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Formulae are facts or rules written in a mathematical way. They help you work out a quantity using other amounts connected to it.

Sometimes, you see letters being used to represent numbers. Letters are used so numbers can then be substituted (swapped in) depending on the rule.

For example, to find the volume of a cuboid, you multiply the length by the width by the height.

As a formula, you would write:

Volume = length x width x height

or, for short:

Volume = lwd

The letters are written next to each other to show that you have to multiply them.

Algebra doesn’t use an ‘x’ to show multiplication because it can get confusing since other letters are being used!

Not all cuboids have the same dimensions - this is where substitution comes in.

Depending on the measurements each time, the numbers representing the letters will change, but the rule to find the volume never will.

Look at this cuboid.

This time the length = 4cm, width = 2cm, and the height = 3cm.

To find the volume, you’d use the formula:

Volume = lwd

So:

4cm x 2cm x 3cm = 24cm³.

What if the length changed to 3cm, the width to 2cm, and the height to 2cm?

You would still use the same formula to find the volume, but swap in the different measurements.

3cm x 2cm x 2cm = 12cm³

Sometimes, formulae use words instead of just letters.

Example:

Libby works in a pet shop. She earns £30 a day plus £10 for every pet she sells.

The formula to show this would be:

Amount earned = £30 + (10 x number of pets sold).

How much would Libby earn if she sold 5 pets on Monday?

To solve this, you would have to substitute the number 5 into the “number of pets sold.”

Amount earned = £30 + (10 x 5)

So Libby would earn £80 on Monday.

How much would Libby earn if she sold 9 pets on Tuesday?

This time, you would substitute the number 9 into the “number of pets sold.”

Amount earned = £30 + (10 x 9).

Libby would earn £120 this time.

Equations and formulae quiz

See if you can use what you learned to solve these equations.

Put your knowledge of substitutions in algebra to the test with this quiz.