Question

Tell whether the fractions in each pair are equivalent, and explain how you know. \(\frac{4}{8}\) and \(\frac{15}{30}\)

Short answer

Yes, they are equivalent. \(\frac{4}{8}\) and \(\frac{15}{30}\) both simplify to \(\frac{1}{2}\).

Step-by-step solution

Simplify the First Fraction

To check for equivalence, first simplify the fraction \(\frac{4}{8}\). Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 4. So, \(\frac{4 \div 4}{8 \div 4} = \frac{1}{2}\). The simplified form of \(\frac{4}{8}\) is \(\frac{1}{2}\).

Simplify the Second Fraction

Next, simplify \(\frac{15}{30}\). The GCD of 15 and 30 is 15. Divide both the numerator and the denominator by their GCD. \(\frac{15 \div 15}{30 \div 15} = \frac{1}{2}\). The simplified form of \(\frac{15}{30}\) is \(\frac{1}{2}\).

Compare Simplified Fractions

Now compare the simplified forms of both fractions. \(\frac{1}{2}\) is equal to \(\frac{1}{2}\). Since both simplified fractions are the same, \(\frac{4}{8}\) and \(\frac{15}{30}\) are equivalent.

Key concepts

Simplifying Fractions

Simplifying fractions is an important skill in math. It helps to make fractions easier to work with and compare.

When you simplify a fraction, you find an equivalent fraction with the smallest possible numbers in the numerator and the denominator.
To do this, divide both the top and bottom numbers by their greatest common divisor (GCD).
For example, in the fraction \(\frac{4}{8} \), the GCD of 4 and 8 is 4.
So, you divide both 4 and 8 by 4.
This gives you the simplified fraction \(\frac{1}{2}\).
This process of simplifying fractions makes it easier to compare and work with them in math problems.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide both the numerator and the denominator of a fraction without leaving a remainder.

Finding the GCD is crucial for simplifying fractions. You can find the GCD using different methods:

• List the factors of both numbers and find the largest one they have in common.
• Use the Euclidean algorithm, which involves repeated division.
  
  For example, if you want to find the GCD of 15 and 30, you can list the factors:
  • Factors of 15: 1, 3, 5, 15
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
    The common factors are 1, 3, 5, and 15. The largest common factor is 15.
    
    So, 15 is the GCD of 15 and 30.
    Once you know the GCD, you can divide both the numerator and the denominator by it to simplify the fraction.

Comparing Fractions

Comparing fractions helps to determine whether they are equivalent or to see which is larger or smaller. Simplifying fractions to their lowest terms is a helpful step.

To compare fractions, follow these steps:

• Simplify both fractions to their lowest terms.
• Check if the simplified fractions are the same.

If the simplified fractions are equal, the original fractions are equivalent.

For example, in the exercise, the fractions \(\frac{4}{8} \) and \(\frac{15}{30} \) are simplified to \(\frac{1}{2} \) and \(\frac{1}{2} \), making them equivalent.
If the simplified fractions are different, compare them using a common denominator or by cross-multiplication. Simplifying and comparing fractions is very useful in many areas of math and everyday life.