Fraction Comparison: Identifying the Smallest Fraction

To determine the smallest fraction among \(\frac{7}{12}\), \(\frac{8}{15}\), \(\frac{17}{24}\), and \(\frac{13}{18}\), we need to compare their values. A common and effective method for comparing fractions is to convert them to equivalent fractions with a common denominator. This allows for a direct comparison of their numerators.

Step-by-Step Fraction Comparison

1. Listing the Fractions

The fractions provided for comparison are:

• \(\frac{7}{12}\)
• \(\frac{8}{15}\)
• \(\frac{17}{24}\)
• \(\frac{13}{18}\)

2. Finding the Least Common Multiple (LCM) of Denominators

The denominators are 12, 15, 24, and 18. We need to find their Least Common Multiple (LCM). The LCM will be our common denominator.

Let's find the prime factorization of each denominator:

• \(12 = 2 \times 2 \times 3 = 2^2 \times 3^1\)
• \(15 = 3 \times 5 = 3^1 \times 5^1\)
• \(24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1\)
• \(18 = 2 \times 3 \times 3 = 2^1 \times 3^2\)

To find the LCM, we take the highest power of all prime factors present:

• Highest power of 2 is \(2^3 = 8\)
• Highest power of 3 is \(3^2 = 9\)
• Highest power of 5 is \(5^1 = 5\)

Therefore, the LCM of 12, 15, 24, and 18 is \(2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 72 \times 5 = 360\).

3. Converting Fractions to a Common Denominator

Now, we convert each fraction to an equivalent fraction with a denominator of 360:

• For \(\frac{7}{12}\): To get 360 from 12, we multiply by \(360 \div 12 = 30\).

  \(\frac{7}{12} = \frac{7 \times 30}{12 \times 30} = \frac{210}{360}\)

• For \(\frac{8}{15}\): To get 360 from 15, we multiply by \(360 \div 15 = 24\).

  \(\frac{8}{15} = \frac{8 \times 24}{15 \times 24} = \frac{192}{360}\)

• For \(\frac{17}{24}\): To get 360 from 24, we multiply by \(360 \div 24 = 15\).

  \(\frac{17}{24} = \frac{17 \times 15}{24 \times 15} = \frac{255}{360}\)

• For \(\frac{13}{18}\): To get 360 from 18, we multiply by \(360 \div 18 = 20\).

  \(\frac{13}{18} = \frac{13 \times 20}{18 \times 20} = \frac{260}{360}\)

4. Comparing the Numerators

Now that all fractions have the same denominator (360), we can easily compare their numerators:

Comparing the numerators (210, 192, 255, 260), the smallest numerator is 192.

5. Identifying the Smallest Fraction

Since \(\frac{192}{360}\) has the smallest numerator, it corresponds to the smallest original fraction. The original fraction for \(\frac{192}{360}\) is \(\frac{8}{15}\).

Conclusion

By finding a common denominator (LCM) and converting all the fractions, we found that \(\frac{8}{15}\) is the smallest fraction among the given options.