Problem 56 - 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. \(107 \%\) of 12.6

Step-by-Step Solution

Answer

The estimated value is 13.91, and the exact value is 13.482. The estimate is reasonable.

Step 1: Understand the Problem

We need to first find 107% of 12.6. A percentage over 100 indicates that we are looking for a value slightly more than 12.6.

Step 2: Round the Numbers for Estimation

Round 12.6 to a nearby whole number for easier estimation. 12.6 is close to 13. Therefore, we round 12.6 to 13.

Step 3: Estimate 107% of Rounded Number

To find 107% of 13, first calculate 100% of 13, which is 13. Then find 7% of 13, which can be estimated as 0.91 by calculating 10% of 13 (1.3) and taking 70% of that (0.91). Add these to estimate 107% of 13 as approximately 13.91.

Step 4: Calculate the Exact Value

Calculate the exact value of 107% of 12.6. First, convert 107% into a decimal, which is 1.07. Then, multiply 1.07 by 12.6: \[ 1.07 \times 12.6 = 13.482 \]

Step 5: Compare Estimated to Exact Value

Compare 13.91 (estimated rounded calculation) to 13.482 (exact calculation). Our estimate is slightly higher but close to the exact value.

Key Concepts

Percentage Calculation

Understanding percentage calculations is crucial for many math problems. A percentage indicates a proportion out of 100. For example, 107% is a bit more than the whole value since it exceeds 100%.
To calculate a specific percentage of a number, convert the percentage into a decimal. Do this by dividing the percentage by 100. For 107%, the decimal equivalent is 1.07.
Using a simple multiplication, you can find the percentage of a number. For instance, take 107% of 12.6, and you do this by multiplying 12.6 by 1.07. This multiplication gives the exact portion that 107% represents of 12.6.

Estimation Techniques

Estimation techniques are handy when an exact calculation isn't necessary or feasible. They give a quick, approximate result and often start with rounding. Rounding simplifies numbers by adjusting them to the nearest whole or significant number.
For example, to estimate 107% of 12.6, rounding 12.6 to 13 makes the calculation simpler. This technique is particularly useful in mental math, as it reduces complexity.

• Round up or down to the nearest whole number or ten
• Use simpler numbers for percentage estimates (e.g., round off percentages to multiples of ten)

This helps to quickly approximate results without detailed calculations.

Exact Calculations

Exact calculations provide precise results. Unlike estimation, this approach doesn't involve rounding. Each step uses the actual numbers as they are.
To find the exact value of 107% of 12.6, you convert the percentage to decimal form (1.07) and multiply it by 12.6. The multiplication gives you an answer accurate to several decimal places.
Exact calculations are ideal for tasks that require precision, such as scientific measurements, financial calculations, or academic problem-solving, where accuracy is non-negotiable.

Comparing Estimates to Exact Values

After estimating and accurately calculating, it’s useful to compare the two results. This comparison ensures that your estimation method is reasonable and serves its intended purpose.
In our example, the estimated 107% of 12.6 was 13.91, while the exact calculation was 13.482. The estimated value is slightly higher but close, showcasing the effectiveness of estimation.
Comparing these values helps recognize any substantial differences, ensuring the estimate is within an acceptable range and guiding adjustments in estimation techniques if necessary.