Problem 24 - Elementary Algebra

Question

\(\frac{3}{h}+\frac{5}{2 h}=1\)

Step-by-Step Solution

Answer

The solution is \(h = \frac{11}{2}\).

Step 1: Identify the Common Denominator

Start by finding a common denominator for the fractions. Here, the denominators are \(h\) and \(2h\). The common denominator will be \(2h\).

Step 2: Rewrite Each Fraction

Express both fractions with the common denominator \(2h\): \[ \frac{3}{h} = \frac{6}{2h} \quad \text{and} \quad \frac{5}{2h} = \frac{5}{2h} \]

Step 3: Combine the Fractions

Since both fractions now have the same denominator, combine them:\[ \frac{6}{2h} + \frac{5}{2h} = \frac{11}{2h} \]

Step 4: Set the Combined Fraction Equal to 1

Set the equation \(\frac{11}{2h} = 1\) and solve for \(h\).

Step 5: Solve for h

Multiply both sides of the equation by \(2h\) to get \(11 = 2h\).

Step 6: Isolate h

Solve for \(h\) by dividing both sides by 2:\[ h = \frac{11}{2} \]

Key Concepts

Fractions

Fractions are a way to express a part of a whole. They consist of a numerator, which is the top part, and a denominator, which is the bottom part. In algebra, fractions often appear in equations to represent division. For example, in the expression \(\frac{3}{h}\), the numerator is 3, and the denominator is \(h\). This format means 3 divided by \(h\). Understanding fractions is crucial, especially when dealing with algebraic equations. They can look complex, but breaking them down into the numerator and denominator makes them easier to understand.

• Numerator: The top part of the fraction.
• Denominator: The bottom part of the fraction. It indicates the total number of equal parts.

Grasping this structure aids in recognizing how to perform operations like addition or subtraction of fractions by bringing them to a common denominator.

Common Denominator

When working with fractions, especially in equations, a common denominator is essential. It unifies the fractions, allowing them to be added or subtracted directly. A common denominator is just a shared multiple of the original denominators.
In the equation given, the fractions \(\frac{3}{h}\) and \(\frac{5}{2h}\) have different denominators \(h\) and \(2h\). To add these fractions, we determine a common denominator. In this case, \(2h\) is the common denominator since it is the larger denominator that can incorporate both original denominators.

• To find a common denominator, consider the least common multiple (LCM) of the denominators.
• Convert each fraction to have this shared denominator.
• Once both fractions have the same denominator, they can easily be combined.

Establishing a common denominator simplifies the equation solving process by allowing us to focus on the numerators.

Solving Equations

Solving algebraic equations often involves isolating the variable you're solving for. This process can include manipulating both sides of the equation through multiplication, division, and other operations. For the equation \(\frac{6}{2h} + \frac{5}{2h} = 1\), once you have a single fraction, you equate it to the value on the other side of the equation and solve for the unknown variable.
To solve:

• First, simplify the fraction so that it has one numerator or form on one side.
• To eliminate the fraction, multiply both sides by the denominator, which clears the fraction.
• Isolate the variable (in this case \(h\)) by performing inverse operations, such as division, to solve for it.

In our specific exercise, \(\frac{11}{2h} = 1\) simplified to \(11 = 2h\). Dividing both sides by 2 gives the solution \(h = \frac{11}{2}\). Following these steps systematically ensures that the process is logical and the solution is correct.