Question

Find three fractions equivalent to each given fraction. \(\frac{12}{54}\)

Short answer

\(\frac{4}{18}\), \(\frac{6}{27}\), \(\frac{8}{36}\)

Step-by-step solution

- Simplify the Given Fraction

First, simplify the fraction \(\frac{12}{54}\). Find the greatest common divisor (GCD) of 12 and 54. The GCD is 6. Divide both the numerator and the denominator by this GCD to get the simplified fraction. \[ \frac{12 \div 6}{54 \div 6} = \frac{2}{9} \]

- Find the First Equivalent Fraction

Multiply both the numerator and the denominator of the simplified fraction \(\frac{2}{9}\) by 2 to get an equivalent fraction. \[ \frac{2 \cdot 2}{9 \cdot 2} = \frac{4}{18} \]

- Find the Second Equivalent Fraction

Multiply both the numerator and the denominator of the simplified fraction \(\frac{2}{9}\) by 3 to get another equivalent fraction. \[ \frac{2 \cdot 3}{9 \cdot 3} = \frac{6}{27} \]

- Find the Third Equivalent Fraction

Multiply both the numerator and the denominator of the simplified fraction \(\frac{2}{9}\) by 4 to get yet another equivalent fraction. \[ \frac{2 \cdot 4}{9 \cdot 4} = \frac{8}{36} \]

Key concepts

simplifying fractions

Simplifying fractions is a crucial skill in math. It makes calculations easier and solutions clearer. To simplify a fraction, you reduce it to its simplest form by dividing both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD). For instance, the fraction \(\frac{12}{54}\) simplifies to \(\frac{2}{9}\) by dividing both 12 and 54 by their GCD, which is 6.

A fraction in its simplest form has no common divisors other than 1 between the numerator and denominator. Simplifying helps when comparing fractions or finding equivalent ones.

greatest common divisor

The greatest common divisor (GCD) is the largest number that can evenly divide both the numerator and the denominator of a fraction. To find the GCD of two numbers:

• List the factors of each number.
• Identify the common factors.
• Choose the largest factor that both numbers share.

For example, to find the GCD of 12 and 54: The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54. The common factors are 1, 2, 3, and 6. Hence, their GCD is 6.
Using the GCD, you can simplify fractions efficiently.

multiplying fractions

Multiplying fractions involves multiplying both the numerators and the denominators. To create equivalent fractions, you multiply both parts of a fraction by the same number. This doesn't change the value of the fraction. For example, starting with \(\frac{2}{9}\), you can get:

• \(\frac{2 \times 2}{9 \times 2} = \frac{4}{18} \)
• \(\frac{2 \times 3}{9 \times 3} = \frac{6}{27} \)
• \(\frac{2 \times 4}{9 \times 4} = \frac{8}{36} \)

The value remains the same, but the fractions look different. Multiplying fractions is useful for finding equivalents or adjusting fractions for comparison.