How to choose the proportion that is true

Evaluating 3/8 = 15/40

Cross-multiply to verify:

Since both cross-products are equal, this proportion is true.

Evaluating 12/13 = 2/3

Cross-multiply to verify:

The cross-products are not equal, so this proportion is not true.

Evaluating 9/13 = 27/36

Cross-multiply to verify:

The cross-products are not equal, so this proportion is not true.

Evaluating 16/49 = 4/7

Cross-multiply to verify:

The cross-products are not equal, so this proportion is not true.

Conclusion

The only true proportion among the given examples is:

Understanding Cross-Multiplication

Cross-multiplication is a mathematical technique often used to solve equations that involve two ratios or fractions. It's a useful method for finding the value of an unknown variable in proportion equations.

The basic idea is to convert an equation of the form:

into a simple equation without fractions. Here's how it's done:

1. Multiply the numerator of the first fraction by the denominator of the second fraction:

2. Multiply the denominator of the first fraction by the numerator of the second fraction:

3. Set the two products equal to each other. This gives:

Example

Suppose you have the equation:

Using cross-multiplication, you can solve for $x$:

1. Multiply the numerator of the first fraction (3) by the denominator of the second fraction (8):

2. Multiply the denominator of the first fraction (4) by the numerator of the second fraction ( $x$ ):

3. Set the two products equal to each other:

4. Solve for $x$ by dividing both sides of the equation by 4:

Key Points

• Cross-multiplication is particularly useful when dealing with proportions and ratios.
• It simplifies complex fraction equations into basic linear equations.
• Always ensure to set the cross-products equal to each other to find the unknown variable.

By following these steps, you can easily use cross-multiplication to solve equations involving ratios!

Concept Overview

Equality of ratios occurs when two or more ratios are equivalent to each other. Ratios are a way of comparing two quantities by division. If ratios are in equality, their cross-products are equal.

Basic Definition

If we have two ratios, $\frac{a}{b}$ and $\frac{c}{d}$, they are equal if and only if the following condition holds:

This means that the product of the outer terms (extremes) is equal to the product of the inner terms (means).

Example

Consider the two ratios $\frac{2}{3}$ and $\frac{4}{6}$. To check if these ratios are equal, we use the cross-multiplication method:

Since both products are equal, the ratios $\frac{2}{3}$ and $\frac{4}{6}$ are indeed equal.

Application

Equality of ratios has numerous applications in mathematics, including proportion problems, similarity of geometric shapes, and scale models. Understanding this concept is essential in solving problems where maintaining proportionality is crucial.

For example, if we know that two lengths in a scale model ($L_1$ and $L_2$) are proportional to the corresponding actual lengths ($A_1$ and $A_2$), we can set up the equation:

By ensuring the equality of ratios, we can solve for unknown quantities in various practical scenarios.