To determine the minimum value of the expression \(\left(\frac{x}{y}\right)\), we must consider how the values of \(x\) (numerator) and \(y\) (denominator) influence the overall size of the fraction. For a fraction to be as small as possible, we need to achieve two conditions simultaneously:

• The numerator (\(x\)) should be as small as its allowed range permits.
• The denominator (\(y\)) should be as large as its allowed range permits.

Let's analyze the given ranges for \(x\) and \(y\):

• For \(x\): The inequality \(3 \le x \le 10\) means that \(x\) can be any real number from 3 up to 10, including 3 and 10.
• For \(y\): The inequality \(5 \le y \le 15\) means that \(y\) can be any real number from 5 up to 15, including 5 and 15.

Minimum Value Selection for Numerator (x)

To make the fraction \(\left(\frac{x}{y}\right)\) as small as possible, we should choose the smallest possible value for \(x\). Based on the range \(3 \le x \le 10\), the absolute minimum value \(x\) can take is 3.

Maximum Value Selection for Denominator (y)

Conversely, to minimize the fraction \(\left(\frac{x}{y}\right)\), we should choose the largest possible value for \(y\). From the range \(5 \le y \le 15\), the absolute maximum value \(y\) can take is 15.

Calculating the Minimum Value of the Ratio

Now, we substitute these chosen extreme values of \(x\) and \(y\) into the expression \(\left(\frac{x}{y}\right)\):

\(\text{Minimum value of } \left(\frac{x}{y}\right) = \frac{\text{Smallest value of } x}{\text{Largest value of } y}\)

\(\text{Minimum value of } \left(\frac{x}{y}\right) = \frac{3}{15}\)

Simplifying the Resulting Fraction

The fraction \(\frac{3}{15}\) can be simplified. Both the numerator (3) and the denominator (15) are divisible by 3. Dividing both by 3, we get:

\(\frac{3 \div 3}{15 \div 3} = \frac{1}{5}\)

Therefore, the minimum value of \(\left(\frac{x}{y}\right)\) is \(\frac{1}{5}\).