Problem 112 - 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. \(\frac{7}{16}+\frac{1}{24}\)

Step-by-Step Solution

Answer

The estimated sum is \(\frac{1}{2}\); the exact sum is \(\frac{23}{48}\). The estimate is slightly higher.

Step 1: Understanding the Problem

We need to estimate the sum of \(\frac{7}{16}+\frac{1}{24}\) by rounding the fractions and then compare it with the exact sum. To start, let's consider the simplest way to round these fractions.

Step 2: Rounding the Fractions

Round \(\frac{7}{16}\) to the nearest simple fraction. Since \(\frac{7}{16}\approx0.4375\), it can be rounded to \(\frac{1}{2}\). Round \(\frac{1}{24}\) as well; since \(\frac{1}{24}\approx0.0417\), it can be rounded to \(0\).

Step 3: Estimating the Sum

Add the rounded fractions: \(\frac{1}{2} + 0 = \frac{1}{2}\). Therefore, the estimated sum of \(\frac{7}{16}+\frac{1}{24}\) is \(\frac{1}{2}\).

Step 4: Finding the Exact Sum

To find the exact sum, we need a common denominator. The least common denominator of 16 and 24 is 48. Convert both fractions: \[\frac{7}{16} = \frac{21}{48}\] and \[\frac{1}{24} = \frac{2}{48}\]. So, the exact sum is \(\frac{21}{48} + \frac{2}{48} = \frac{23}{48}\).

Step 5: Comparing Estimated and Exact Values

The estimated sum \(\frac{1}{2}\) is approximately \(0.5\), while the exact sum \(\frac{23}{48}\approx0.4792\). Comparing these values, the estimation is fairly close to the exact value, though slightly higher.

Key Concepts

Estimating Sums

When dealing with fractions and wanting a quick result, estimating sums through rounding can be quite handy. Rounding helps to simplify fractions into easier numbers. To estimate, look at each fraction and round them to a simple, nearby value. Let's take a closer look at how this works in practice by considering the example of the fractions \(\frac{7}{16}\) and \(\frac{1}{24}\).
\(\frac{7}{16}\) is approximately \(0.4375\). A nearby simple fraction is \(\frac{1}{2}\) or 0.5, making it a good candidate for rounding. Similarly, \(\frac{1}{24}\) roughly equals \(0.0417\) and can be approximated as 0. By rounding, you get simple fractions to work with: \(\frac{1}{2}\) and 0, making the estimated sum \(\frac{1}{2}\).
Keep in mind that while estimating through rounding can provide a quick reference, the result won’t be exact. The main goal of estimating is to obtain a general idea of the sum, which is useful for quick checks and real-world applications without precise accuracy needs.

Common Denominator

To find the exact sum of fractions, a common denominator is necessary. This allows fractions to be combined easily into a single expression. To find a common denominator, you'll typically determine the least common multiple (LCM) of the denominators in your fractions.
In the case of \(\frac{7}{16}\) and \(\frac{1}{24}\), the denominators are 16 and 24. Finding the LCM of these two numbers begins with identifying the smallest multiple they both share, which is 48. This becomes the least common denominator.
With a common denominator of 48, you can express both fractions consistently: Convert \(\frac{7}{16}\) to \(\frac{21}{48}\) and \(\frac{1}{24}\) to \(\frac{2}{48}\). Now, you can easily add these fractions because they have the same base denominator, allowing for simple addition across the numerators.

Exact Sum Calculation

Now that we have a common denominator, obtaining the exact sum is straightforward. Using the previously converted fractions, \(\frac{21}{48}\) and \(\frac{2}{48}\), simply add their numerators together since they share the common denominator of 48.
This provides the exact sum: \(\frac{21}{48} + \frac{2}{48} = \frac{23}{48}\). This fraction \(\frac{23}{48}\) represents the precise sum of the original fractions \(\frac{7}{16}\) and \(\frac{1}{24}\).
Once calculated, compare this exact sum with your estimate. For instance, the estimated sum from rounding was \(\frac{1}{2}\), approximately 0.5, whereas the exact result, \(\frac{23}{48}\), is around 0.4792. This shows the estimation was quite close, illustrating the usefulness of both estimation for speed and exact calculations for accuracy.