Question

Give two fractions that are equivalent to each given fraction. $$\frac{1}{5}$$

Short answer

\( \frac{2}{10} \) and \( \frac{3}{15} \) are equivalent to \( \frac{1}{5} \).

Step-by-step solution

Understanding Equivalent Fractions

Two fractions are equivalent if they have the same value when simplified. To find equivalent fractions, multiply the numerator and the denominator by the same number.

Choose the First Multiplier

Multiply both the numerator and the denominator of \( \frac{1}{5} \) by 2. \( \frac{1 \times 2}{5 \times 2} = \frac{2}{10} \)

Choose a Different Multiplier

Multiply both the numerator and the denominator of \( \frac{1}{5} \) by 3. \( \frac{1 \times 3}{5 \times 3} = \frac{3}{15} \)

Verification

Check that both new fractions represent the same value as the original fraction. \( \frac{2}{10} = \frac{1}{5} \) and \( \frac{3}{15} = \frac{1}{5} \)

Key concepts

Fractions

Fractions represent parts of a whole. They consist of two main parts: the numerator and the denominator. The fraction line, which separates these two numbers, indicates division. For example, in the fraction \( \frac{1}{5} \), 1 is the numerator and 5 is the denominator. Fractions are used whenever we need to denote a part of a whole, whether it's a slice of pizza or a piece of a chocolate bar.

Understanding fractions is crucial in various math topics and everyday life. It helps in measurements, dividing tasks, and even in financial calculations. In short, mastering fractions can simplify many aspects of learning and life.

Simplifying Fractions

Simplifying fractions means to make the fraction as simple as possible, without changing its value. This is done by finding the greatest common divisor (GCD) of both the numerator and the denominator and then dividing them both by this number. A fraction is considered simplified if no number other than 1 can evenly divide both the numerator and the denominator.

For example, consider the fraction \( \frac{6}{9} \). The greatest common divisor of 6 and 9 is 3. Dividing both the numerator and the denominator by 3, we get: \( \frac{6 \, \div \, 3}{9 \, \div \, 3} = \frac{2}{3} \). This is the simplified form of \( \frac{6}{9} \). Simplifying fractions reduces the numbers involved, making calculations easier and clearer.

Numerator and Denominator

The numerator and denominator are fundamental parts of a fraction.

• The numerator is the top part of the fraction and indicates how many parts we are considering.
• The denominator is the bottom part and shows into how many equal parts the whole is divided.

For instance, in the fraction \( \frac{3}{15} \), 3 is the numerator, and 15 is the denominator. The fraction tells us that we are considering 3 parts out of a total of 15.
Modifying both the numerator and the denominator by the same number does not change the value of the fraction. This property is useful in finding equivalent fractions. For example, multiplying both parts of \( \frac{1}{5} \) by 2 gives us \( \frac{2}{10} \), and multiplying both by 3 gives us \( \frac{3}{15} \). These are equivalent fractions because the overall value remains the same.