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Magazine Chapter 10: Problem 37
Solve the proportion: \(\frac{5}{9}=\frac{60}{x}\)
Short Answer
Expert verified
The solution to the proportion is \( x = 108 \).
Step by step solution
01
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 \( x \).
02
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:\[ 5 \cdot x = 9 \cdot 60 \]
03
Simplify the Equation
Perform the multiplication on the right side of the equation:\[ 5x = 540 \]
04
Solve for x
To isolate \( x \), divide both sides of the equation by 5:\[ x = \frac{540}{5} \]
05
Simplify the Division
Calculate the division on the right side of the equation:\[ x = 108 \]
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \( a \cdot d = b \cdot c \).
5x = 540. This equation makes it straightforward to find \( x \).
- 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.
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.
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鈥攍ike cross-multiplication鈥攊s necessary to solve the question.
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.
