Question

Find three fractions equivalent to each given fraction. \(\frac{6}{13}\)

Short answer

The three equivalent fractions are \( \frac{12}{26} \), \( \frac{18}{39} \), and \( \frac{24}{52} \).

Step-by-step solution

- Identify the original fraction

The given fraction is \( \frac{6}{13} \).

- Multiply numerator and denominator

To find an equivalent fraction, multiply both the numerator and the denominator by the same number. Start with multiplying by 2: \( \frac{6 \times 2}{13 \times 2} = \frac{12}{26} \).

- Find a second equivalent fraction

Next, multiply both the numerator and the denominator by 3: \( \frac{6 \times 3}{13 \times 3} = \frac{18}{39} \).

- Find a third equivalent fraction

Finally, multiply both the numerator and the denominator by 4: \( \frac{6 \times 4}{13 \times 4} = \frac{24}{52} \).

Key concepts

Fractions

A fraction represents a part of a whole. It consists of two integers separated by a slash. The number above the slash is called the numerator, and the number below the slash is called the denominator. Fractions can describe any part of a whole, from one whole item to multiple parts.

For example, in the fraction \(\frac{3}{4}\), '3' is the numerator, and '4' is the denominator.

Fractions can be classified into proper fractions, improper fractions, and mixed numbers:

• Proper fractions: The numerator is less than the denominator, e.g., \(\frac{3}{4}\).
• Improper fractions: The numerator is greater than or equal to the denominator, e.g., \(\frac{7}{4}\).
• Mixed numbers: A combination of a whole number and a fraction, e.g., 1\(\frac{3}{4}\).

Fractions can be compared, added, subtracted, and multiplied just like whole numbers. Simplifying fractions or finding equivalent fractions helps in comparing and performing operations more easily.

Numerator and Denominator

In any given fraction, the top number is called the numerator, and the bottom number is the denominator. Understanding these components is essential in creating equivalent fractions or performing arithmetic operations with fractions.

Numerator:

• Represents the number of parts taken from the whole.
• For example, in \(\frac{6}{13}\), '6' is the numerator, indicating that 6 parts of a total 13 are considered.

Denominator:

• Indicates the total number of equal parts the whole is divided into.
• In the fraction \(\frac{6}{13}\), '13' is the denominator, showing the whole is divided into 13 parts.

If you want to find an equivalent fraction, you'll need to multiply or divide both the numerator and the denominator by the same number. This does not change the value of the fraction, as the ratio between the numerator and denominator remains the same.

For example, multiplying both by 2 in \(\frac{6}{13}\):
\(\frac{6 \times 2}{13 \times 2} = \frac{12}{26}\)

This creates an equivalent fraction with the same value.

Multiplication

Multiplication plays a crucial role in creating equivalent fractions. To find fractions that are equivalent, you multiply both the numerator and the denominator by the same number.

Step-by-step process to find an equivalent fraction by multiplication:

• Identify the original fraction, e.g., \(\frac{6}{13}\).
• Choose a number to multiply both the numerator and denominator by. Common choices are small integers like 2, 3, or 4.
• Multiply: If you choose 2, multiply as follows:

\(\frac{6 \times 2}{13 \times 2} = \frac{12}{26}\)

Repeat with different numbers to find more equivalent fractions:
Multiplying by 3: \(\frac{6 \times 3}{13 \times 3} = \frac{18}{39}\)
Multiplying by 4: \(\frac{6 \times 4}{13 \times 4} = \frac{24}{52}\)

Equivalent fractions can be useful in various mathematical calculations, including simplifying problems or comparing fractions. Practicing these steps helps in understanding the consistency of ratios and maintaining proportional relationships in fractions.