Question

If \(1 \leq x \leq 3\) and \(2 \leq y \leq 4\), what is the maximum value of \(\left(\frac{x}{y}\right)\) ? (a) \(\frac{2}{3}\) (b) 4 (c) \(\frac{3}{2}\) (d) 2

Short answer

Answer: (c) $\frac{3}{2}$.

Step-by-step solution

List extreme points from the intervals

The extreme points for x are 1, 3, and for y are 2, 4. The combinations we will check are (x,y) pairs: (1, 2), (1, 4), (3, 2), and (3, 4).

Evaluate \(\frac{x}{y}\) for each pair

For the (x, y) pairs, calculate \(\frac{x}{y}\). For (1, 2): \(\frac{1}{2}\) For (1, 4): \(\frac{1}{4}\) For (3, 2): \(\frac{3}{2}\) For (3, 4): \(\frac{3}{4}\)

Compare and find maximum value

Compare the four values obtained: \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{3}{2}\), and \(\frac{3}{4}\). The maximum value is \(\frac{3}{2}\).

Match with the answer choice

The maximum value, \(\frac{3}{2}\), matches with the answer choice (c). So the correct answer is (c) \(\frac{3}{2}\).

Key concepts

Extreme Points

When solving maximum value problems, identifying the extreme points of the given intervals is crucial. In this particular exercise, we had two intervals: \(1 \leq x \leq 3\) and \(2 \leq y \leq 4\). Extreme points refer to the smallest and largest values in each interval.
For the interval involving \(x\):

• The extreme points are 1 and 3.

For the interval involving \(y\):

• The extreme points are 2 and 4.

These extreme values define the boundaries of each interval. When determining the maximum or minimum value of a function of two variables over these intervals, you must evaluate the function at all combinations of these extreme points.

Intervals

Understanding intervals is key to solving problems that involve ranges for variables. An interval represents a set of numbers between two endpoints. In our exercise, the intervals were given in a standard notation, \([a, b]\), where \(a\) is the lower bound and \(b\) is the upper bound.

• The interval \([1, 3]\) means \(x\) can be any value from 1 to 3.
• The interval \([2, 4]\) means \(y\) can be any value from 2 to 4.

By focusing on these intervals, we can pinpoint the exact pairs that need to be evaluated to find the maximum or minimum values of the function. This simplification is crucial for efficient problem-solving when dealing with bounded variable scenarios.

Fractional Comparison

Fractional comparison involves examining different fractions to determine which is larger or smaller. To find the maximum value of \(\frac{x}{y}\) using the extreme points, we need to compare:\(\frac{1}{2}, \frac{1}{4}, \frac{3}{2}, \frac{3}{4}\).
When comparing fractions:

• Convert them to a common denominator.
• Use decimal approximations.

For example, converting to decimals:

• \(\frac{1}{2} = 0.5\)
• \(\frac{1}{4} = 0.25\)
• \(\frac{3}{2} = 1.5\)
• \(\frac{3}{4} = 0.75\)

Clearly, \(\frac{3}{2} (1.5)\) is the largest among all the calculated values. Comparing fractions is an essential skill, often used in maximizing or minimizing fractional expressions.

Numerical Reasoning

Numerical reasoning is the logical process used to derive conclusions from numerical patterns and relationships. In this exercise, numerical reasoning helped in systematically comparing values.

• First, listing possible combinations of extreme points led to examining all possible outcomes of \(\frac{x}{y}\).
• Next, calculating the specific fraction for each combination provided a direct path to comparison.

Numerical reasoning is applied when interpreting numerical data, making deductions, and identifying the best option based on analysis. By logically structuring the steps of listing, calculating, comparing, and finally selecting, we can reliably determine the maximum value for scenarios like these. This skill becomes valuable in mathematics and real-world applications alike, facilitating efficient decision-making based on numerical data.