Problem 43 - Fundamentals of Mathematics

Question

(Section 5.3) Find the sum of \(\frac{4}{5}+\frac{3}{8}\).

Step-by-Step Solution

Answer

The sum is \( 1 \frac{7}{40} \).

Step 1: Identify the Problem

We need to find the sum of two fractions: \( \frac{4}{5} \) and \( \frac{3}{8} \).

Step 2: Find the Common Denominator

The denominators of the fractions are 5 and 8. The least common multiple of 5 and 8 is 40, which will be our common denominator.

Step 3: Convert Fractions

Convert each fraction to an equivalent fraction with a denominator of 40.\[ \frac{4}{5} = \frac{4 \times 8}{5 \times 8} = \frac{32}{40} \] \[ \frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40} \]

Step 4: Add the Fractions

Now add the two fractions with a common denominator:\[ \frac{32}{40} + \frac{15}{40} = \frac{32+15}{40} = \frac{47}{40} \]

Step 5: Simplify the Result

The sum \( \frac{47}{40} \) is an improper fraction. We can convert it into a mixed number: \( 1 \frac{7}{40} \).

Key Concepts

Understanding the Least Common Multiple

When working with fractions that have different denominators, it's important to find a common ground where they can coexist. This is where the concept of Least Common Multiple (LCM) comes into play.

The LCM of two numbers is the smallest number that is a multiple of both. To find the LCM of two denominators, like 5 and 8 from our original exercise, list the multiples of each number:

• Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40...
• Multiples of 8: 8, 16, 24, 32, 40...

As you can see, 40 is the first multiple that appears in both lists, making it the LCM.

Knowing how to find the LCM is crucial because it allows us to convert fractions into equivalent fractions with the same denominator. This makes it possible to add, subtract, or compare fractions effectively.

Breaking Down Improper Fractions

An improper fraction is when the numerator (the top number) is larger than the denominator (the bottom number). For instance, in our solution, we reached an improper fraction: \( \frac{47}{40} \).

Improper fractions are perfectly fine, but often, teachers or textbooks prefer them to be expressed as mixed numbers for ease of understanding. To tackle this, one might convert an improper fraction into a mixed number.

To do this, divide the numerator by the denominator. The quotient becomes the whole number part of the mixed number, while the remainder becomes the numerator of the fraction part.

In our case, dividing 47 by 40 gives 1 with a remainder of 7, hence:

• The whole number: 1
• The remainder: 7
• Thus, the mixed number is \( 1 \frac{7}{40} \)

Improper fractions are important in mathematics because they offer a way to express a quantity that surpasses a whole unit.

The Role and Conversion of Mixed Numbers

Mixed numbers combine whole numbers with fractions. They are an intuitive way to express numbers greater than one, often appearing naturally in real-life situations, like splitting a pizza.

In our exercise, after converting the improper fraction \( \frac{47}{40} \) into \( 1 \frac{7}{40} \), we arrive at a mixed number. This involves:

• The whole number (1 in this case) represents the full groups of 40 that fit into 47.
• The fraction (\( \frac{7}{40} \)) expresses the part of the group we have beyond our full whole.

Mixed numbers can also be converted back to improper fractions if needed. Simply multiply the whole number by the denominator, add the numerator, and place over the original denominator. This reverse process maintains consistency across mathematical operations, like addition or subtraction. Understanding mixed numbers helps smooth the transition between different fractional representations.