If 3 ≤ x ≤ 10 and 5 ≤ y ≤ 15 , then maximum value of \(\left(\frac{x}{y}\right)\) is-
Maximum Value of a Fraction with Given Ranges
To determine the maximum value of the expression \(\left(\frac{x}{y}\right)\) given the ranges for \(x\) and \(y\), we need to consider the properties of fractions. For a fraction to achieve its largest possible value, its numerator (the top part) should be as large as possible, and its denominator (the bottom part) should be as small as possible.
Understanding the Given x and y Ranges
The problem provides specific ranges for the variables \(x\) and \(y\):
- For x: The range is \(3 \le x \le 10\). This means \(x\) can be any number from 3 to 10, inclusive.
- For y: The range is \(5 \le y \le 15\). This means \(y\) can be any number from 5 to 15, inclusive.
Identifying Optimal Values for Maximum Value Calculation
To maximize the fraction \(\left(\frac{x}{y}\right)\):
- We must select the maximum value for the variable \(x\). Looking at the range \(3 \le x \le 10\), the highest possible value for \(x\) is 10.
- We must select the minimum value for the variable \(y\). Looking at the range \(5 \le y \le 15\), the lowest possible value for \(y\) is 5.
Calculating the Maximum Value of the Fraction
Now, we substitute these identified optimal values into the expression \(\left(\frac{x}{y}\right)\):
Maximum value of \(\left(\frac{x}{y}\right)\) = \(\frac{\text{Maximum value of } x}{\text{Minimum value of } y}\)
Substituting the specific values:
Maximum value = \(\frac{10}{5}\)
Maximum value = \(2\)
Conclusion on Maximum Value
Therefore, the maximum value of the expression \(\left(\frac{x}{y}\right)\), based on the given ranges for \(x\) and \(y\), is 2.
| Variable | Given Range | Value Chosen for Maximum \(\left(\frac{x}{y}\right)\) |
|---|---|---|
| \(x\) | \(3 \le x \le 10\) | \(10\) (Maximum of its range) |
| \(y\) | \(5 \le y \le 15\) | \(5\) (Minimum of its range) |
This method ensures we obtain the largest possible ratio by maximizing the numerator and minimizing the denominator within their respective constraints.
Important Questions from Maxima and Minima
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