# Transposing and Rearranging Formulas

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In a previous lesson we showed how to Solve Equations using the work down through the “Onion Skins” Method.

Knowledge of that lesson is needed as a background, before doing our lesson on Rearranging Formulas.

In our Transposing Formulas lesson we will review the use of the “Onion Skin” method, and show how it can be applied to Transposing (or rearranging) Algebra Formula Equations.

Review of Onion Skin Method

Here are the steps that we follow to solve an Equation using Onion Skins.

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Here is an example of solving a simple Equation using the Onion Method.

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Onion Skins for Transposing

The steps we do for Transposing an Equation into a new equation with a different letter variable as the subject, is basically the same as we do for Solving Equations.

Transposing Equations is the same as Solving Equations, except that we have a set of letter variables to deal with, and very few if any numbers present.

Here is a simple Transposing Example that is very similar to our previous Solving Equation example.

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Solving a Two Step Equation Using Onion Skins

The following example shows how to solve a two step equation.

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We always put our first innermost circle around the letter variable we are solving for, in this case it was “h”.

In the above example, the variable letter “h” is x2 and +3.

If you are not sure which of these two operations gets circled first, remember we have to do them in BODMAS/PEMDAS order.

This means we circle the 2h for x2 before we circle the 2h + 3 for the +3 operation.

Transposing a Two Step Formula Using Onion Skins

Let’s now look at a Formula that is very similar to the Equation we just solved, but contains lots of letters instead of numbers.

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Formulas Containing Exponents and Fractions

The following examples show to transpose formulas that contain exponents such as power of 2 squares.

Squaring comes under the “O” for other things in BODMAS, and under the “E” for Exponents in PEMDAS.

Because it is the next operation to follow brackets, it is usually one of the very first onion skins we draw around our letter.

Square Root is the opposite of Squaring, and so when we peel the Onion we do square root to undo squaring.

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The next example contains a Square power of 2 exponent, but because it is not on the letter we are transposing to, we do not have to reverse it out with a square root.

The formula has a Fraction and we need to move it into the bottom of our Algebra Formula BEFORE we draw our Onion Skins.

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In the next example, we are using the same formula, but this time we are transposing to make “v” the subject.

This “v” that we are transposing to does have a square on it, so we will have to do a square root reversal of it when we “peel” our onion.

(Squaring and Square Roots are opposites, just like + and – are opposites).

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Transposing Financial Mathematics Formulas

Financial Mathematics has a lot of Cost Price, Selling Price, Markup, Discount, Profit and Loss Formulas.

Rather than memorising every single possible formula we may need, it is much easier to only know the main formulas, and then be able to transpose these to get any other formulas we need.

In the following example, we start with the Formula for applying a discount “D”, to a Marked Price on an item “M” to work out the selling price “S”.

We transpose the formula to an M= formula, so that we could work out the original Marked Price required on an item if it is going to sell for a certain amount when discounted.

Notice that we follow “BODMAS/PEMDAS” order, and onion circle the brackets item before circling the dividing part of the formula.

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Transposing a Multi-Variable Formula

In this next example we have quite an involved formula, which results in many onion skins that have been drawn outwards from our subject letter of “c”, in BODMAS/PEMDAS order.

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Transposing a Square Root Formula

The following formula contains a square root, and when we reverse this out in the peeling process we use squaring as its opposite.

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The above transposing is probably the most complicated example we have covered.

Note that because the k/l is all under the Square Root sign it is circled before the actual Square Root sign.

This does not seem to follow BODMAS/PEMDAS, because the “D” dividing should come after the “O” other things square root.

The k/l being under the Square Root sign is kind of like it being in brackets, and it must be considered first, before a Square root can be taken.

Eg. Square Root of 27/3 would be worked out as Square Root of 27/3 = Square Root of 9 = 3; and not as Square Root of 27 then divided by 3.

Transposing When Subject is in Denominator

If our desired subject letter varibale is inthe bottom of a Fraction, we cannot directly apply the onion skins.

We must first flip over both sides of the Equation, and then do our Onion Skins.

This is mathematically Possible because things like 12/6 = 6/3 are still true when we flip both sides and have 6/12 = 3/6.

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Related Items

Solving Equations Using Onion Skins

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