Problem 62 - 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. $$ 73.73+72.9 $$

Step-by-Step Solution

Answer

The estimated sum is 147; the exact sum is 146.63. They are close, differing by 0.37.

Step 1: Round Each Number

To estimate the sum, first round each number to the nearest whole number. 73.73 rounds up to 74 because the digit after the decimal is 7, which is 5 or more. Similarly, 72.9 rounds up to 73 because the digit after the decimal is 9, which is also 5 or more.

Step 2: Calculate the Estimated Sum

Add the rounded numbers to find the estimated sum. The estimated sum is:\[74 + 73 = 147\]

Step 3: Find the Exact Sum

Now calculate the exact sum of the original numbers. The exact sum is:\[73.73 + 72.9 = 146.63\]

Step 4: Compare Estimated and Exact Values

Compare the estimated sum to the exact sum. The estimated value of 147 is close to the exact value of 146.63, differing by 0.37.

Key Concepts

Estimating Sums

Estimating sums by rounding numbers is an excellent way to quickly get a sense of how large a result you might expect without having to do detailed calculations. This skill becomes particularly useful in your day-to-day life, such as when budgeting, grocery shopping, or planning your time.

• Start by identifying the numbers you need to sum up.
• Round each number to the nearest whole number.
• After rounding, add the numbers to find an estimated sum.

This estimation method gives you a quick insight, which can be valuable for determining whether your real calculations are in the ballpark of what you expect. However, always remember that rounding off numbers means some level of precision is lost, making this method useful for approximate answers only.

Whole Numbers

Whole numbers are the numbers without fractions or decimals. They include numbers like 0, 1, 2, 3, and so on. When estimating sums, rounding numbers to the nearest whole number simplifies the arithmetic and lets you handle larger numbers with ease.

• If a decimal is 5 or more, round up to the next whole number.
• If a decimal is less than 5, round down, keeping the whole number the same.

In the context of our problem, 73.73 becomes 74 and 72.9 becomes 73 when rounded to the nearest whole number. By working with whole numbers, you streamline arithmetic operations while retaining a view on the approximate total or outcome.

Exact Values

Exact values are the precise sums derived without rounding off any of the numbers. While estimating can be helpful, finding exact values is essential in situations where precision matters, such as scientific calculations or financial accounting.
In our example, the numbers 73.73 and 72.9 are added exactly to get the sum of 146.63.

• Use exact numbers when accuracy is critical.
• Always compare exact values with estimated values to understand the discrepancy.

Understanding both the estimated sums through rounding and exact values helps you gauge the level of accuracy needed for different situations. Therefore, while using rounding for quick estimates, ensure to calculate exact values when precision is significant.