Problem 115 - 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. \(\frac{13}{25}+\frac{7}{30}\)

Step-by-Step Solution

Answer

Estimated sum is 0.75, exact sum is approximately 0.753.

Step 1: Understand the Fractions

First, break down each fraction. We have \(\frac{13}{25}\) and \(\frac{7}{30}\). These will need to be estimated using rounding, and then calculated exactly to find the sum.

Step 2: Round Each Fraction to Simplify

Round \(\frac{13}{25}\) to the nearest simple fraction. Since \(\frac{13}{25} \approx 0.52\), it can be rounded to \(\frac{1}{2}\) or \(0.5\). Round \(\frac{7}{30}\). Since \(\frac{7}{30} \approx 0.233\), it is closer to \(\frac{1}{4}\) or \(0.25\) when rounded.

Step 3: Estimate the Sum

Add the rounded values: \(\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}\). So, the estimated sum is \(0.75\).

Step 4: Find the Exact Value

To find the exact sum, first find a common denominator for \(\frac{13}{25}\) and \(\frac{7}{30}\). The least common multiple of 25 and 30 is 150. Convert each fraction: \(\frac{13}{25} = \frac{78}{150}\) and \(\frac{7}{30} = \frac{35}{150}\). Add them: \(\frac{78}{150} + \frac{35}{150} = \frac{113}{150}\). The exact sum is \(\frac{113}{150} \approx 0.753\).

Step 5: Compare Estimated and Exact Values

The estimated sum was \(0.75\) and the exact sum is approximately \(0.753\). Both values are very close, indicating that the rounding estimation was a reasonable approximation of the actual sum.

Key Concepts

Rounding Fractions

Rounding fractions is a useful strategy when estimating sums. It simplifies calculations by converting fractions into numbers that are easier to work with.
For example, if a fraction is close to a simple benchmark value like \( \frac{1}{2} \) or \( \frac{1}{4} \), you can round it to these values.
This process involves comparing the decimal equivalent of a fraction to see which benchmark it's closest to.

• Consider \( \frac{13}{25} \approx 0.52 \). It is rounded to \( \frac{1}{2} \) since 0.52 is closer to 0.5.
• Similarly, \( \frac{7}{30} \approx 0.233 \) becomes \( \frac{1}{4} \) as 0.233 is close to 0.25.

Rounding simplifies the addition process, making it an accessible method to quickly estimate sums.

Common Denominator

When adding fractions, ensuring a common denominator is essential.
A common denominator is a shared multiple of the denominators of the fractions involved in the operation.
By using a common denominator, fractions become easier to add because it transforms them into equivalent fractions that share the same base.

• Consider fractions like \( \frac{13}{25} \) and \( \frac{7}{30} \). To add them, you first need to find a denominator that both 25 and 30 can divide into without leaving a remainder.
• Using the least common multiple (LCM) as the common denominator is a helpful approach and will be explained next.

Once you have re-expressed the original fractions with this shared denominator, it becomes straightforward to combine them.

Fraction Addition

The operation of fraction addition requires common denominators. Once aligned, you simply add the numerators while keeping the denominator the same.
Let’s use our exercise example: convert \( \frac{13}{25}\) and \( \frac{7}{30} \) using a common denominator so you can add them smoothly.

• Our common denominator is 150. Hence, \( \frac{13}{25} \) changes to \( \frac{78}{150} \), and \( \frac{7}{30} \) changes to \( \frac{35}{150} \).
• With this conversion, add the numerators: \( 78 + 35 = 113 \).
• Combine the results: \( \frac{113}{150} \), which rounds to approximately 0.753 in decimal form.

This method ensures accuracy and is essential when exact sums are necessary.

Least Common Multiple

Finding the least common multiple (LCM) is crucial when dealing with problems requiring a common denominator.
The LCM of two numbers is the smallest number that both can evenly divide into. It greatly simplifies the process of finding a common denominator.
For instance, determining the LCM of 25 and 30 involves evaluating multiples:

• List a few multiples: 25, 50, 75, 100, 125, 150 for 25 and 30, 60, 90, 120, 150 for 30.
• You can see 150 appears in both lists first, so it is the least common multiple.

Using the LCM can dramatically reduce the complexity of fraction addition by minimizing the size of the fractions being calculated. It ensures that you can easily compute the exact values when working with multiple fractions.