Problem 83 - Elementary Algebra

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? What advice would you offer your friend?

Step-by-Step Solution

Answer

The answer is correct, and your friend followed the right process.

Step 1: Verify the Calculation of the Numerator

Check the calculation of the numerator in the addition. Your friend used the formula \(a(\text{denominator of } b) + b(\text{denominator of } a)\). For the expression \(\frac{5}{8}+\frac{7}{12}\), the numerator should be \(5(12) + 7(8) = 60 + 56 = 116\). This part is correct.

Step 2: Verify the Calculation of the Denominator

Your friend calculated the denominator by multiplying the denominators directly: \(8 \times 12 = 96\). This is also correct because when adding fractions, we use the product of the denominators if we're doing it manually without reducing to the Lowest Common Multiple (LCM) first.

Step 3: Check the Simplification

Now check if the resulting fraction \(\frac{116}{96}\) is simplified correctly. Divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 116 and 96 is 4, so \(\frac{116\div 4}{96\div 4} = \frac{29}{24}\), which is correct.

Step 4: Ensure the Use of Simplest Form

It's important to always express the final answer in its simplest form. Here, \(\frac{29}{24}\) is already in its simplest form.

Key Concepts

Numerator Calculation

When adding fractions, the numerator of the resulting fraction is calculated by multiplying each fraction's numerator by the opposing denominator, then adding those products together. Let's break down what your friend did with an example.For two fractions, like \( \frac{5}{8} + \frac{7}{12} \), the steps are:

• Multiply the numerator of the first fraction (5) by the denominator of the second fraction (12), giving you \(5 \times 12 = 60\).
• Multiply the numerator of the second fraction (7) by the denominator of the first fraction (8), which results in \(7 \times 8 = 56\).
• Add these two results together to get the new numerator. So, \(60 + 56 = 116\).

Using this approach ensures each part of the fractions has been properly cross-multiplied, leading to the correct numerator for the fraction sum.

Denominator Calculation

Determining the denominator when adding fractions can seem a bit tricky at first, but it follows a straightforward rule. Your friend's approach assumed multiplying the denominators directly, which can work, but let's understand why.Consider \( \frac{5}{8} + \frac{7}{12} \):

• We directly multiply the denominators: \(8 \times 12 = 96\).

This approach is simplistic but doesn't always result in the least complex calculation. Ideally, one should find the Least Common Multiple (LCM) of the denominators (8 and 12) for simpler arithmetic. However, multiplying the denominators directly remains a valid (though sometimes cumbersome) manual calculation method.In this case, both methods yield a numerically correct common denominator, allowing the addition to proceed accurately.

Simplifying Fractions

After calculating the sum of fractions like \(\frac{116}{96}\), it's important to simplify it to its lowest form for clarity and ease. Your friend wisely simplified this step, but here's how it's thoroughly done.Simplification involves:

• Finding the greatest common divisor (GCD) of the numerator and denominator. Here, the GCD of 116 and 96 is found to be 4.
• Divide both the numerator (116) and the denominator (96) by this GCD: \(116 \div 4 = 29\) and \(96 \div 4 = 24\). This results in the simplified fraction \(\frac{29}{24}\).

Ensuring fractions are in their simplest form not only makes them easier to work with but also gives a cleaner, clearer final answer. This final expression can't be reduced further, confirming that \(\frac{29}{24}\) is indeed the simplest form.