Cross Multiply to Solve Equations with Fractions

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Specific Cross Multiplication Example : image by Mike DeHaan

Specific Cross Multiplication Example : image by Mike DeHaan

Solving General Equations with Fractions

Next, try solving fractions in the general case where ‘a’, ‘b’, ‘c’ and ‘d’ are non-zero integers. Again, solve for X starting in either the numerator or denominator.

X/b = c/d

Multiply both sides by b

X/b * b = c/d * b

Cancel out the /b * b on the left side, and consolidate the right side of the equation.

X = (b*c)/d

The cross multiplication would be more obvious in a similar case:

a*X/b = c/d

Multiply both sides by b/a

a*X/b * b/a = c/d * b/a

Cancel out “a/b * b/a” on the left side of the equation, and associate the numerator (‘c’ & ‘b’) and denominator (‘d’ & ‘a’) on the right.

X = c*b/d*a

The next example starts with X in the denominator and solves the most complicated example, c/d = a/(b*X).

c/d = a/(b*X)

Multiply both sides by X * (d/c).

(c/d) * X * (d/c)  = a/(b*X) * X * (d/c)

Cancel out “c/d * d/c” on the left and “/X * X” on the right.

X = (a*d)/(b*c)

A Five Step Summary for Solving Equations with Fractions

Simplest Cross Multiplication of Fractions : image by Mike DeHaan

Simplest cross multiplication of fractions. Image by Mike DeHaan

From the above examples, you’ve seen how to cross multiply fractions: Take the denominator of one fraction and multiply both sides of the equation by that value.

One side of the equation is simplified because the denominator is cancelled out, and the other side increases by the same ratio, so the equation remains balanced.

The five key steps for solving equations with fractions are:

  1. If needed, change any mixed fractions to improper fractions. For example, change 2 1/3 to 7/3.
  2. Identify the simplest method of isolating the unknown x to only one side of the equation.
  3. Cross multiply both sides by the same values. For example, pick the denominator from one side to be the numerator of the new multiplier on both sides.
  4. If you need to simplify the equation one step at a time, do so by repeating step 3 as needed.
  5. If needed, change any remaining improper fractions to mixed fractions.

Remember to take a step-by-step approach as you learn how to solve the equation; with practice, you can take shortcuts to solve fractions equations.

More Help for Cross Multiplying Equations with Fractions

You might need more help or practice solving equations with fractions or with cross multiplying. If you’re taking math at school, the first suggestion is to ask your teacher to help you learn how to solve fraction problems. Also, you might find a math tutor online, or through local newspaper ads. Online tutoring in math, or online¬†math fractions worksheet drills, can be cost effective ways to improve your math skills.


Online Math Learning. Cross Multiply. Accessed February 7, 2013.

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© Copyright 2013 Mike DeHaan, All rights Reserved. Written For: Decoded Science

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