Problem 19 - 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. $$ 9 \frac{1}{15}+6 \frac{4}{5} $$

Step-by-Step Solution

Answer

The estimated sum is 16, and the exact sum is \( 15 \frac{13}{15} \).

Step 1: Understand the Problem

We are given the sum of two mixed numbers: \( 9 \frac{1}{15} \) and \( 6 \frac{4}{5} \). We need to first estimate the sum by rounding each number and then find the exact sum.

Step 2: Round the Mixed Numbers

Round each mixed number to the nearest whole number. \( 9 \frac{1}{15} \) is close to 9, and \( 6 \frac{4}{5} \) is close to 7. So the estimated sum will be \( 9 + 7 = 16 \).

Step 3: Convert Mixed Numbers to Improper Fractions

Convert \( 9 \frac{1}{15} \) and \( 6 \frac{4}{5} \) into improper fractions. \[9 \frac{1}{15} = \frac{9 \times 15 + 1}{15} = \frac{136}{15}\] \[6 \frac{4}{5} = \frac{6 \times 5 + 4}{5} = \frac{34}{5}\]

Step 4: Find a Common Denominator

Find a common denominator to add the fractions \( \frac{136}{15} \) and \( \frac{34}{5} \). The least common denominator of 15 and 5 is 15. Convert \( \frac{34}{5} \) to \( \frac{102}{15} \).

Step 5: Add the Improper Fractions

Add the fractions with a common denominator: \[\frac{136}{15} + \frac{102}{15} = \frac{238}{15}\]

Step 6: Convert the Sum Back to a Mixed Number

Convert \( \frac{238}{15} \) into a mixed number by dividing 238 by 15. The quotient is 15 and the remainder is 13: \[238 \div 15 = 15 \text{ R } 13\] Thus, \( \frac{238}{15} = 15 \frac{13}{15} \).

Step 7: Compare the Estimated Sum with the Exact Sum

The estimated sum was 16, and the exact sum we calculated is \( 15 \frac{13}{15} \). The estimated sum is very close to the exact sum.

Key Concepts

Mixed Numbers

A mixed number is a number made up of a whole number and a fraction. It is a convenient way to express quantities that are more than whole but not as complex as improper fractions. For example, the mixed number \(9 \frac{1}{15}\) tells us there are 9 full parts and a little more than 1/15 of another part.
Mixed numbers are often encountered in real-life situations such as measurements in recipes or distances, where exact amounts are expressed as parts of a whole. Converting mixed numbers to improper fractions makes it easier to perform mathematical operations like addition or subtraction.

Improper Fractions

An improper fraction is a type of fraction where the numerator is larger than the denominator, representing a value greater than one, such as \(\frac{136}{15}\). Unlike mixed numbers, improper fractions make it easier to handle mathematical calculations.
Converting between mixed numbers and improper fractions is essential. To convert a mixed number into an improper fraction, multiply the whole number by the denominator and add the numerator. Place the result over the original denominator. This transformation is crucial for performing operations, like addition, that require a common format. Conversely, simplifying back to a mixed number involves dividing the numerator by the denominator, giving a whole number quotient and a remainder.

Common Denominator

Finding a common denominator between fractions is essential when adding or subtracting them. A common denominator ensures that the fractions refer to the same whole, allowing for straightforward addition or subtraction.
In our original exercise, the fractions \(\frac{136}{15}\) and \(\frac{34}{5}\) needed a common denominator. The least common denominator (LCD) here is 15, which allows both fractions to be expressed with the same bottom number.

• Multiply the numerator and denominator of \(\frac{34}{5}\) by 3 to reach a denominator of 15, resulting in \(\frac{102}{15}\).
• Add \(\frac{136}{15}\) and \(\frac{102}{15}\) to complete the operation.

This commonality facilitates addition or subtraction by aligning the parts of each fraction to be on equal footing.

Estimating Sums

Estimating sums by rounding can be very useful when a quick approximation is needed, especially in practical applications like budgeting or measuring. Rounding mixed numbers to the nearest whole number aids in creating a simple estimate. For example, \(9 \frac{1}{15}\) rounds down to 9, and \(6 \frac{4}{5}\) rounds up to 7. Adding these yields an estimated sum of 16.
This step provides a useful check against more complex calculations, ensuring that the detailed arithmetic is on the right track and within a reasonable range. It also offers insight into whether the expected result aligns with the performed calculations, helping to spot errors early. However, while estimation helps with accuracy, always remember it doesn't replace calculation, just simplifies it.