Problem 72 - Algebra for College Students

Question

Suppose that your friend does an addition problem as follows: $$ \frac{5}{8}+\frac{7}{12}=\frac{5(12)+8(7)}{8(12)}=\frac{60+56}{96}=\frac{116}{96}=\frac{29}{24} $$ Is this answer correct? If not, what advice would you offer your friend?

Step-by-Step Solution

Answer

The final answer \( \frac{29}{24} \) is correct, but your friend should first find a common denominator.

Step 1: Understanding the Problem

Our goal is to add the fractions \( \frac{5}{8} \) and \( \frac{7}{12} \) correctly. We need to check if the provided solution follows the proper steps for adding fractions.

Step 2: Checking the Given Solution

The solution given was \( \frac{5}{8} + \frac{7}{12} = \frac{60 + 56}{96} = \frac{116}{96} = \frac{29}{24} \). The addition should be performed by finding a common denominator, not by directly cross multiplying and summing the numerators and denominators as shown.

Step 3: Finding a Common Denominator

The common denominator for 8 and 12 is their least common multiple (LCM), which is 24. Hence, we convert \( \frac{5}{8} \) and \( \frac{7}{12} \) to equivalent fractions with denominator 24.

Step 4: Converting Fractions to Common Denominator

To convert \( \frac{5}{8} \) to a fraction with a denominator of 24, we multiply the numerator and the denominator by 3, which gives \( \frac{15}{24} \). Similarly, to convert \( \frac{7}{12} \), multiply both numerator and denominator by 2 to get \( \frac{14}{24} \).

Step 5: Adding the Converted Fractions

Now, we can add the fractions: \( \frac{15}{24} + \frac{14}{24} = \frac{15 + 14}{24} = \frac{29}{24} \).

Step 6: Evaluating the Final Solution

The final result \( \frac{29}{24} \) is correct, but the process used in the original solution is incorrect. The correct process involves converting the fractions to a common denominator first before adding.

Key Concepts

Common Denominator

When adding fractions, it's crucial to have a common denominator. This means both fractions need to have the same bottom number, so covering the same kind of division of the whole.
The common denominator serves as the shared reference point for the fractions, allowing direct addition of the numerators. To find a suitable common denominator, identify a number that both denominators can divide evenly, ensuring no change in the value of the fractions.
In the exercise shared, the fractions \( \frac{5}{8} \) and \( \frac{7}{12} \) were added incorrectly in the original attempt, as they didn't have a common denominator. Therefore, identifying a correct common denominator is essential to add these two fractions accurately.

Least Common Multiple

The least common multiple (LCM) is an essential concept in adding fractions since it helps us find the smallest common denominator for two or more fractions. The LCM of two numbers is the smallest number that both numbers perfectly divide into without a remainder.
For example, when finding the LCM for 8 and 12, consider which multiples they share:

• Multiples of 8: 8, 16, 24, 32, ...
• Multiples of 12: 12, 24, 36, ...

As we can see, 24 is the smallest number found in both lists, making it the LCM of 8 and 12.
Once the least common multiple is identified, it becomes easier to convert the fractions, ensuring they have the same denominator, facilitating their addition.

Converting Fractions

Once the common denominator is determined, fractions must be adjusted to share this denominator, a process called converting fractions. This step is crucial for adding fractions accurately.
To do this, multiply the numerator and denominator of each fraction by the same number so that the denominator matches the common denominator.
In our exercise:

• Convert \( \frac{5}{8} \) to be over 24: Multiply both the numerator (5) and denominator (8) by 3 to get \( \frac{15}{24} \).
• Convert \( \frac{7}{12} \) to be over 24: Multiply both the numerator (7) and denominator (12) by 2 to get \( \frac{14}{24} \).

Now, with matching denominators, adding the fractions becomes straightforward. Add the numerators and keep the common denominator for a correct solution: \( \frac{15}{24} + \frac{14}{24} = \frac{29}{24} \).
This adjustment method ensures that even though the fractions are initially dissimilar, they are directly comparable and ready to be added together.