Problem 1 - Algebra 2 and Trigonometry

Question

If \(\frac{a}{b}=\frac{c}{d},\) then is \(\frac{a}{c}=\frac{b}{d} ?\) Justify your answer.

Step-by-Step Solution

Answer

Yes, \( \frac{a}{c} = \frac{b}{d} \) is true if \( \frac{a}{b} = \frac{c}{d} \). This is verified by cross-multiplication.

Step 1: Understanding the Given Proportion

We are given that \( \frac{a}{b} = \frac{c}{d} \). This implies a relationship where the ratio of \(a\) to \(b\) is equal to the ratio of \(c\) to \(d\). We need to determine if this equality implies that \( \frac{a}{c} = \frac{b}{d} \).

Step 2: Cross-Multiplying the Original Proportion

Cross-multiply the given proportion to verify it: \( a \times d = c \times b \). This confirms the relationship between \(a, b, c,\) and \(d\).

Step 3: Checking the New Proportion

Consider the new proportion \( \frac{a}{c} = \frac{b}{d} \). Cross-multiply to see if this proportion is valid: \( a \times d = b \times c \).

Step 4: Comparing Cross-Multiplication Results

From Step 2, we have \( a \times d = c \times b \). In Step 3, we found \( a \times d = b \times c \). These two are identical equations, indicating that \( \frac{a}{c} = \frac{b}{d} \) holds true under the given condition.

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a powerful technique in mathematics, especially useful when dealing with proportions or comparison of two ratios. When you have two fractions set equal, like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction. This results in the equation \( a \cdot d = b \cdot c \).

This process helps us verify if two fractions or ratios are equivalent by transforming them into a simple equation. By cross-multiplying, you eliminate the fractions and can directly compare the products, providing a clearer understanding of the relationship between the four values involved. This is why cross-multiplication is an essential tool in solving proportion problems.

Exploring Ratios

Ratios are a way to express a relationship between two numbers. They are comparable to fractions in that they demonstrate how many times one number contains another. The ratio \( \frac{a}{b} \) tells us how much of \( a \) there is per unit of \( b \).

Ratios can be written in several forms:

• As a fraction, \( \frac{a}{b} \)
• With a colon, \( a:b \)
• In words, "a to b"

Understanding ratios is crucial because they allow us to compare quantities in a meaningful way and are used in a variety of real-world scenarios, such as maps, recipes, and mixing solutions.

Defining Equivalent Fractions

Equivalent fractions are fractions that represent the same value or proportion, even though they may appear different. For example, \( \frac{1}{2} \) and \( \frac{2}{4} \) are equivalent because they both equal 0.5 when simplified.

To determine if two fractions are equivalent, we can use cross-multiplication. If \( \frac{a}{b} = \frac{c}{d} \) and cross-multiplying gives us the equation \( a \cdot d = b \cdot c \), then the fractions are equivalent.

• This is because their simplified forms result in the same number.
• Understanding equivalent fractions is key in topics like simplifying fractions, finding common denominators, and solving proportion problems.

Recognizing equivalent fractions helps enhance our numerical reasoning and problem-solving abilities.