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How to Solve Multi-Step Equations with Fractions & Decimals

How to Solve Multi-Step Equations with Fractions & Decimals
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  • 0:03 Take the Sting Out of…
  • 0:39 Equations with Fractions
  • 1:30 Eliminating the Denominator
  • 2:59 Decimals
  • 5:18 Mixed Numbers
  • 6:03 Lesson Summary
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Lesson Transcript
Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Fractions and decimals can definitely be annoying when you're trying to solve multi-step equations. In this lesson, we'll figure out how to get them out of the way, allowing us to solve the equations more easily.

Take the Sting out of Fractions

In this lesson, we are going to learn how to deal with multi-step equations with fractions and decimals. These are problems where we not only have to do more than one step to get to an answer, but we also have to deal with decimals and/or fractions that show up along the way! It can get a little annoying - kind of like a bee buzzing around you - but at the end of this lesson, you'll have a sure-fire way to get to the right answer with the least amount of personal agony - no sting!

Equations with Fractions

Let's look at an example problem that uses fractions, then we'll discuss the differences when decimals or mixed numbers show up.

First, a couple of reminders:

  1. The number above the division line is called the numerator, and the number below the division line is called the denominator.
  2. When multiplying fractions, you just multiply the numerators together, multiply the denominators together, and then simplify the result.
  3. Remember, we can add, subtract, multiply, or divide anything we want to on one side of an equation so long as we do the same thing on the other side.

Okay, so let's look at our first problem:


graphic Step 1

A problem like this can be a little scary, but the steps are pretty easy to take, and it gets a lot simpler after the first step. When we have fractions in a problem like this, often the easiest approach is to get rid of the fractions by canceling out all of the denominators. Here's how you do that:

Eliminating the Denominator

1. Find a common denominator. This is a number that will divide evenly by each of the denominators in the problem. In this equation, the only denominators that appear are 2 and 4, and 4 divides evenly by 2, so we'll use 4.

2. Then, multiply both sides of the equation by the common denominator. Once you've done that, you'll be able to cancel out the denominators. In this case, when we multiply everything times 4, this gives us the equation in the middle. We'll simplify in the next step.


image Step 2

3. Let's simplify the fractions by dividing all of the numerators by their denominators. Now the problem is much simpler.


image Step 3

Combining Like Terms

All right, now that we got rid of those ugly fractions, we can take the next step: combining like terms. We need to pull all of the x terms to the left side, and all of the number terms to the right side. Subtracting 2x and 2 from both sides of the equation allows us to isolate x.


image Step 4

So x = 1! See, that wasn't so bad. You can always get rid of the denominators using the same approach, and this makes the problem much easier.

Decimals

So what if they're decimals instead of fractions? Well, remember that a decimal number is merely another way to write a fraction. The decimal expression 0.1 is said as 'one tenth' and means 1/10, 0.01 is said as 'one hundredth' and means 1/100, and 0.001 is said as 'one thousandth' and means 1/1000. Each position farther to the right of the decimal point means you're dividing by a larger multiple of 10. Another way to think about it is that you are using a power of 10 that has a 0 for each digit that is to the right of the decimal.

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