Problem 7 - Fundamentals of Mathematics

Question

Estimate each sum or difference using the method of rounding. After you have made an estimate, find the exact value of the sum or difference and compare this result to the estimated value. Result may vary. \(\frac{9}{10}+\frac{3}{5}\)

Step-by-Step Solution

Answer

Estimated sum: 2, exact sum: 1.5. Estimate is greater.

Step 1: Convert Fractions to Decimal Form for Rounding

First, we convert each fraction to its decimal form. \(\frac{9}{10}\) is 0.9 and \(\frac{3}{5}\) is 0.6. Rounding these values to the nearest whole number, we round 0.9 up to 1 and 0.6 up to 1.

Step 2: Estimate the Sum Using Rounded Values

Using the rounded values from Step 1, we calculate the estimated sum: \(1 + 1 = 2\). This is our estimate for \(\frac{9}{10} + \frac{3}{5}\).

Step 3: Find the Exact Sum of the Fractions

Now, let's find the exact sum. First, find a common denominator for the fractions \(\frac{9}{10}\) and \(\frac{3}{5}\). The common denominator is 10. Convert \(\frac{3}{5}\) to \(\frac{6}{10}\). So, \(\frac{9}{10} + \frac{6}{10} = \frac{15}{10}\). Simplifying, \(\frac{15}{10} = 1.5\).

Step 4: Compare Estimated Sum to Exact Sum

The estimated sum was 2, and the exact sum is 1.5. The estimate of 2 is greater than the exact answer, but it was a reasonable estimation given our rounded values.

Key Concepts

Rounding Numbers

Rounding numbers is a technique used to simplify numbers, making them easier to work with. The process involves converting a given number to the nearest specified place value. When you round decimals like 0.9 or 0.6, you might round them to the nearest whole number to make mental calculations simpler.

• For example, 0.9 is very close to 1, so we round it up to 1.
• Similarly, 0.6 is closer to 1 than to 0, so it is also rounded up to 1.

Rounding is particularly helpful in estimation. It allows you to quickly approximate sums, differences, or other calculations without the need for complex arithmetic. This process requires you to decide the level of accuracy needed: are you rounding to the nearest whole number, tenth, or hundredth? The choice depends on the context of the problem or the degree of precision required.

Fraction Addition

Adding fractions may seem challenging at first, but it becomes easy once you understand the steps. Before you add fractions, you need to ensure they have a common denominator. This means both fractions need to have the same number at the bottom. A common denominator is the first step in fraction addition. For example, to add fractions like \( \frac{9}{10} \) and \( \frac{3}{5} \), find a number both denominators can divide into evenly. In this case, it is 10.

• Convert \( \frac{3}{5} \) by multiplying the numerator and the denominator by 2, resulting in \( \frac{6}{10} \).
• Now, add \( \frac{9}{10} \) and \( \frac{6}{10} \), resulting in \( \frac{15}{10} \).

Simplifying the fraction gives a cleaned-up result of \( 1.5 \). Understanding this ensures accurate addition, whether for an estimation or deriving an exact result.

Decimal Conversion

Converting fractions to decimals is a fundamental skill that simplifies many math tasks, including estimation and comparison. A fraction can be turned into a decimal by dividing the numerator by the denominator.

• Take \( \frac{9}{10} \) as an example, where dividing 9 by 10 results in 0.9.
• This process applies similarly to \( \frac{3}{5} \). Dividing 3 by 5 gives you 0.6.

These conversions help in making estimations easier. Once fractions are in decimal form, you can quickly round or add them as shown in the exercise above. Decimal conversion is particularly useful because it allows individuals to apply operations universally understood, like addition and subtraction, without dealing with various denominators.