Problem 40 - Fundamentals of Mathematics

Question

(Section 7.3) Solve the proportion: \(\frac{7}{15}=\frac{x}{90}\).

Step-by-Step Solution

Answer

The value of \( x \) is 42.

Step 1: Understand the Proportion

We are given the proportion \( \frac{7}{15} = \frac{x}{90} \). This means the two ratios are equal, and we need to find the value of \( x \) that makes this equation true.

Step 2: Cross Multiply

To solve the proportion, cross-multiply the terms. This means multiplying the numerator of one ratio by the denominator of the other ratio.\( 7 \times 90 = 15 \times x \) or \( 630 = 15x \).

Step 3: Solve for x

Now, solve for \( x \) by dividing both sides of the equation by 15:\[ x = \frac{630}{15} \]So, \( x = 42 \).

Step 4: Verify the Solution

To ensure our solution is correct, substitute \( x = 42 \) back into the original proportion:\( \frac{7}{15} = \frac{42}{90} \).Both fractions reduce to \( \frac{7}{15} \). Hence, our solution is verified as correct.

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities by showing the relative size of one quantity to another. It is commonly written in the form of a fraction, like \( \frac{a}{b} \) where \( a \) and \( b \) are quantities being compared. In our exercise, we have the ratio \( \frac{7}{15} \) on one side and \( \frac{x}{90} \) on the other. This indicates that these two separate parts relate to each other proportionally.

Consider this: If you have 7 apples and 15 oranges, the ratio of apples to oranges is \( \frac{7}{15} \). If you want to maintain this same relationship (ratio) but have a total of 90 units of oranges, you'd need to find how many apples correspond to keep the ratio equivalent; that's where the unknown \( x \) comes in.

This concept is incredibly useful in everyday scenarios, like scaling recipes or calculating distances. By understanding ratios, you can navigate problems of proportionality with ease.

Cross-Multiplication Technique

The technique of cross-multiplication is pivotal when finding unknown values in proportions. It involves multiplying the numerator of one ratio by the denominator of the other ratio, and vice versa. This method transforms a proportion into a simple equation that can be easily solved. It is an efficient way to handle proportions because it eliminates the fractions, simplifying the solution process.

In our example, the exercise gave us the proportion \( \frac{7}{15} = \frac{x}{90} \). To use cross-multiplication:

• Multiply 7 (numerator of the first ratio) by 90 (denominator of the second ratio), which equals 630.
• Then, multiply 15 (denominator of the first ratio) by \( x \) (numerator of the second ratio), which provides \( 15x \).

The new equation is then \( 630 = 15x \). By simplifying through cross-multiplication, it leads us directly to our next step of solving for \( x \).

Strategies for Solving Equations

Solving equations is all about finding the value of the unknown variable that makes the equation true. Once you have isolated the equation through cross-multiplication, the aim is to solve for \( x \).

In the proportional equation \( 630 = 15x \), the goal is to isolate \( x \) so you can find its value. Achieve this by performing operations that maintain balance on both sides of the equation. In this instance, you divide both sides by 15:

\[ x = \frac{630}{15} \]

By solving, you'll find \( x = 42 \).

Checking your work is a very important final step in solving equations. Substitute the value of \( x \) back into the original proportion to verify the solution fits:
\( \frac{7}{15} = \frac{42}{90} \), both fractions reduce to the same value, confirming our calculations are correct. This methodical approach ensures your math is spot on, and builds a solid foundation for tackling more complex equations in the future.