Question

One way to prove that two proportions are equivalent is to apply the properties of equality to transform one of the proportions into the other proportion. Give a sequence of steps that transforms the proportion \(\frac{a}{b}=\frac{c}{d}\) into \(\frac{a}{c}=\frac{b}{d}\).

Short answer

The sequence of steps involves setting up the proportion, cross-multiplying the ratios, dividing both sides by 'cd', and rewriting the equation to get the new proportion.

Step-by-step solution

Set Up the Proportion

Start with the given proportion \(\frac{a}{b} = \(\frac{c}{d}\). This states that the ratio of 'a' to 'b' is the same as 'c' to 'd'.

Cross-Multiply

Cross-multiply the two ratios to get \(ad = bc\). This is a valid operation because it maintains the equality and this operation is called the 'Cross-Product Property' of proportions.

Divide both sides by cd

Now, divide both sides of the equation by 'cd'. Note that it's valid to divide each term by the same nonzero number.

Rewrite Equation

Rewrite your equation \(\frac{a}{c} = \frac{b}{d}\). This is the required transformed proportion.

Key concepts

Cross-Multiplication

Understanding cross-multiplication is essential when dealing with proportions. It is a reliable method to compare ratios or fractions and to solve for unknown variables.

To cross-multiply, take the numerator of one fraction and multiply it by the denominator of the other fraction. Do the same with the remaining numerator and denominator. After these multiplications, you set the two products equal to each other. This forms an equation that must be true if the original proportions are equivalent.

For example, given \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication entails calculating \( ad \) and \( bc \) and then equating them as \( ad = bc \). This step checks the initial claim that the ratios \( \frac{a}{b} \) and \( \frac{c}{d} \) are indeed equivalent.

Properties of Equality

The properties of equality are the backbone of algebra and are indispensable in proving and solving equations. These properties allow for manipulating an equation while preserving its equality.

Some fundamental properties include the Reflexive Property (\( a = a \)), the Symmetric Property (\( if \, a = b \, then \, b = a \)), the Transitive Property (\( if \, a = b \, and \, b = c \, then \, a = c \)), and the Substitution Property (replacing a term in an expression with another term that is equal to it). A key operation is performing the same action (like addition, subtraction, multiplication, or division) on both sides of an equation, which keeps the balance intact.

During the solution to the exercise, after cross-multiplying, \( ad = bc \), you're allowed to divide both sides of the equation by the same nonzero quantity without losing equality (\( \frac{ad}{cd} = \frac{bc}{cd} \)), which effectively uses the Division Property of Equality and leads to \( \frac{a}{c} = \frac{b}{d} \).

Proportional Reasoning

Proportional reasoning is a way of thinking that allows one to make sense of ratios and proportional relationships. It underpins many concepts in mathematics and is used frequently in various applications like scaling, determining rates, or solving problems involving similarity.

In proportional reasoning, one seeks to understand how two quantities vary in relation to each other, thereby forming a ratio that is consistent across equivalent comparisons. When two ratios are set equal, as in \( \frac{a}{b} = \frac{c}{d} \), it's saying that 'a' is to 'b' as 'c' is to 'd', maintaining a constant ratio between the quantities. This relationship is fundamental to constructing and understanding proportional relationships.

Using proportional reasoning is not just about cross-multiplying; it also involves recognizing patterns and relationships between numbers to infer missing values or deduce logical conclusions based on proportional relationships.