Problem 45 - 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. $$ 1,007 \div 19 $$

Step-by-Step Solution

Answer

Estimate: 50; Exact: 53; Estimate is slightly lower.

Step 1: Rounding the Dividend

First, we need to round the dividend, 1,007. The nearest round number to 1,007 is 1,000 because both the hundreds digit (0) is less than 5, so we round down to the nearest thousand.

Step 2: Rounding the Divisor

Next, we round the divisor, 19. Since 19 is close to 20, we round it to 20.

Step 3: Estimation Using Rounded Numbers

Now, we use the rounded figures to estimate. We estimate 1,000 divided by 20. \[\frac{1,000}{20} = 50\]So, the estimated value is 50.

Step 4: Calculating the Exact Value

Now, calculate the exact value of the expression. \[\frac{1,007}{19} \approx 53\]By performing the division, we find that the exact value is approximately 53.

Step 5: Comparing Estimates with Exact Values

Compare the estimated value (50) with the exact value (53). The estimated value is slightly lower than the exact value.

Key Concepts

Estimating Division Results

Estimating division results helps simplify complex calculations and provides a quick approximation without the need for precise computation. In many cases, estimating using rounding is a practical way to gauge approximate values, especially in mental math or when precision is not critical. To make an estimation:

• Rounding the dividend and divisor to the nearest ten, hundred, or another convenient unit can streamline the process.
• Calculate the division with these rounded numbers.

For instance, in the exercise given, the numbers 1,007 and 19 were rounded to 1,000 and 20, respectively. This rounding simplifies calculating an estimate, resulting in an easier computation of \[\frac{1,000}{20} = 50\]As a result, the estimated division result is 50, which gives us a quick sense of the answer without performing detailed division.

Exact vs Estimated Values

Understanding the difference between exact and estimated values is essential in mathematics and everyday life. An exact value is obtained from precise calculations, while an estimated value offers an approximation derived from simplified calculations.

The steps to achieve each value vary:

• Exact Value: Perform the division operation traditionally using the original numbers. It's precise and leaves no room for error, as demonstrated by dividing 1,007 by 19, resulting in a more accurate quotient of approximately 53.
• Estimated Value: Round the numbers and perform the division using these simpler figures. Although not precise, it's a valuable tool for quick judgments or when exactness is unnecessary.

In the given exercise, the estimated value of 50 (from rounding) is compared to the exact value of around 53. This comparison highlights that while estimates are handy, they may vary from the precise outcome.

Rounding Dividends and Divisors

Rounding dividends and divisors is a crucial skill to simplify division operations and make estimation more feasible. The process involves altering numbers to their closest significant figures and performing operations on these simplified figures.

When rounding:

• Dividends: Look at the hundreds or thousands place, depending on the number. For example, rounding 1,007 to the nearest thousand simplifies it to 1,000, as the digit in the hundreds place (0) is less than 5.
• Divisors: Consider rounding to the nearest ten or other convenient interval. For instance, 19 rounds up to 20 as the next easy-to-use number.

By rounding both the dividend and divisor to simpler numbers, like turning 1,007 into 1,000 and 19 into 20, calculation becomes less complicated, enabling quicker and more efficient estimation. This technique is useful for mental math or when an approximate value suffices.