Problem 39 - 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. $$ 72.14+21.08 $$

Step-by-Step Solution

Answer

The estimated sum is 93 and the exact sum is 93.22; they are close, so the estimate is reasonable.

Step 1: Round Each Number

First, we round each number to the nearest whole number. For 72.14, we round down to 72. For 21.08, we round down to 21.

Step 2: Estimate the Sum

Add the rounded numbers together. So, 72 + 21 = 93. This is our estimated result for the sum.

Step 3: Calculate the Exact Sum

Now we calculate the exact sum without rounding. Add 72.14 and 21.08 together: \[72.14 + 21.08 = 93.22\]

Step 4: Analyze the Comparison

Compare the estimated sum from Step 2 with the actual sum from Step 3. The estimated sum is 93, while the exact sum is 93.22. The difference is small, only 0.22, indicating the estimate is reasonable.

Key Concepts

Estimation

Estimation in mathematics is a valuable skill that helps simplify complex calculations. It involves approximating a number to make a calculation manageable. The most common way is through rounding. Rounding is adjusting a number to its nearest value of a chosen place value. For instance, rounding 72.14 to the nearest whole number results in 72. Similarly, rounding 21.08 gives us 21.

• Rounding to the nearest whole number simplifies addition.
• It provides an estimate that is often close to the actual result.

Estimation is not just about getting the answer quickly but also understanding how numbers behave. It's a useful tool for verifying more precise calculations and making initial guesses when dealing with complicated figures.

Addition of Decimals

When dealing with the addition of decimals, each number is aligned by the decimal point. This ensures the digits are in the correct place value, making addition seamless. In our exact calculation of 72.14 + 21.08, aligning the decimal points allows us to add each corresponding digit correctly.

• Ensure that numbers are aligned by their decimal points.
• Add digits starting from the rightmost column, the hundredths place in this case.
• Handle any carry forward, just like addition with whole numbers.

The meticulous addition process ensures that our result is accurate, providing the exact sum of 93.22 for the numbers in our example. Being precise with decimal addition aids in achieving accurate computations in various real-world applications.

Comparison of Estimates and Exact Values

After performing both an estimation and an exact calculation, comparing the two results can reveal how reasonable the estimate is. In our example, we estimated the sum of 72.14 and 21.08 to be 93. The exact value calculated was 93.22.

• Compare both the estimated and exact results.
• The estimated value was very close to the actual value, differing by only 0.22.
• A small difference suggests the estimation was reasonable.

Analyzing these comparisons is crucial as it helps in verifying the proximity of the estimate to the exact figure. Estimation acts as a quick check to ensure that calculations are on the right track, providing insight into potential errors or miscalculations if the discrepancy is too large.