Question

In each pair, tell which fraction is closer to 0.5 \(\frac{4}{9}\) or \(\frac{11}{23}\)

Short answer

The fraction \(\frac{11}{23}\) is closer to 0.5.

Step-by-step solution

Find the decimal equivalent of each fraction

Convert each fraction to its decimal form. Divide the numerator by the denominator for each fraction. For \(\frac{4}{9}\): \(\frac{4}{9} = 0.444...\). For \(\frac{11}{23}\): \(\frac{11}{23} \approx 0.478\).

Calculate the difference between each decimal and 0.5

Next, find out how far each fraction's decimal is from 0.5. For \(\frac{4}{9}\): \(|0.5 - 0.444...| = 0.055...|\). For \(\frac{11}{23}\): \(|0.5 - 0.478| = 0.022\).

Compare the differences

Compare the differences calculated in Step 2 to determine which fraction is closer to 0.5. \(\frac{4}{9}\) is approximately 0.055... units away from 0.5, and \(\frac{11}{23}\) is approximately 0.022 units away from 0.5. \(\frac{11}{23}\) is closer to 0.5 since 0.022 < 0.055...

Key concepts

Decimal Conversion

To effectively compare fractions and determine how close they are to a specific value, like 0.5, the first step is decimal conversion. This means we need to convert the fractions into their decimal equivalents. This is done by dividing the numerator (the top number) by the denominator (the bottom number). This conversion helps us to easily compare values without dealing with fractions directly.

For example, let's convert \(\frac{4}{9}\). When we divide 4 by 9, we get approximately 0.444.... Similarly, converting \(\frac{11}{23}\) to a decimal gives us approximately 0.478. By converting both fractions to decimal form, we make the comparison process more straightforward.

Difference Calculation

Once we have the decimal equivalents, the next step is difference calculation, which helps in determining how far each value is from 0.5. To do this, we find the absolute difference between each decimal and 0.5.

For the fraction \(\frac{4}{9}\), its decimal is 0.444..., and the difference calculation is \|0.5 - 0.444...|\, resulting in approximately 0.055.... For the fraction \(\frac{11}{23}\), its decimal is 0.478, and the difference calculation is \|0.5 - 0.478|\, resulting in approximately 0.022.

Finding these differences lets us see how close each fraction's decimal is to 0.5, which is crucial for the next step.

Close to 0.5

Finally, we need to determine which fraction is closer to 0.5. This is done by comparing the differences calculated in the previous steps. A smaller difference means the fraction's decimal is closer to 0.5.

From our calculations:

• The difference for \(\frac{4}{9}\) is 0.055...
• The difference for \(\frac{11}{23}\) is 0.022

Since 0.022 is smaller than 0.055..., this tells us that \(\frac{11}{23}\) is closer to 0.5 than \(\frac{4}{9}\).

Understanding these steps can greatly simplify comparing fractions to a specific value and can be applied to a variety of other similar math problems.