Problem 68 - Elementary Algebra

Question

A sausage recipe includes \(\frac{1}{2}\) pound of ground pork and \(\frac{1}{3}\) pound of ground lamb. Find the total amount of groun meat in this recipe.

Step-by-Step Solution

Answer

The total amount of ground meat is \(\frac{5}{6}\) pound.

Step 1 - Identify the Amounts

The recipe includes \(\frac{1}{2}\) pound of ground pork and \(\frac{1}{3}\) pound of ground lamb.

Step 2 - Find a Common Denominator

To add the fractions, find a common denominator. The denominators are 2 and 3. The least common multiple of 2 and 3 is 6.

Step 3 - Convert Fractions

Convert \(\frac{1}{2}\) and \(\frac{1}{3}\) to have a common denominator of 6. \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{1}{3} = \frac{2}{6}\).

Step 4 - Add the Fractions

Now add the fractions: \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).

Step 5 - Conclusion

The total amount of ground meat in the recipe is \(\frac{5}{6}\) pound.

Key Concepts

least common denominator

When adding fractions, it is essential to have the same denominator for both fractions. This denominator is called the **least common denominator** (LCD). In our example, the fractions are \(\frac{1}{2}\) and \(\frac{1}{3}\). The denominators here are 2 and 3. To find the LCD, you need the smallest number that both denominators can divide into without leaving a remainder.

Here's how you find the LCD:

• List the multiples of each denominator.
• 2: 2, 4, 6, 8,...
• 3: 3, 6, 9, 12,...
• The smallest number that appears in both lists is 6.

This means 6 is the least common denominator for \(\frac{1}{2}\) and \(\frac{1}{3}\). Using the LCD simplifies the process of adding fractions.

fraction conversion

After identifying the least common denominator, the next step is to convert the fractions so they both have this common denominator. For our exercise, we need to convert \(\frac{1}{2}\) and \(\frac{1}{3}\) to have a denominator of 6.

Here’s how to do it:

• To convert \(\frac{1}{2}\) to a fraction with a denominator of 6, multiply both the numerator and the denominator by 3.
• \(\frac{1}{2} \times \frac{3}{3} = \frac{3}{6}\)
• To convert \(\frac{1}{3}\) to a fraction with a denominator of 6, multiply both the numerator and the denominator by 2.
• \(\frac{1}{3} \times \frac{2}{2} = \frac{2}{6}\)

Now, both fractions \(\frac{1}{2}\) and \(\frac{1}{3}\) have been converted to \(\frac{3}{6}\) and \(\frac{2}{6}\), respectively. This makes it easier to add them together.

fraction addition

With both fractions now having the same denominator, adding them is straightforward. Simply add the numerators while keeping the common denominator the same.

Using our example:

• Take \(\frac{3}{6}\) (converted from \(\frac{1}{2}\)) and \(\frac{2}{6}\) (converted from \(\frac{1}{3}\)).
• Combine the numerators: 3 + 2 = 5.
• Keep the denominator the same, which is 6.
• You get \(\frac{5}{6}\).

So, the total amount of ground meat in the recipe is \(\frac{5}{6}\) pounds. This shows how important a common process like finding the least common denominator is for adding fractions effectively. Now, you can confidently add any fractions by following these steps.