Problem 26 - Elementary and Intermediate Algebra

Question

Solve each equation and check the result. If an equation has no solution, so indicate. $$ \frac{5}{9}-\frac{1}{3}=\frac{1}{b} $$

Step-by-Step Solution

Answer

The solution is \( b = \frac{9}{2} \).

Step 1: Find a Common Denominator

To subtract the fractions on the left, we need a common denominator. The denominators are 9 and 3. The least common denominator is 9. So, convert \( \frac{1}{3} \) to have a denominator of 9: \( \frac{1}{3} = \frac{3}{9} \).

Step 2: Subtract the Fractions

Subtract \( \frac{3}{9} \) from \( \frac{5}{9} \) to find the difference: \[ \frac{5}{9} - \frac{3}{9} = \frac{2}{9} \].

Step 3: Set the Equation Equal

Now that the fractions are subtracted, the equation is \( \frac{2}{9} = \frac{1}{b} \).

Step 4: Solve for b

Cross-multiply to solve for \( b \): \[ 2b = 9 \]. Divide both sides by 2 to isolate \( b \): \[ b = \frac{9}{2} \].

Step 5: Check the Solution

Substitute \( b = \frac{9}{2} \) back into the equation to verify the solution. The right side becomes \( \frac{1}{\frac{9}{2}} = \frac{2}{9} \), which matches the left side \( \frac{2}{9} \). The solution is correct.

Key Concepts

Fractions

When working with fractions, it's important to understand that they represent parts of a whole. A fraction consists of a numerator (the top number) and a denominator (the bottom number). The denominator indicates how many equal parts the whole is divided into, while the numerator tells you how many of those parts are considered.
Fractions can often have different denominators, which can make addition, subtraction, or comparison more complex. Instead of trying to directly work with these unlike fractions, we typically find a common denominator. This makes them easier to work with. In this exercise, changing \( \frac{1}{3} \) to \( \frac{3}{9} \) helps make subtraction straightforward. Getting comfortable with converting fractions is critical to becoming proficient with them.
Understanding fractions also involves simplifying them. A fraction like \( \frac{9}{3} \) is not in its simplest form because both numbers are divisible by 3, and it simplifies to 3/1 or just 3.

Cross-Multiplication

Cross-multiplication is a technique used to solve equations where two fractions are set equal to each other. When we say "cross-multiply," we mean we will multiply across the equal sign diagonally. This is particularly useful when you have fractions like \( \frac{2}{9} = \frac{1}{b} \).
To cross-multiply:

• Multiply the numerator of the first fraction by the denominator of the second.
• Multiply the numerator of the second fraction by the denominator of the first.

In this exercise, cross-multiplying \( 2 \times b = 1 \times 9 \) turns the equation into a simpler one: \( 2b = 9 \). This process quickly facilitates solving for a variable without fractions, which can be complex. Cross-multiplication is an essential skill for efficiently solving these types of fractional equations.

Common Denominator

Finding a common denominator is a key step when you need to add or subtract fractions. It involves ensuring that the fractions you're working with share the same denominator. This makes it possible to easily perform operations on them.
For example, when considering the fractions \( \frac{5}{9} \) and \( \frac{1}{3} \), we notice their denominators are different. To create a common denominator, we find the smallest number that both original denominators divide into evenly, which, in this case, is 9.

• Convert \( \frac{1}{3} \) to \( \frac{3}{9} \).
• This conversion allows you to directly subtract \( \frac{5}{9} - \frac{3}{9} = \frac{2}{9} \).

Using a common denominator simplifies operations and ensures accuracy in calculations. Mastery of this technique will bolster your ability to work efficiently with fractions.