Problem 55 - Fundamentals of Mathematics

Question

Write the proportion in fractional form: " 9 is to 8 as \(x\) is to 7 ""

Step-by-Step Solution

Answer

The proportion in fractional form is \( \frac{9}{8} = \frac{x}{7} \).

Step 1: Understand the Proportion

A proportion is a statement that two ratios are equal. The phrase "9 is to 8 as \(x\) is to 7" suggests that we have the ratio 9:8 and it equals the ratio \(x\):7.

Step 2: Set Up the Proportion as an Equation

Convert the verbal statement into a mathematical equation: \( \frac{9}{8} = \frac{x}{7} \). Here, 9 corresponds to \(x\) and 8 corresponds to 7.

Step 3: Solve for x by Cross-Multiplying

To solve the proportion \( \frac{9}{8} = \frac{x}{7} \), cross-multiply to eliminate the fractions: \( 9 \times 7 = 8 \times x \). This simplifies to \( 63 = 8x \).

Step 4: Isolate x

Solve for \(x\) by dividing both sides of the equation by 8: \( x = \frac{63}{8} \).

Key Concepts

Ratio Equality Basics

In mathematics, ratio equality is a fundamental concept that involves expressing two ratios as equal. By using ratios, we can compare two quantities directly. For example, the sentence "9 is to 8 as \( x \) is to 7" forms a ratio equality. Here, 9:8 and \( x \):7 are your two ratios. Ratio equality is expressed mathematically as \( \frac{9}{8} = \frac{x}{7} \). Breaking this down: the idea is that the first pair of numbers relate in the same way as the second pair. Understanding ratio equality is crucial, as it is often the foundation for solving mathematical problems involving proportions. When you equate ratios, you can solve for unknown quantities, making it an incredibly useful tool in everyday math problems.

Cross-Multiplication Method

Cross-multiplication is a powerful method used to solve proportions where two ratios are set equal to each other. In our example, we have the equation \( \frac{9}{8} = \frac{x}{7} \). This is a classic setup for cross-multiplying:

• Multiply the numerator of the first ratio by the denominator of the second ratio: 9 \( \times \) 7.
• Multiply the numerator of the second ratio by the denominator of the first ratio: \( x \times 8 \).
• Set these two products equal to each other: 9 \( \times \) 7 = 8 \( \times \) x.

This gives us the equation 63 = 8x. By getting rid of the fractions, cross-multiplication simplifies the process of solving proportions. This method makes it easier to find the value of unknown variables in a single step, transforming a proportion into a straightforward multiplication problem.

Solving Equations

Once you've used cross-multiplication to reach an equation like 63 = 8x, the next step is solving for the unknown variable, in this case, \( x \). This involves isolating \( x \) on one side of the equation. You do this by dividing both sides of the equation by 8:

• Divide 63 by 8 to simplify the expression: \( x = \frac{63}{8} \).

This calculation provides the solution for \( x \) in a form that can be further simplified or converted into a decimal if needed. Solving equations is an essential skill in algebra that enables you to find unknown quantities by rearranging and manipulating mathematical expressions. By mastering this process, you can solve various types of equations beyond just proportions.