Problem 35 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{63}{g}=\frac{9}{2} $$

Step-by-Step Solution

Answer

The solution is \( g = 14 \).

Step 1: Understand Proportions

A proportion states that two ratios are equal. Here, \( \frac{63}{g} = \frac{9}{2} \) indicates the ratios are equal, and our goal is to find the unknown variable \( g \).

Step 2: Cross-Multiply

In proportions, you can use cross-multiplication to solve for the unknown variable. Cross-multiply by multiplying opposite terms across the equal sign, yielding: \( 63 \times 2 = g \times 9 \).

Step 3: Simplify the Equation

Carry out the multiplication: \( 63 \times 2 = 126 \). So, the equation becomes \( 126 = g \times 9 \).

Step 4: Solve for g

To isolate \( g \), divide both sides of the equation by 9: \( g = \frac{126}{9} \).

Step 5: Calculate the Result

Perform the division: \( \frac{126}{9} = 14 \). Thus, \( g = 14 \).

Key Concepts

Cross-Multiplication: A Handy Tool

Cross-multiplication is a key technique for solving proportions. It's a useful method when you have an equation involving two ratios set equal to each other. The basic idea is to cross-multiply, which helps simplify and eventually solve for the unknown variable. Here's how it works in simple terms:

• Identify the two sides of the proportion. Each side is a ratio—like in our example, \( \frac{63}{g} = \frac{9}{2} \).
• Multiply across the equal sign. This means multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa. For our case, that is \( 63 \times 2 \) and \( 9 \times g \).

When you finish the multiplication, you end up with an equation that no longer looks like a fraction. Instead, it becomes a much simpler algebraic equation, which you can solve using basic algebra techniques. Cross-multiplication turns a complex fractional equation into an easier multiplication problem, providing clarity and ease in the problem-solving process.

Understanding and Using Ratios

Ratios are just comparisons between two quantities. They provide a way to express how much of one thing there is compared to another. A ratio can be written in three different forms: as a fraction, using a colon, or with the word 'to'. Consider the ratio \( 9:2 \), which can also be written as \( \frac{9}{2} \) or "9 to 2". This means for every 9 units of one thing, there are 2 units of another.

• The two ratios in the proportion \( \frac{63}{g} = \frac{9}{2} \) mean that the two set of quantities are in the same relative amounts.
• This equivalence allows us to use proportions and cross-multiplication to find missing values.

By setting two ratios equal to each other, you create a proportion—a powerful tool in mathematics. By understanding ratios, you also enhance your ability to assess relationships in diverse scenarios, from simple math problems to real-world applications like recipes or scaling models.

Solving Equations for Unknown Variables

Finding unknown variables is an essential skill in mathematics. Once cross-multiplication gives you a straightforward equation, your task is to solve for the variable—denoted here as \( g \). Solving for \( g \) involves basic algebraic steps:

• Once you have \( 126 = g \times 9 \), the next step is to isolate \( g \).
• To do that, perform the opposite operation of what's affecting \( g \). Since it's being multiplied by 9, you'll divide both sides by 9 to isolate \( g \).

So, solving the equation \( 126 = g \times 9 \) turns into \( g = \frac{126}{9} \), which simplifies to \( g = 14 \). Solving equations this way enables you to find unknowns with precision and confidence, a skill that applies across all branches of mathematics and various practical fields.