Problem 3 - Elementary Algebra

Question

Solve each of the equations. $$\frac{5}{12}=\frac{n}{24}$$

Step-by-Step Solution

Answer

The solution is \( n = 10 \).

Step 1: Understand the Equation

We are given the equation in the form \( \frac{5}{12} = \frac{n}{24} \). This is a proportion, where two ratios are set equal to each other.

Step 2: Use Cross-Multiplication

To solve the proportion, we use cross-multiplication. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction. This gives us: \( 5 \times 24 = 12 \times n \).

Step 3: Perform the Multiplication

Calculate the products from Step 2:- \( 5 \times 24 = 120 \).- On the right side, we have \( 12 \times n \) which remains as \( 12n \). So our equation now is: \( 120 = 12n \).

Step 4: Solve for n

To isolate \( n \), divide both sides of the equation by 12:\[ n = \frac{120}{12} \]Perform the division to find \( n \).

Step 5: Calculate the Division

Divide 120 by 12 to find the value of \( n \).\( n = 10 \)

Step 6: Verify the Solution

Substitute \( n = 10 \) back into the original equation to verify:\( \frac{10}{24} \) simplifies to \( \frac{5}{12} \) which confirms our solution is correct, as both sides of the equation are equal.

Key Concepts

Understanding Proportions

Proportions are mathematical statements that show two ratios or fractions being equal to each other. They are useful for comparing quantities and understanding relationships between different variables. In the given exercise, we are dealing with a proportion because the equation \( \frac{5}{12} = \frac{n}{24} \) involves two equal fractions.
To solve proportions, especially when an unknown variable is present like \( n \) here, a common method is cross-multiplication. This technique is used when you have an equation of the form \( \frac{a}{b} = \frac{c}{d} \). By cross-multiplying, you solve for the unknown by transforming the equation into a simpler form.
This technique simplifies fractions and reveals how one number relates to another. It's a powerful tool for solving those pesky equations involving proportions!

Comprehending Fractions

Fractions are numbers that represent parts of a whole. They consist of a numerator and a denominator. The fraction \( \frac{5}{12} \) indicates that 5 is the numerator and 12 is the denominator.
In fractions, the numerator is the top number, and it tells us how many parts are being considered. The denominator, the bottom number, shows how many equal parts make up the whole. When solving equations involving fractions, it is essential to treat numerators and denominators separately to ensure calculations are accurate.
Fractions can be simplified by finding common factors, as demonstrated in the verification step of the exercise. Simplifying \( \frac{n}{24} \) to equal \( \frac{5}{12} \) validates the solution. Understanding fractions' basic properties helps in comparing, simplifying, and solving such equations.

Solving Equations with Variables

Equations often involve variables, which are symbols like \( n \) in our exercise. Solving an equation means finding the value of the variable that makes the equation true.
Here, the original equation \( \frac{5}{12} = \frac{n}{24} \) is simplified to \( 120 = 12n \) after cross-multiplying. The goal is to isolate \( n \) to solve the equation. This is done by performing operations equally on both sides, such as dividing by 12 in this example.
Once \( n \) is isolated, we find \( n = 10 \). Substituting back this value into the original equation allows for verification. Solving equations requires attention to detail in maintaining equality on both sides. Understanding these basic operations facilitates solving many algebraic expressions efficiently.