Question

In each pair, tell which fraction is closer to 0.5 \(\frac{4}{9}\) or \(\frac{6}{9}\)

Short answer

\(\frac{4}{9}\)

Step-by-step solution

Understand 0.5 as a Fraction

First, convert 0.5 to a fraction. It is equivalent to \(\frac{1}{2}\).

Find Common Denominator

To compare the given fractions \(\frac{4}{9}\) and \(\frac{6}{9}\) to \(\frac{1}{2}\), convert \(\frac{1}{2}\) to have a common denominator with the other fractions. Since the denominators are 9, find the equivalent fraction of \(\frac{1}{2}\): \(\frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18}\).

Convert Given Fractions to the Same Denominator

Convert the given fractions to have the same denominator of 18: \(\frac{4}{9} = \frac{4 \times 2}{9 \times 2} = \frac{8}{18}\) and \(\frac{6}{9} = \frac{6 \times 2}{9 \times 2} = \frac{12}{18}\).

Compare to Find the Difference

Compare the differences of \(\frac{8}{18}\) and \(\frac{12}{18}\) from \(\frac{9}{18}\): \(\frac{9}{18} - \frac{8}{18} = \frac{1}{18}\) and \(\frac{12}{18} - \frac{9}{18} = \frac{3}{18}\).

Determine Which is Closer

Since \(\frac{1}{18}\) is less than \(\frac{3}{18}\), \(\frac{4}{9}\) (which is equivalent to \(\frac{8}{18}\)) is closer to \(\frac{9}{18}\) (or 0.5) than \(\frac{6}{9}\).

Key concepts

Common Denominators

When two fractions need to be compared, it's often helpful to have a common denominator. This makes the fractions easier to compare because the denominators match. Imagine you are comparing the lengths of two pieces of string. If one piece is measured in inches and the other in centimeters, it would be challenging. But if both pieces are measured in inches, the comparison becomes straightforward.

For example, in our exercise, we compared \(\frac{4}{9}\) and \(\frac{6}{9}\) to 0.5 (which is \(\frac{1}{2}\)). To make the comparison easier, we converted \(\frac{1}{2}\) to a fraction with the same denominator as our given fractions, 9. This results in \(\frac{9}{18}\).

To find a common denominator for two or more fractions, you can multiply the denominators together, or you can find the least common denominator (LCD), which is often smaller and more manageable.

Fraction Conversion

Conversion is a key step in fraction comparison. By converting fractions to have the same denominator, we can easily see which fraction is larger or smaller. In our given exercise, we converted fractions to the common denominator of 18.

Take the fraction \(\frac{4}{9}\). By multiplying both the numerator and the denominator by 2, we convert it to \(\frac{8}{18}\). Similarly, \(\frac{6}{9}\) converts to \(\frac{12}{18}\). Now, all fractions have the denominator of 18, making them easier to compare directly with \(\frac{9}{18}\) (the equivalent of 0.5). This process ensures that you're comparing the same-sized pieces.

Mathematical Comparison

Once fractions are converted to have a common denominator, mathematical comparison becomes simpler. You only need to look at the numerators since the denominators are identical.

In the exercise, we wanted to find out which fraction, \(\frac{4}{9}\) or \(\frac{6}{9}\), is closer to 0.5. After converting, we compared \(\frac{8}{18}\) and \(\frac{12}{18}\) against \(\frac{9}{18}\). By subtracting, we found the differences:

• \(\frac{9}{18} - \frac{8}{18} = \frac{1}{18}\)
• \(\frac{12}{18} - \frac{9}{18} = \frac{3}{18}\)

Since \(\frac{1}{18}\) is smaller than \(\frac{3}{18}\), \(\frac{8}{18}\) (or \(\frac{4}{9}\)) is closer to \(\frac{9}{18}\) (or 0.5).

Thus, by comparing numerators, you can determine which fraction is larger, smaller, or closer to another value.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. For example, \(\frac{1}{2}\) is equivalent to \(\frac{2}{4}\), \(\frac{3}{6}\), and \(\frac{9}{18}\). These all simplify to \(\frac{1}{2}\).

In our exercise, we converted 0.5 into its fractional form, \(\frac{1}{2}\), and then made it equivalent to \(\frac{9}{18}\). Understanding equivalent fractions is important because it allows us to simplify or compare fractions easily.

To make a fraction equivalent, multiply or divide both the numerator and the denominator by the same number. This doesn't change the value, just the appearance. For instance:

• \(\frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18}\).

Using equivalent fractions allows you to directly compare or combine fractions without changing their values.