Cross Multiply to Solve Equations with Fractions

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Equations with fractions: Image by Decoded Science

Equations with fractions: Image by Decoded Science

Fractions aren’t as hard as you think. If you want to learn how to solve equations with fractions by cross-multiplying, such as solving “x/7 = 2/3″ and “7/3 = 2/x” for x, you’re in the right place.

Solving Equations: The Golden Rule

The goal in solving an equation for x is to finish with a statement in which x equals something that does not include an x.

The golden rule for solving equations is to apply the same operation to both sides of the equation. This may involve adding, subtracting or multiplying both sides by the same amount.

For example, if x/37 = 5, you can multiply both sides by 37. Since x divided by 37 times 37 just equals x, then x = 5*37, which  = 185.

Solving Equations with Sample Fractions

Here is a sample equation, and a step-by-step process for solving it:

Sample equation: x/7 = 2/3

Step One: Multiply both sides by 7.

x/7 * 7 = 2/3 * 7

Multiply 2 and 7 to get 14.

x /7 * 7 = 14/3

Cancel out /7 * 7 to leave x, and convert to solve for x.

x = 14/3 = 4 2/3

You multiply both sides by 7 to eliminate the denominator on the left side of the equation. The 7s cancel out on the left. Then you solve the equation for x. As a final step, you convert the improper fraction 14/3 to the mixed fraction 4 and 2/3.

Note that you only cross multiplied the 7 in the denominator. If you had started with 23*x/7 = 2/3, you would have cross multiplied both sides by 7/23 instead of just 7. This would leave only x on the left side.

Mixed Fractions: Another Example

Specific Cross Multiplication Example : image by Mike DeHaan

Specific Cross Multiplication Example : Image by Mike DeHaan

In a second example, “7/3 = 2/x”, the mixed fraction is already converted from 2 1/3 to 7/3. It’s much easier to solve fraction equations that have improper fractions than equations with mixed fractions.

Notice that this example has x in the denominator. You should explicitly state the assumption that x is not zero.

Sample equation: 7/3 = 2/x

Multiply both sides by x so that x will be in the numerator.

7/3 * x = 2/x * x

Cancel out the /x * x.

7*x/3 = 2

Multiply both sides by 3/7 to cancel out the 7/3.

x * (7/3) * (3/7) = 2 * (3/7)

Cancel out 7/3 with 3/7 and multiply to solve for x.

x = 2*3/7 = 6/7

If you cross multiply all in one step, it would be:

7/3 = 2/x
(7/3) * (3*x/7) = (2/x) * (3*x/7)
x = 6/7

It’s easier to recognize cross multiplication when the numerators and denominators travel across the equals sign to multiply.

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.

Reference

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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