Problem 82 - Algebra A Combined Function

Question

For Exercises \(79-82,\) determine whether the work is correct or incorrect. If incorrect, find the error and correct. See the Concept Check in this section. $$ \frac{16}{28}=\frac{2 \cdot 5+6 \cdot 1}{2 \cdot 5+6 \cdot 3}=\frac{1}{3} $$

Step-by-Step Solution

Answer

The work is incorrect. Correct simplification of \(\frac{16}{28}\) is \(\frac{4}{7}\).

Step 1: Understand the Problem

We are given the problem of determining whether the simplification \( \frac{16}{28} = \frac{2 \cdot 5 + 6 \cdot 1}{2 \cdot 5 + 6 \cdot 3} \) leading to \( \frac{1}{3} \) is correct. There is an expression that appears to use an unusual method to simplify a fraction.

Step 2: Verify Multiplication and Addition

Verify the numerators: \( 2 \cdot 5 + 6 \cdot 1 = 10 + 6 = 16 \) and the denominators: \( 2 \cdot 5 + 6 \cdot 3 = 10 + 18 = 28 \). This calculation is correct, matching the original fraction of \( \frac{16}{28} \).

Step 3: Simplify the Fraction Correctly

The given method doesn't simplify correctly. Instead, simplify \( \frac{16}{28} \) by finding the greatest common divisor (GCD) of 16 and 28, which is 4. So, divide both the numerator and the denominator by 4: \( \frac{16}{4} = 4 \) and \( \frac{28}{4} = 7 \), resulting in \( \frac{4}{7} \).

Step 4: Conclusion

The original simplification attempt showing \( \frac{1}{3} \) is incorrect. The correct simplification of \( \frac{16}{28} \) is \( \frac{4}{7} \). It appears the given work was an invalid method, mistakenly implying correct results with incorrect arithmetic.

Key Concepts

Understanding the Greatest Common Divisor

The greatest common divisor (GCD) plays a crucial role in fraction simplification. It is the largest number that divides both the numerator and the denominator without leaving a remainder.
This makes it the key element for reducing fractions to their simplest form.

• Take the numbers 16 and 28 from our exercise as an example.
• First, list the factors of each number: 16 has 1, 2, 4, 8, 16; 28 has 1, 2, 4, 7, 14, 28.
• Identify the greatest factor they have in common, which is 4.

By dividing both the numerator and the denominator by this GCD, we ensure that the fraction is reduced correctly. This essential step helps avoid errors in fraction simplification.

Role of Numerator and Denominator

Fractions are composed of two parts: the numerator and the denominator. The numerator is the top number, representing the number of parts we have. The denominator is the bottom number, indicating into how many parts the whole is divided.
To simplify a fraction, reducing both these components using the GCD preserves the value of the fraction while minimizing its form.

• In our solved example, 16 is the numerator, and 28 is the denominator.
• When simplified using the GCD of 4, the fraction changes to \( \frac{4}{7} \).

Understanding these parts helps in applying the correct method for fraction simplification, sidestepping errors like incorrect arithmetic.

Ensuring Correct Arithmetic Methods

Using correct arithmetic methods is essential for accurate mathematical computation. In the context of fraction simplification, it involves correctly identifying and applying the GCD to simplify a fraction without altering its value.
In our given exercise, incorrect arithmetic methods initially led to a misleading simplification of \( \frac{1}{3} \).To clarify:

• Follow the correct method by checking calculations carefully.
• Ensure the original value is preserved by reducing both the numerator and denominator appropriately.
• Always verify the final result by checking if there is any possibility of further simplification.

By adhering to these arithmetic methods, mathematical integrity is maintained, and mistakes such as assuming unusual strategies are avoided.