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Chapter 11: Problem 37

Solve proportion. \(\frac{9}{15}=\frac{n}{5}\)

Short Answer

Expert verified
The solution is \(n = 3\).

Step by step solution

01

Understand the Cross Multiplication Method

In a proportion, the product of the means equals the product of the extremes. For the equation \(\frac{9}{15}=\frac{n}{5}\), perform cross multiplication to set up the equation. This means you multiply the numerator of the first fraction by the denominator of the second and vice versa. This will give you: \(9 \times 5 = 15 \times n\).
02

Perform the Multiplication

Calculate the products from the cross multiplication. Multiply 9 by 5 to get 45, and set it equal to 15 times \(n\). So, the equation becomes: \(45 = 15n\).
03

Solve for \(n\)

To isolate \(n\), you need to divide both sides of the equation by 15. This will give you \(n = \frac{45}{15}\).
04

Simplify the Expression

Divide 45 by 15 to simplify the expression. This calculation gives \(n = 3\). Thus, the solution to the proportion is \(n = 3\).

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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 a useful tool when working with proportions. It involves two fractions set equal to each other, just like in our problem \( \frac{9}{15} = \frac{n}{5} \). To solve for the unknown \( n \), we cross multiply. This means:
  • Multiply the numerator of the first fraction by the denominator of the second fraction.
  • Multiply the numerator of the second fraction by the denominator of the first fraction.
Hence, you get \( 9 \times 5 = 15 \times n \).
Cross multiplication turns a proportion into an equation without fractions, making it much easier to solve.
Equation Solving
Once you've cross multiplied, you will have a basic equation to solve. In our example, this equation is \( 45 = 15n \). Let's break down the steps to solve it:
  • Identify the variable part of the equation (in this case, \( 15n \) ).
  • To isolate the variable \( n \), divide both sides by the coefficient of \( n \).
  • In our example, divide both sides by 15, leading to \( n = \frac{45}{15} \).
Equation solving is often just a matter of reversing the arithmetic that has been done to \( n \), ensuring you simplify to find \( n = 3 \).
This step-by-step approach helps manage the equation and ensures accuracy.
Simplifying Expressions
After isolating \( n \), we simplify the expression \( \frac{45}{15} \). Simplifying expressions involves:
  • Performing division if both the numerator and denominator are numbers.
  • Reducing fractions to their simplest form.
Simplifying \( \frac{45}{15} \) means solving the division, which gives us 3.
This simplification confirms our result that \( n = 3 \) is the correct solution.
Always simplify expressions whenever possible to arrive at the cleanest and most understandable form of your answer.

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