Problem 113 - Fundamentals of Mathematics

Question

Estimate each sum using the method of rounding fractions. After you have made an estimate, find the exact value. Compare the exact and estimated values. Results may vary. \(\frac{7}{15}+\frac{13}{30}\)

Step-by-Step Solution

Answer

Estimated sum is 1, exact sum is \( \frac{9}{10} \).

Step 1: Round Fractions

First, let's round each fraction to the nearest half. The fraction \( \frac{7}{15} \) is close to \( \frac{1}{2} \) because \( \frac{7}{15} \approx 0.47 \). The fraction \( \frac{13}{30} \) is close to \( \frac{1}{2} \) because \( \frac{13}{30} \approx 0.43 \). So both fractions can be estimated as \( \frac{1}{2} \).

Step 2: Estimate the Sum

Add the rounded fractions: \( \frac{1}{2} + \frac{1}{2} \). This equals \( 1 \). Therefore, our estimated sum of the two fractions is \( 1 \).

Step 3: Find a Common Denominator

To find the exact sum, we need a common denominator. The denominators 15 and 30 have a least common multiple of 30. So, we'll convert \( \frac{7}{15} \) and \( \frac{13}{30} \) to fractions with a denominator of 30.

Step 4: Convert Fractions

Convert \( \frac{7}{15} \) to \( \frac{30}{30} \) form: \( \frac{7}{15} = \frac{14}{30} \), since \( 7 \times 2 = 14 \). \( \frac{13}{30} \) is already in \( \frac{30}{30} \) form: \( \frac{13}{30} \).

Step 5: Add the Exact Fractions

Now add \( \frac{14}{30} + \frac{13}{30} = \frac{27}{30} \).

Step 6: Simplify the Fraction

Simplify \( \frac{27}{30} \) by dividing the numerator and the denominator by their greatest common divisor, which is 3. \( \frac{27}{30} = \frac{9}{10} \).

Step 7: Compare Estimated and Exact Values

The estimated value was \( 1 \) and the exact value is \( \frac{9}{10} \), which is close to \( 1 \), hence confirming our estimate was reasonable.

Key Concepts

Estimating Sums

Estimating sums can be a helpful skill, especially when dealing with complex numbers or fractions. It simplifies calculations and helps us to quickly check the plausibility of an answer. The method of rounding fractions is one common way to estimate sums.

When rounding fractions to estimate the sum, you simply round each fraction to the nearest benchmark fraction. For instance, common benchmarks include 0, 1/4, 1/2, 3/4, and 1. In the given example:

• The fraction \( \frac{7}{15} \approx 0.47 \) was rounded to \( \frac{1}{2} \).
• The fraction \( \frac{13}{30} \approx 0.43 \) was also rounded to \( \frac{1}{2} \).

After rounding, simply add the resulting fractions. Here, \( \frac{1}{2} + \frac{1}{2} = 1 \) was the estimated sum.

Estimating is quick and beneficial, often used in various real-world scenarios, such as budgeting or cooking. Estimations provide a rough idea of what to expect and if performed correctly, they should be fairly close to the exact result.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions, as it enables us to combine the fractions seamlessly. A common denominator is a shared multiple of the denominators of the fractions involved. To find it, you can identify the least common multiple (LCM) of the denominators.

In the example problem, 15 and 30 were the denominators:

• The LCM of 15 and 30 is 30.

The conversion process involved changing fractions to equivalent fractions with this common denominator. For \( \frac{7}{15} \):

• Multiply the numerator and the denominator by 2, converting it to \( \frac{14}{30} \).

In cases where one fraction already has the common denominator, like \( \frac{13}{30} \), no conversion is necessary.

Once all fractions involved have the same denominator, they can be directly added or subtracted. This step is crucial for obtaining precise sums and differences.

Simplifying Fractions

After adding fractions with a common denominator, the resulting fraction may need simplification. Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1.

In the example, after finding the sum \( \frac{14}{30} + \frac{13}{30} = \frac{27}{30} \), the fraction \( \frac{27}{30} \) can be simplified. To simplify:

• Identify the greatest common divisor (GCD). For 27 and 30, the GCD is 3.
• Divide both the numerator (27) and the denominator (30) by their GCD.

This results in \( \frac{9}{10} \), the simplified form of \( \frac{27}{30} \).

Simplifying fractions makes them easier to work with and understand, revealing their true proportions. It's a key mathematical skill, enhancing the clarity and simplicity of your calculations.