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When transposing equations that involve fractions, the process is similar to transposing terms connected by plus or minus signs, but with an extra step. The goal is to eliminate the fraction by multiplying both sides of the equation by the denominator of the fraction.

For instance, consider the equation \( \frac{x}{3} = 4 \). To transpose the fraction, you multiply both sides of the equation by the denominator, which in this case is 3. This gives you \( x = 4 \times 3 \), or \( x = 12 \).

If the equation involves multiple fractions, like \( \frac{y}{5} + 2 = 7 \), start by isolating the fraction. Subtract 2 from both sides to get \( \frac{y}{5} = 5 \). Then, multiply both sides by 5 to eliminate the fraction, resulting in \( y = 5 \times 5 \), or \( y = 25 \).

In cases where the equation has fractions on both sides, like \( \frac{a}{4} = \frac{b}{2} \), you can eliminate the fractions by cross-multiplying. This means multiplying the numerator of one fraction by the denominator of the other, leading to \( 2a = 4b \). You can then simplify or transpose further as needed.

The key when dealing with fractions is to always multiply by the appropriate denominator to eliminate the fraction, ensuring the equation remains balanced.

To transpose a formula and find the subject of the formula when it involves fractions, follow these steps:

Transposing Equations and Transpose formula

General Approach

1. Identify the Fractional Formula:
Start with a formula where the subject is contained within fractions. For example:
\[ \frac{a}{b} = c + \frac{d}{e} \]

2. Isolate the Fraction Containing the Subject:
Rearrange the equation to isolate the fraction or term that contains the variable you want to solve for. For the example formula:
\[ \frac{a}{b} – \frac{d}{e} = c \]

3. Clear Fractions by Multiplying:
Multiply through by the denominators to eliminate the fractions. This step simplifies the equation and helps isolate the variable.

4. Solve for the Subject:
Continue isolating the subject by using algebraic operations such as addition, subtraction, multiplication, and division.

Worked Example 1: Simple Case

Formula:
\[ \frac{2x}{3} = y + 5 \]

Objective:
Find \( x \) in terms of \( y \).

Step 1: Isolate the Fraction Containing the Subject

Subtract 5 from both sides to isolate the fraction:
\[ \frac{2x}{3} = y + 5 \]

Step 2: Clear the Fraction

Multiply both sides by 3 to get rid of the fraction:
\[ 3 \times \frac{2x}{3} = 3 \times (y + 5) \]

Simplify:
\[ 2x = 3y + 15 \]

Step 3: Solve for \( x \)

Divide both sides by 2 to isolate \( x \):
\[ x = \frac{3y + 15}{2} \]

Solution:
The formula for \( x \) in terms of \( y \) is:
\[ x = \frac{3y + 15}{2} \]

Worked Example 2: More Complex Case

Formula:
\[ \frac{a}{x} + \frac{b}{y} = c \]

Objective:
Find \( x \) in terms of \( a \), \( b \), \( y \), and \( c \).

Step 1: Isolate the Fraction Containing the Subject

Subtract \( \frac{b}{y} \) from both sides:
\[ \frac{a}{x} = c – \frac{b}{y} \]

Step 2: Clear the Fraction

To eliminate the fraction involving \( x \), multiply both sides by \( x \):
\[ a = x \left(c – \frac{b}{y}\right) \]

Step 3: Solve for \( x \)

Divide both sides by \( \left(c – \frac{b}{y}\right) \):
\[ x = \frac{a}{c – \frac{b}{y}} \]

Solution:
The formula for \( x \) in terms of \( a \), \( b \), \( y \), and \( c \) is:
\[ x = \frac{a}{c – \frac{b}{y}} \]

By following these steps, you can isolate and solve for the subject of any formula involving fractions.