When transposing equations where the terms are connected by plus (\(+\)) or minus (\(-\)) signs, you’re essentially moving a term from one side of the equation to the other. To do this correctly, you must change the sign of the term as it crosses the equals sign (\(=\)).

For example, if you have an equation like \( x + 5 = 12 \), and you want to move the \(+5\) to the other side, it becomes \( x = 12 – 5 \). Notice how the \(+5\) changes to \(-5\) when it moves across the equals sign.

Similarly, if you have an equation like \( y – 3 = 7 \), and you want to move the \(-3\) to the other side, it becomes \( y = 7 + 3 \). Here, the \(-3\) changes to \(+3\) when transposed.

Remember, this rule applies to any terms connected by plus or minus signs. The key is to always reverse the sign of the term you’re transposing to maintain the equality of the equation.

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:

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:

\[ x = \frac{3y + 15}{2} \]

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.

Mixed numbers, proper fractions, improper fractions, fraction addition, fraction subtraction, and more...

Convert decimal to fraction, significant figures, decimal places, adding and subtracting decimal,...

Percentage error, changes, quantity, percentage conversions and more worksheets...

Ratio, direct proportion, inverse proportion, Hooke’s law, Charles’s and Ohm’s law

Laws of indices, power, roots, square roots, equation with indices, indexes...

Brackets, order of precedence, division, multiplication and...

Transposing Formula Calculator is designed to simplify this process, giving you step-by-step guidance to manipulate equations with confidence. Whether you're dealing with algebraic formulas or complex equations, our calculator ensures you get the right answer every time—completely free!

Transposing a formula means rearranging it to solve for a different variable. It's an essential skill in math and science, helping you isolate variables and make equations more manageable. For example, if you have an equation like \( y = mx + b \) and you want to solve for \( x \), transposing allows you to rearrange the formula to make \( x \) the subject.