Problem 22 - Elementary Algebra

Question

Solve each of the equations. $$\frac{5}{6}=\frac{n}{n+1}$$

Step-by-Step Solution

Answer

\(n = 5\)

Step 1: Cross-Multiply

To solve the equation \(\frac{5}{6}=\frac{n}{n+1}\), begin by cross-multiplying. This involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa:\[5\cdot(n+1) = 6\cdot n\].

Step 2: Distribute

Next, distribute the 5 on the left side of the equation:\[5n + 5 = 6n\].

Step 3: Rearrange the Equation

Reorganize the equation to isolate one of the terms involving \(n\). Subtract \(5n\) from both sides:\[5 = 6n - 5n\].

Step 4: Simplify the Equation

Combine like terms on the right side of the equation:\[5 = n\].

Step 5: Verify the Solution

To ensure that \(n = 5\) is correct, substitute \(n = 5\) back into the original equation:\[\frac{5}{6} = \frac{5}{5+1}\],which simplifies to \[\frac{5}{6} = \frac{5}{6}\]. This confirms that the solution is correct.

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve equations in which two ratios are set equal to each other, also known as a proportion. It simplifies the process of solving by removing the fractions from the equation altogether.
When you have an equation in the form \( \frac{a}{b} = \frac{c}{d} \), cross multiplication involves multiplying the numerator (top number) of each fraction by the denominator (bottom number) of the other fraction.
This yields the equation:

• \( a \times d = b \times c \)

This new equation no longer contains fractions, making it easier to solve for the unknown.
For instance, in the original exercise with \( \frac{5}{6} = \frac{n}{n+1} \), cross multiplication transforms it into \( 5(n+1) = 6n \), which is simpler to manipulate.
Using this method helps in controlling the fractions and progresses efficiently towards finding the value of the unknown.

Distributive Property

The distributive property is a useful algebraic principle that allows you to simplify expressions. It states that multiplying a single term by a sum or difference inside parentheses is the same as multiplying each individual term inside the parentheses by the single term outside it.
In mathematical terms, it is expressed as:

• \( a(b + c) = ab + ac \)

This property ensures that every term inside the parentheses is affected uniformly by the multiplier.
In our exercise, after cross multiplication gives you \( 5(n+1) = 6n \), applying the distributive property transforms it to \( 5n + 5 = 6n \).
By distributing the 5 over the \( n+1 \) term, we obtain a simple linear equation, thus breaking down complex expressions to simpler, more manageable components.

Equation Solving Steps

Successfully solving algebraic equations involves a consistent approach that generalizes to various problems. Here are the key steps:
1. **Eliminate Fractions:** Start by using cross multiplication to clear any fractions, so the equation becomes simpler to handle.
2. **Distribute and Simplify:** Use the distributive property wherever applicable to expand expressions and gather like terms.
3. **Isolate the Variable:** Rearrange the equation to bring all terms involving the unknown variable on one side. For our given example, it involved subtracting \( 5n \) from both sides to isolate \( n \).
4. **Simplify Further:** Combine like terms and solve for the variable. In this problem, it was simply calculating \( 6n - 5n = n \).
5. **Verify the Solution:** Lastly, substitute the obtained solution back into the original equation to confirm its correctness. This helps to rule out any calculation errors along the way.
Following these structured steps ensures an organized approach to tackling any algebraic equation systematically.