Question

Are \(\frac{34}{64}\) and \(\frac{18}{36}\) in the same fraction family? Explain how you know.

Short answer

No, \(\frac{34}{64}\) and \(\frac{18}{36}\) are not in the same fraction family because their simplified forms are different.

Step-by-step solution

Reduce the first fraction

Simplify \(\frac{34}{64}\). Find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 34 and 64 is 2. Divide both the numerator and the denominator by 2 to get: \(\frac{34 \/ 2}{64 \/ 2} = \frac{17}{32}\). So, \(\frac{34}{64} = \frac{17}{32}\).

Reduce the second fraction

Simplify \(\frac{18}{36}\). Find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 18 and 36 is 18. Divide both the numerator and the denominator by 18 to get: \(\frac{18}{18}{36}{18} = \frac{1}{2}\). So, \(\frac{18}{36} = \frac{1}{2}\).

Compare the simplified fractions

Compare the simplified fractions to see if they are the same. \(\frac{17}{32}\) is not equal to \(\frac{1}{2}\). Since the two simplified fractions are not the same, \(\frac{34}{64}\) and \(\frac{18}{36}\) are not in the same fraction family.

Key concepts

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that can exactly divide both of them. To find the GCD, one can use the Euclidean algorithm or factorize both numbers and find the largest common factor. For example, to simplify \(\frac{34}{64}\), we need to find the GCD of 34 and 64.
By factorizing, we get:

• Factors of 34 are 1, 2, 17, 34
• Factors of 64 are 1, 2, 4, 8, 16, 32, 64

The greatest common factor is 2. Similarly, for 18 and 36:

• Factors of 18 are 1, 2, 3, 6, 9, 18
• Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36

The greatest common factor is 18.
This step helps in reducing the fractions to their simplest forms, making comparisons easier.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest forms by dividing the numerator and the denominator by their GCD. For instance, with \(\frac{34}{64}\), we divide both 34 and 64 by their GCD, which is 2:

\(\frac{34 \/ 2}{64 \/ 2} = \frac{17}{32}\)

This gives us the simplified fraction of \(\frac{17}{32}\).
For \(\frac{18}{36}\), the GCD is 18. Dividing both by 18, we get:

\(\frac{18 \/ 18}{36 \/ 18} = \frac{1}{2}\)

This simplifies to \(\frac{1}{2}\).
In summary, simplifying fractions reduces them to their most basic form, making them easy to work with and compare.

Fraction Comparison

Once fractions are simplified, they can be easily compared. In the given exercise, after simplification:

\(\frac{34}{64}\) simplifies to \(\frac{17}{32}\), and \(\frac{18}{36}\) simplifies to \(\frac{1}{2}\).
To compare these fractions, we need to observe if they are equal or ascertain which one is larger or smaller.
Here, \(\frac{17}{32}\) is clearly not equal to \(\frac{1}{2}\). Thus, these fractions are not from the same fraction family.
Comparing fractions involves converting them to a common form or using their decimal equivalents. For accurate comparisons, always ensure fractions are in their simplest forms.