Finding the Smallest Fraction

To find the smallest fraction among a given set, we need to compare them. There are several ways to compare fractions. One common method is to convert each fraction into its decimal equivalent and then compare the decimal values. Another method is to find a common denominator for all fractions and then compare the numerators.

Comparing Fractions by Decimal Conversion

Let's convert each of the given fractions into decimal form:

• The first fraction is \(\frac{4}{5}\). Converting this to a decimal gives \(4 \div 5 = 0.8\).
• The second fraction is \(\frac{7}{8}\). Converting this to a decimal gives \(7 \div 8 = 0.875\).
• The third fraction is \(\frac{6}{7}\). Converting this to a decimal gives \(6 \div 7 \approx 0.8571\).
• The fourth fraction is \(\frac{5}{6}\). Converting this to a decimal gives \(5 \div 6 \approx 0.8333\).

Now, we have the decimal values for each fraction:

• \(\frac{4}{5} = 0.8\)
• \(\frac{7}{8} = 0.875\)
• \(\frac{6}{7} \approx 0.8571\)
• \(\frac{5}{6} \approx 0.8333\)

Comparing these decimal values: \(0.8\), \(0.875\), \(0.8571\), \(0.8333\).

The smallest decimal value is \(0.8\).

This decimal value corresponds to the fraction \(\frac{4}{5}\).

Comparing Fractions Using a Common Denominator

Alternatively, we can find the least common multiple (LCM) of the denominators (5, 8, 7, 6). The LCM of 5, 8, 7, and 6 is \(5 \times 8 \times 7 \times 3 = 840\). Note that 6 = 2 * 3, 8 = 2^3, 5 and 7 are prime. LCM = \(2^3 \times 3 \times 5 \times 7 = 8 \times 3 \times 5 \times 7 = 24 \times 35 = 840\).

Now, convert each fraction to an equivalent fraction with a denominator of 840:

• \(\frac{4}{5} = \frac{4 \times 168}{5 \times 168} = \frac{672}{840}\)
• \(\frac{7}{8} = \frac{7 \times 105}{8 \times 105} = \frac{735}{840}\)
• \(\frac{6}{7} = \frac{6 \times 120}{7 \times 120} = \frac{720}{840}\)
• \(\frac{5}{6} = \frac{5 \times 140}{6 \times 140} = \frac{700}{840}\)

Now compare the numerators: 672, 735, 720, 700.

The smallest numerator is 672.

This corresponds to the fraction \(\frac{672}{840}\), which is equivalent to the original fraction \(\frac{4}{5}\).

Conclusion

Both methods show that the smallest fraction among \(\frac{4}{5}, \frac{7}{8}, \frac{6}{7}, \frac{5}{6}\) is \(\frac{4}{5}\).