Problem 65 - Fundamentals of Mathematics

Question

Estimate each value using the method of rounding. After you have made an estimate, find the exact value. Compare the exact and estimated values. Results may vary. $$ 174.6+97.2 $$

Step-by-Step Solution

Answer

Estimated sum: 270. Exact sum: 271.8. The estimate is slightly lower than the exact value.

Step 1: Round the Numbers

We start by rounding each number to the nearest ten. The number 174.6 is rounded to 170, and 97.2 is rounded to 100.

Step 2: Estimate the Sum

Add the rounded numbers: 170 + 100 = 270. This is our estimated sum.

Step 3: Calculate the Exact Sum

Add the original numbers: 174.6 + 97.2. First, line up the numbers by their decimal points. \[ \begin{array}{c} 174.6 \+ \; 97.2 \\hlinetotal \end{array} \]Calculate:\[ 174.6 + 97.2 = 271.8 \] The exact sum is 271.8.

Step 4: Compare the Estimated and Exact Values

Compare the estimated sum of 270 to the exact sum of 271.8. The estimate is close to the exact value, with a difference of 1.8.

Key Concepts

Estimating Sums

Estimating sums is a valuable mathematical strategy that involves simplifying calculations by rounding numbers before adding them. This approach is particularly helpful when quick approximations are needed. Rounding numbers means adjusting them to the nearest ten, hundred, or another convenient value, which makes mental arithmetic easier. For instance, if we have 174.6, it can be rounded to 170, and 97.2 can be rounded to 100. The sum of these rounded numbers, 170 + 100, gives us an estimated total of 270.

• Use rounding to simplify calculations.
• Estimates are useful for quick checks.
• Estimates are not exact but should be reasonably close.

Keep in mind that while estimating sums, the purpose is not to find the exact answer but to get a sense of what the sum might be around.

Addition

Addition is one of the most fundamental operations in mathematics. It involves combining numbers to find a sum. To add, numbers should be aligned by their place values, particularly when dealing with decimals. In our example, we started with 174.6 and 97.2. Align these numbers by their decimal points: \[\begin{array}{c}174.6 \+ \,97.2 \hlinetotal\end{array}\] To perform the addition, take each column separately:

• 6 + 2 = 8 in the decimal place.
• 4 + 7 = 11, write 1 and carry over 1.
• 7 + 9 + 1 (carried) = 17, write 7 and carry over 1.
• 1 + 1 (carried) = 2.

Finally, the exact sum of 174.6 and 97.2 is 271.8. Practicing addition and verifying with estimations strengthens your mathematical intuition.

Comparison of Values

Comparing values involves looking at the difference between numbers to understand how close or far apart they are. In the context of our exercise, we first estimated the sum to be 270 by rounding, and the exact calculation gave us 271.8. To compare these:

• Subtract the estimate from the exact value: 271.8 - 270 = 1.8.

This shows a difference of 1.8 between the estimated and exact sums. Comparison helps to assess whether an estimate is reasonable. When estimating and comparing, small differences are generally acceptable because the purpose is to simplify calculation. Larger discrepancies might mean you should check your rounding or calculations again.

Exact vs. Estimated Values

Understanding the difference between exact and estimated values is crucial in mathematics. Exact values are the true results obtained via precise calculations, while estimated values result from rounding and approximations. In our scenario, the exact sum (174.6 + 97.2) calculated as 271.8 gives the precise answer. In contrast, the estimated total of 270 was derived from rounding.

• Exact values provide precise measurements.
• Estimated values offer quick and handy approximations.
• Often used to check whether calculations are in the right ballpark.

While the difference between exact and estimated values in our example was only 1.8, the choice between using one or the other depends on the requirement for precision in your calculations. Estimations are perfect for forecasts, budgeting, and situations where a precise number isn't required.