Problem 36 - Prealgebra

Question

Find two counterexamples for the statement Two ratios always form a proportion.

Step-by-Step Solution

Answer

Ratios \( \frac{3}{4} \) and \( \frac{2}{5} \); and \( \frac{7}{3} \) and \( \frac{5}{2} \) are counterexamples.

Step 1: Understanding Proportions

A proportion states that two ratios are equal. If we have two ratios \( \frac{a}{b} \) and \( \frac{c}{d} \), they form a proportion if \( \frac{a}{b} = \frac{c}{d} \), which can be rewritten as \( a \cdot d = b \cdot c \).

Step 2: Formulating Counterexamples

To find counterexamples, we need two ratios where \( a \cdot d eq b \cdot c \). We'll form two ratios and check if they meet this condition.

Step 3: Counterexample 1

Consider the ratios \( \frac{3}{4} \) and \( \frac{2}{5} \). Calculating their cross products: \( 3 \times 5 = 15 \) and \( 4 \times 2 = 8 \). Since 15 is not equal to 8, these ratios do not form a proportion.

Step 4: Counterexample 2

Take the ratios \( \frac{7}{3} \) and \( \frac{5}{2} \). Calculating their cross products: \( 7 \times 2 = 14 \) and \( 3 \times 5 = 15 \). Since 14 is not equal to 15, these ratios do not form a proportion.

Key Concepts

Ratios

Ratios are a way to compare two quantities by showing the relative sizes of different values. You can think of a ratio as a fraction, \(\frac{a}{b}\), where \(a\) and \(b\) are numbers or quantities being compared. Ratios can be used in various situations, like comparing the number of apples to oranges in a basket or the speed of two cars.

Here are some key points about ratios:

• Ratios represent a relationship between two numbers.
• They can be written in three ways: \(a:b\), \(\frac{a}{b}\), or "a to b".
• The order in a ratio is important; \(a:b\) is not the same as \(b:a\).

For example, if there are 2 apples and 3 oranges, the ratio of apples to oranges is \(\frac{2}{3}\). It's important to remember that ratios do not have units, which makes them versatile for comparison.

Understanding ratios helps you grasp more complex mathematical concepts, like proportions, making it essential to master their basics.

Cross-Multiplication

When it comes to checking if two ratios form a proportion, cross-multiplication is a useful tool. This process helps determine if two ratios are indeed equal.

Let's explore cross-multiplication:

• If you have ratios \(\frac{a}{b}\) and \(\frac{c}{d}\), you use cross-multiplication to check their equality.
• The cross products, \(a \cdot d\) and \(b \cdot c\), are compared.
• If \(a \cdot d = b \cdot c\), then the ratios are proportional.

For instance, with ratios \(\frac{3}{4}\) and \(\frac{2}{5}\), we find that the cross products are 15 and 8, respectively, showing they do not form a proportion as these products aren't equal.

Cross-multiplication simplifies the task of identifying proportions without fully converting ratios into decimals, making it a powerful technique in both academic and practical scenarios.

Counterexamples

Learning about disproving general statements using counterexamples enriches your understanding of mathematical concepts. A counterexample is a specific case that shows a statement to be false.

For instance, the statement "Two ratios always form a proportion" can be proven wrong by finding ratios that aren't proportional. Here’s how you can construct a counterexample:

• Choose two ratios, such as \(\frac{7}{3}\) and \(\frac{5}{2}\).
• Calculate cross products: \(7 \cdot 2 = 14\) and \(3 \cdot 5 = 15\).
• Since \(14 eq 15\), the ratios do not form a proportion.

This disproves the universal claim, serving as a counterexample. Counterexamples are vital in mathematics since they demonstrate the boundaries of statements and assumptions, giving students deeper insights into the nature of mathematical truths and falsehoods.