You can use the reciprocal of the number that you’re trying to “get rid of” if a fraction is multiplying the variable. You solve linear equations with reciprocals when you see a fraction — it's easier than using multiplication or division.

Here is an example of a fraction multiplying the variable:


Look at the following examples of reciprocals — after the reciprocals, you can see how each, when multiplied together, equals 1.

  • 5 and 1/5 are reciprocals:

  • −3/7 and −7/3 are reciprocals:


Solving equations in the fewest possible steps is usually preferable.



Use the following steps to solve for the variable, using reciprocals.

  1. Multiply each side by the reciprocal.

    In this example, the variable is multiplied by 4/5. So each side of the equation needs to be multiplied by the reciprocal 5/4.

  2. Reduce and simplify.


    On the left side, the 5s and 4s cancel each other out, and on the right side, 12 divided by 4 equals 3, resulting in:

    a = 5 × 3

    a = 15

Decimals can be made into fractions, which are much easier to deal with and avoid the problem of inadvertently misplacing the decimal when performing operations.