the diffrence between the largest and the smallest of the rationals, 5/8,7/12,1/3,2/5

1. Break Down the Problem

We need to identify the largest and smallest values among the given rational numbers: $\frac{5}{8}, \frac{7}{12}, \frac{1}{3}, \frac{2}{5}$. After identifying these values, we will calculate the difference between them.

2. Relevant Concepts

To compare the rational numbers, we can convert them to a common denominator or convert them to decimal form.

3. Analysis and Detail

1. Convert to Decimal Form:

   • $\frac{5}{8} = 0.625$
   • $\frac{7}{12} \approx 0.5833$
   • $\frac{1}{3} \approx 0.3333$
   • $\frac{2}{5} = 0.4$

2. Identify Largest and Smallest:

   • Largest: $\frac{5}{8} = 0.625$
   • Smallest: $\frac{1}{3} \approx 0.3333$

3. Calculate the Difference: $\text{Difference} = \text{Largest} - \text{Smallest} = \frac{5}{8} - \frac{1}{3}$

   • First, find a common denominator. The least common multiple of 8 and 3 is 24.

   • Convert the fractions: $\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}$ $\frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24}$

   • Now, substract: $\frac{15}{24} - \frac{8}{24} = \frac{15 - 8}{24} = \frac{7}{24}$

4. Verify and Summarize

We have calculated the difference between the largest and the smallest rational numbers, confirming that the difference is indeed $\frac{7}{24}$.

Final Answer

The difference between the largest and the smallest of the rationals $\frac{5}{8}, \frac{7}{12}, \frac{1}{3}, \frac{2}{5}$ is $\frac{7}{24}$.