Question

Find three fractions equivalent to each given fraction. \(\frac{7}{9}\)

Short answer

Equivalent fractions: \( \frac{14}{18} \), \( \frac{21}{27} \), \( \frac{28}{36} \).

Step-by-step solution

- Understand Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. To find equivalent fractions, we multiply or divide both the numerator and the denominator by the same non-zero number.

- Multiply by 2

Multiply both the numerator and the denominator of \(\frac{7}{9}\) by 2: \[ \frac{7 \times 2}{9 \times 2} = \frac{14}{18} \]

- Multiply by 3

Multiply both the numerator and the denominator of \(\frac{7}{9}\) by 3: \[ \frac{7 \times 3}{9 \times 3} = \frac{21}{27} \]

- Multiply by 4

Multiply both the numerator and the denominator of \(\frac{7}{9}\) by 4: \[ \frac{7 \times 4}{9 \times 4} = \frac{28}{36} \]

- Summarize Results

The three equivalent fractions to \(\frac{7}{9}\) are \( \frac{14}{18} \), \( \frac{21}{27} \), and \( \frac{28}{36} \).

Key concepts

fraction multiplication

To understand equivalent fractions, it's important to get comfortable with multiplying fractions. This means multiplying both the numerator (the top number) and the denominator (the bottom number) of a fraction by the same number. This operation doesn't change the value of the fraction, but it creates a new fraction that looks different. For example, when we multiply both the numerator and the denominator of \(\frac{7}{9}\) by 2, we get: \[ \frac{7 \times 2}{9 \times 2} = \frac{14}{18} \]
This operation is like zooming in on or scaling up the fraction. Even though it looks different, it represents the same amount as the original fraction. Multiplying by any other number, such as 3 or 4, follows the same steps. Remember, the key to maintaining the value of the fraction is to use the same factor for both the numerator and the denominator.

numerator and denominator

The numerator and the denominator are the parts that make up a fraction. The numerator is the top number, and it represents how many parts of the whole we have. The denominator is the bottom number, and it tells us into how many equal parts the whole is divided. For instance, in the fraction \(\frac{7}{9}\), the 7 is the numerator and the 9 is the denominator.

Understanding these two components is crucial when working with fractions. When creating equivalent fractions, we change both the numerator and the denominator by the same amount to keep the value of the fraction the same. For example, multiplying the numerator and the denominator of \(\frac{7}{9}\) by 3 will result in: \[ \frac{7 \times 3}{9 \times 3} = \frac{21}{27} \]
This keeps the relationship between the parts (numerator) and the whole (denominator) consistent.

simplifying fractions

Simplifying fractions involves reducing them to their simplest form. This is done by finding the greatest common divisor (GCD) for both the numerator and the denominator and dividing them by it.

In the context of equivalent fractions, simplifying helps check that the fractions are indeed equivalent. For example, if we start with the fraction \(\frac{14}{18}\) and find its GCD, which is 2, we divide both the numerator and the denominator by 2: \[ \frac{14 \div 2}{18 \div 2} = \frac{7}{9} \]
This confirms that \(\frac{14}{18}\) is equivalent to \(\frac{7}{9}\). Simplifying fractions is a good way to ensure you're correctly finding equivalent fractions and can make fractions easier to work with in calculations.