Question

Tell whether each fraction is closest to \(0, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4},\) or \(1 .\) Explain how you decided. $$\frac{5}{18}$$

Short answer

Closest to \( \frac{1}{4} \)

Step-by-step solution

Understand the problem

Determine which of the fractions (0, \( \frac{1}{4} \), \( \frac{1}{3} \), \( \frac{1}{2} \), \( \frac{2}{3} \), \( \frac{3}{4} \), or 1) is closest to \( \frac{5}{18} \). This involves comparing \( \frac{5}{18} \) to each of these fractions.

Convert fractions to decimals

Convert \( \frac{5}{18} \) and the given fractions to decimals for easier comparison: \ 0 = 0.0, \ \( \frac{1}{4} \) = 0.25, \ \( \frac{1}{3} \) \( \approx \) 0.333, \ \( \frac{1}{2} \) = 0.5, \ \( \frac{2}{3} \) \( \approx \) 0.667, \ \( \frac{3}{4} \) = 0.75, \ 1 = 1.0, \ \( \frac{5}{18} \) \( \approx 0.278 \).

Compare decimals

Compare 0.278 to the converted decimals: \ 0.278 is closest to 0.25 (\( \frac{1}{4} \)) since 0.278 is greater than 0.25 but much closer to 0.25 than it is to the other fractions.

Conclusion

Based on the comparison, determine that \( \frac{5}{18} \) is closest to \( \frac{1}{4} \).

Key concepts

Fraction Conversion to Decimals

When comparing fractions, it can be helpful to convert them into decimal form. This can make it easier to see which numbers are closer to each other.
For example, let's convert the fraction \(\frac{5}{18}\) into a decimal. We do this by dividing the numerator by the denominator: $$\frac{5}{18} \approx 0.278.$$
To compare this with other fractions such as 0, \(\frac{1}{4}\), \(\frac{1}{3}\), \(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{3}{4}\), and 1, we convert each of them into decimals as follows:

• 0 = 0.0
• \(\frac{1}{4} = 0.25\)
• \(\frac{1}{3} \approx 0.333\)
• \(\frac{1}{2} = 0.5\)
• \(\frac{2}{3} \approx 0.667\)
• \(\frac{3}{4} = 0.75\)
• 1 = 1.0

Now that everything is in decimal form, we can easily compare these numbers to determine which is closest to 0.278.

Decimal Comparison

Comparing decimals is straightforward. Look at each digit from left to right to determine which number is larger or smaller.
Let's say we want to compare 0.278 to the other decimal forms we converted:

• 0.278 vs. 0.0 - 0.278 is clearly larger.
• 0.278 vs. 0.25 - 0.278 is slightly larger.
• 0.278 vs. 0.333 - 0.278 is smaller.
• 0.278 vs. 0.5 - 0.278 is smaller.
• 0.278 vs. 0.667 - 0.278 is smaller.
• 0.278 vs. 0.75 - 0.278 is smaller.
• 0.278 vs. 1.0 - 0.278 is smaller.

From this comparison, we can see that 0.278 is closest to 0.25 since 0.278 is greater than 0.25 but still much closer to it than to any of the larger numbers.

Approximation of Fractions

Approximating fractions involves finding a close, but simpler, fraction form that is easier to understand. This is helpful when you need to quickly determine the value of a fraction in comparison to others.
For our example, \(\frac{5}{18}\) can be approximated by comparing its decimal form 0.278 with common fractions.
You notice that 0.278 is closest to 0.25 (\(\frac{1}{4}\)) among the given fractions. This approximation helps us to understand that \(\frac{5}{18}\) is just a bit less than \(\frac{1}{3}\), but closer to one-fourth.
By converting fractions to decimals and then comparing these decimals, you can determine which fractions they are closest to, aiding in quick approximations and understanding of their values.