Dividing Letters in Algebra
(KS3, Year 7)

The Lesson

Letters can be divided, multiplied with numbers, other letters and the same letter.

• A letter can be divided by a number. Write the letter as the numerator of a fraction, and the number as the denominator.
• A letter can be divided by a different letter. Write the letter you are dividing by (b) under the letter you are dividing (a).

Dividing Letters to Make Terms

A term is a collection of letters and numbers multiplied and/or divided together. In the examples above, the letter a has been divided by a number and the letter b to make terms. These divisions can be combined to make a more complicated term: Read more about how to divide terms

Dividing a Letter with the Same Letter

Dividing a letter with itself equals 1: Letters sometimes have exponents, which tell you how many times the letter is multiplied by itself. For example, a¹ = a, a² = a × a and a³ = a × a × a. When a letter with an exponent is divided by that same letter, we must subtract the exponents.

• For example, imagine we wanted to divide a² ÷ a. (Don't forget: if a letter does not have an exponent, it has an implicit exponent of 1): We can see why this works if we write out the term in full, rather than using exponent notation, remembering that a² = a × a. Each a on the denominator cancels out an a on the numerator, leaving only one a:
• Imagine we wanted to divide a⁴ ÷ a²: We can see why this works if we write out the term in full, remembering that
  a⁴ = a × a × a × a and that a² = a × a. Each a on the denominator cancels out an a on the numerator, leaving two a's:

Read more about how to divide letters with exponents

Beware

Be Careful with Signs

Letters can have different signs: a + sign if they are positive, and a - sign if they are negative. Remember the rules for dividing different signs: Same signs give a plus: Different signs give a minus: