Problem 47 - Algebra A Combined Function

Question

To find the average of two numbers, we find their sum and divide by \(2 .\) For example, the average of 65 and 81 is found by simplifying \(\frac{65+81}{2} .\) This simplifies to \(\frac{146}{2}=73 .\) Find the average of \(\frac{1}{3}\) and \(\frac{3}{4}\).

Step-by-Step Solution

Answer

The average of \(\frac{1}{3}\) and \(\frac{3}{4}\) is \(\frac{13}{24}\).

Step 1: Find the sum of the fractions

To find the average, we first calculate the sum of the two fractions: \(\frac{1}{3}\) and \(\frac{3}{4}\). We need a common denominator to add them. The least common denominator of 3 and 4 is 12. Convert \(\frac{1}{3}\) to \(\frac{4}{12}\) and \(\frac{3}{4}\) to \(\frac{9}{12}\). Add them: \(\frac{4}{12} + \frac{9}{12} = \frac{13}{12}\).

Step 2: Divide the sum by 2

Now, we need to find the average by dividing the sum \(\frac{13}{12}\) by 2. Dividing by 2 is equivalent to multiplying by the reciprocal, \(\frac{1}{2}\): \(\frac{13}{12} \times \frac{1}{2} = \frac{13}{24}\).

Key Concepts

Adding Fractions

Adding fractions can be tricky if you're not used to working with different denominators. The key step is to find a common denominator, which allows you to add the numerators directly.

• First, let's understand what denominators and numerators are. In a fraction, the numerator is the top number, expressing how many parts you're taking into account. The denominator is the bottom number, indicating the total number of equal parts something is divided into.
• To add fractions, you need both denominators to be the same. Why? Because fractions like and are different slices of wholes. Simply put, you can't add unlike things without converting them into like things first.
• Once you get both fractions to have the same denominator, just add the numerators, and keep the denominator the same. This gives you a single fraction that incorporates both values.

This is what we did when we converted to and added them. Remember, the denominator stays the same, only numerators are added.

Least Common Denominator

The least common denominator (LCD) is essential in adding or comparing fractions. It helps standardize different fractions so they can be easily added or compared.

• The LCD is the smallest number that both denominators can divide into without leaving a remainder.
• To find the LCD, you can list the multiples of each denominator and identify the smallest common multiple. For instance, the multiples of 3 are 3, 6, 9, 12, 15, while the multiples of 4 are 4, 8, 12, 16. Here, 12 is the smallest number that appears in both lists.
• Another approach to finding the LCD is using the greatest common divisor, but it's not necessary when dealing with small numbers. Using the LCD makes adding and subtracting fractions far more manageable.

In our problem, this led us to convert and so that their terms are compatible for addition.

Dividing Fractions

Dividing fractions can initially seem challenging, but it becomes straightforward once you understand the process. Unlike regular division, dividing by a fraction involves multiplying by its reciprocal.

• What is a reciprocal? A reciprocal is simply flipping the numerator and the denominator of the fraction. So, the reciprocal of is .
• When you need to divide by a fraction, instead of performing division, multiply by the reciprocal of the fraction.
• This is known as "invert and multiply" and is the principle behind dividing fractions. It means you multiply the first fraction by the reciprocal of the second fraction.

For instance, when dividing by 2, you find the reciprocal of 2, which is , and multiply by this instead.