Question

Tell whether the fractions in each pair are equivalent, and explain how you know. \(\frac{4}{12}\) and \(\frac{8}{32}\)

Short answer

The fractions \(\frac{4}{12}\) and \(\frac{8}{32}\) are not equivalent.

Step-by-step solution

Simplify the First Fraction

To determine if the fractions are equivalent, first simplify \(\frac{4}{12}\). Find the greatest common divisor (GCD) of 4 and 12, which is 4. Divide both the numerator and the denominator by the GCD: \(\frac{4 \div 4}{12 \div 4} = \frac{1}{3}\).

Simplify the Second Fraction

Next, simplify \(\frac{8}{32}\). Find the greatest common divisor (GCD) of 8 and 32, which is 8. Divide both the numerator and the denominator by the GCD: \(\frac{8 \div 8}{32 \div 8} = \frac{1}{4}\).

Compare the Simplified Fractions

Now compare the simplified forms of both fractions: \(\frac{1}{3}\) and \(\frac{1}{4}\). Since \(\frac{1}{3} eq \frac{1}{4}\), the fractions \(\frac{4}{12}\) and \(\frac{8}{32}\) are not equivalent.

Key concepts

simplifying fractions

Fractions might look complicated, but they can be made simpler by reducing them. This process is called simplifying fractions. To simplify a fraction, you divide its numerator and its denominator by their greatest common divisor (GCD).

For example, let's take the fraction \(\frac{4}{12}\). The GCD of 4 and 12 is 4. When you divide both the numerator (4) and the denominator (12) by their GCD (4), you get \(\frac{1}{3}\). This means \(\frac{4}{12}\) simplifies to \(\frac{1}{3}\).

This method works for any fraction. It's very useful because it allows us to work with smaller, simpler numbers which makes comparing and understanding fractions much easier.

greatest common divisor

The greatest common divisor (GCD) is an important concept when working with fractions. The GCD of two numbers is the largest number that can divide both of them without leaving a remainder. To find the GCD, you can list all the factors of each number and choose the biggest one that appears in both lists.

For instance, consider the numbers 8 and 32. The factors of 8 are 1, 2, 4, and 8. The factors of 32 are 1, 2, 4, 8, 16, and 32. The largest number that is common in both lists is 8. So, the GCD of 8 and 32 is 8.

Using the GCD is key to simplifying fractions. By dividing both the numerator and the denominator of a fraction by their GCD, you ensure that you are getting the simplest form of that fraction.

comparing fractions

To determine if two fractions are equivalent, simplifying them can help. Once fractions are in their simplest form, they are much easier to compare. Here’s how you can do it:

Let's take the example fractions from the exercise: \(\frac{4}{12}\) and \(\frac{8}{32}\). Simplifying \(\frac{4}{12}\) gives \(\frac{1}{3}\), and simplifying \(\frac{8}{32}\) gives \(\frac{1}{4}\).

Now, we compare \(\frac{1}{3}\) and \(\frac{1}{4}\). Since these two fractions are not the same (because \(\frac{1}{3} eq \frac{1}{4}\)), the original fractions \(\frac{4}{12}\) and \(\frac{8}{32}\) are also not equivalent.

So, remember: first simplify the fractions and then compare their simplest forms. This way, you can easily check if fractions are equivalent or not.