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