Problem 53 - Fundamentals of Mathematics

Question

Estimate each calculation using the method of rounding. After you have made an estimate, find the exact value and compare this to the estimated result to see if your estimated value is reasonable. Results may vary. (17.31)(.003)

Step-by-Step Solution

Answer

The exact product is 0.05193, which is close to the estimated 0.051.

Step 1: Round Each Number

First, round each number to one significant figure. The number 17.31 can be rounded to 17, and the number 0.003 can be rounded to 0.003. Notice that 0.003 is already at one significant figure.

Step 2: Multiply Rounded Numbers

Now, multiply the rounded numbers together. Multiply 17 by 0.003 to get an estimated product:\[ 17 \times 0.003 = 0.051. \]

Step 3: Calculate Exact Product

Calculate the exact product of the original numbers without rounding:\[ 17.31 \times 0.003 = 0.05193. \]

Step 4: Compare Estimates and Exact Values

Compare the estimated product, which is 0.051, with the exact product, which is 0.05193. The estimated product is quite close to the exact product, showing a reasonable estimate.

Key Concepts

Significant Figures

Significant figures are crucial in mathematical calculations because they reflect the precision of numbers. When rounding numbers, particularly for estimation, you round them to a specific number of significant figures. In the step-by-step solution provided, 17.31 was rounded to 17, implying rounding to two significant figures. However, since 0.003 is already at one significant figure, it remains the same.

• Significant figures convey which digits in a number are reliable and contribute to its precision.
• When rounding to significant figures, you often choose the number that retains the most meaningful precision for your calculation.
• Rounding can simplify complex calculations, especially when a rough estimate suffices.

Rounding to significant figures allows you to maintain a balance between accuracy and simplicity in mathematical work.

Estimation

Estimation is a valuable skill that can save time and effort when performing calculations. By rounding numbers to significant figures, you produce an approximation that can quickly be calculated. The exercise uses this approach for multiplication. The estimated result here was obtained by rounding: 17 becomes 17, and 0.003 stays the same, given it's already one significant figure, leading to an estimated product of 0.051.

• Estimates provide a quick check of your work and reveal if your answer is reasonable.
• They are especially useful in verifying if further detailed calculations are accurate.
• While estimates aren't precise, they give a sense of the order of magnitude of a result.

Using estimation through rounding simplifies initial calculations, offering a ballpark figure before performing exact calculations.

Exact Calculation

An exact calculation uses the entire numbers without rounding, ensuring maximum precision. The exact calculation for this problem is computed by multiplying 17.31 by 0.003 to get 0.05193.

• Exact calculations are critical in situations where precision is paramount and estimations are insufficient.
• They provide the most reliable result when precise values are necessary, like scientific measurements.
• In practice, estimates and exact results often complement each other; the former gives a quick reference, while the latter ensures accuracy when needed.

By comparing exact values to estimates, you confirm the reasonableness of the rounded numbers and maintain accuracy in important work.

Multiplication

When multiplying numbers, whether using estimates or exact values, the procedure remains the same. Multiplication combines quantities, and even with rounded numbers, it follows the same operation. In the example, the rounded multiplication of 17 by 0.003 resulted in an estimated product of 0.051. The exact multiplication of 17.31 and 0.003 yielded 0.05193.

• Multiplication, as a fundamental arithmetic operation, consolidates numbers into a single, larger product.
• It functions the same whether dealing with whole, decimal, or rounded numbers.
• Understanding multiplication's process helps apply it correctly regardless of numerical precision.

Grasping the basics of multiplication ensures you can easily navigate between exact calculations and estimates, enhancing your ability to assess the complexity of numerical problems.