Problem 37 - Fundamentals of Mathematics

Question

Solve the proportion: \(\frac{5}{9}=\frac{60}{x}\)

Step-by-Step Solution

Answer

The solution to the proportion is \( x = 108 \).

Step 1: Understand the Proportion

The given proportion is \( \frac{5}{9} = \frac{60}{x} \). This means that the ratio of 5 to 9 is the same as the ratio of 60 to \( x \). Our goal is to find the value of \( x \).

Step 2: Cross-Multiply the Ratios

To solve for \( x \), cross-multiply the two fractions. This means multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa:\[ 5 \cdot x = 9 \cdot 60 \]

Step 3: Simplify the Equation

Perform the multiplication on the right side of the equation:\[ 5x = 540 \]

Step 4: Solve for x

To isolate \( x \), divide both sides of the equation by 5:\[ x = \frac{540}{5} \]

Step 5: Simplify the Division

Calculate the division on the right side of the equation:\[ x = 108 \]

Step 6: Verify the Solution

Check that the solution is correct by substituting \( 108 \) back into the original proportion:\( \frac{5}{9} = \frac{60}{108} \). Simplify \( \frac{60}{108} \) to \( \frac{5}{9} \) confirming the solution is correct.

Key Concepts

Cross-Multiplication

Cross-multiplication is like a magical tool when it comes to solving proportions. It lets you find the unknown value in a fraction equation easily. When you have a proportion, which is an equation where two ratios are equal, cross-multiplication helps solve it by simplifying the comparison. For example, if you have the equation \( \frac{a}{b} = \frac{c}{d} \), cross-multiplying leads to \( a \cdot d = b \cdot c \).

• This method works because if two fractions are equal, their cross-products must also be equal.
• It's especially handy because once you cross-multiply, you're dealing with a simple equation without fractions.

Let's apply cross-multiplication to the exercise \( \frac{5}{9} = \frac{60}{x} \). By cross-multiplying, we multiply 5 by \( x \) and 9 by 60, leading to the equation:
5x = 540. This equation makes it straightforward to find \( x \).

Ratio

A ratio shows the relative size of two or more values. It gives you a comparison between two numbers, showing how much of one thing there is compared to another. In the original exercise, the ratio \( \frac{5}{9} \) means we have 5 parts of something for every 9 parts of another thing.

• Ratios play a crucial role in proportions. A proportion, essentially, is two equivalent ratios.
• They are everywhere: in recipes, maps, and even in scale models.

Recognizing that the problem \( \frac{5}{9} = \frac{60}{x} \) maintains a constant relationship helps us understand the balance between the quantities involved. By equating these two ratios, we’re tasked with finding the unknown part of the second ratio, which is \( x \). Understanding ratios provides the foundation for solving proportions.

Fractions

Fractions are numbers that represent a part of a whole. They include two parts: a numerator (the number on top) and a denominator (the number on the bottom). In proportions like \( \frac{5}{9} \) and \( \frac{60}{x} \), these fractions denote a specific relationship between the numerator and the denominator.

• Fractions are a key part of solving proportions because they help us understand the ratios involved.
• Proper understanding of fractions aids in identifying what operation—like cross-multiplication—is necessary to solve the question.

Smarter use of fractions—like simplifying them—can make solving equations easier. Once we solve for \( x \), verifying by substituting it back as a fraction \( \frac{60}{108} \), and reducing to \( \frac{5}{9} \) confirms our solution.

Solution Verification

Solution verification ensures that your answer is accurate. When you solve a mathematical equation, it's always a good idea to check if your solution makes sense.

• The best way to verify a solution in a proportion is to substitute it back into the original ratio and simplify.
• Simplification should yield the same result on both sides of the equation if the answer is correct.

In our example, after finding \( x = 108 \) from \( \frac{5}{9} = \frac{60}{x} \), we substitute \( x \) back into the proportion as \( \frac{60}{108} \). Simplifying this fraction gives us \( \frac{5}{9} \), matching the original ratio. This confirms our solution is indeed correct, as both sides of the equation are equal when simplified.