimate their product.

- Round the Numbers

Round the numbers to the nearest value that simplifies the multiplication. 3.6 can be rounded to 3.5 or 4, and 4.85 can be rounded to 5.

- Evaluate the Answer Choices

Consider each answer choice for how closely the rounded numbers match the original numbers. The answer choices are: (A) 3.5 × 5 (B) 3.5 × 4.5 (C) 4 × 4 (D) 4 × 5

- Compare the Magnitudes

Check how close the product of the rounded numbers is to the original multiplication. Option A: 3.5 × 5 = 17.5 Option B: 3.5 × 4.5 = 15.75 Option C: 4 × 4 = 16 Option D: 4 × 5 = 20

- Select the Closest Approximation

The actual product of 3.6 × 4.85 is approximately 17.46. Considering the options, 3.5 × 5 = 17.5 is the closest to 17.46.

Key concepts

Rounding Numbers

Rounding numbers is a fundamental estimation technique in mathematics. The process simplifies calculations by reducing the number of digits. For example, the number 3.6 can be rounded to 3.5 or 4 depending on the context. Similarly, the number 4.85 can be rounded to 5. Rounding involves identifying the closest simplified value that still represents the original number accurately. This simplification makes mental math more manageable and helps in providing quicker estimates for answers without needing precise calculations.

Multiplication Estimation

When estimating products in multiplication, it's crucial to round the numbers first. This approach allows for a quick and easy estimation. For example, estimating the product of 3.6 and 4.85 involves first rounding these values. By rounding 3.6 to 3.5 or 4 and 4.85 to 5, the multiplication becomes simpler: 3.5 × 5 or 4 × 5. These simplified products give us a close approximation of the actual value, helping us understand the magnitude without precise calculations.

Thus, umbers rounded to their nearest convenient figures simplify the multiplication task considerably.

Answer Comparison

After estimating the multiplication, compare the calculated product with the possible answer choices. This step ensures the closest approximation is selected. For the problem 3.6 × 4.85, rounded values like 3.5 × 5 or 4 × 5 result in 17.5 and 20, respectively. By comparing these products to the actual value of 17.46, we recognize that 17.5 is the closest. Thus, 3.5 × 5 is a more accurate estimate. Always select the estimation that aligns most closely with the exact value, ensuring minimal error in approximation.

Product Approximation

Product approximation in mathematics involves identifying how close an estimated product is to the actual product. For instance, the actual product of 3.6 × 4.85 is approximately 17.46. Through rounding and comparing different products (like 3.5 × 5 or 4 × 5), we find that 17.5 (from 3.5 × 5) is the nearest approximation. This method helps in quick decision-making, especially when answer choices have small differences. Select the option with the product closest to the exact value, ensuring accurate and effective estimation.