Simplifying & Solving Equations With Decimals & Mixed Numbers

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will review decimals and mixed numbers, and it will discuss how to convert between the two. We will then use examples to learn how to solve equations involving these types of numbers.

Decimals and Mixed Numbers

Speedy the snail is going to be competing in an upcoming race! He is out training, and he is able to cover 6 and 2/5 feet in an hour. In mathematics, we would say that Speedy is traveling at 6 2/5 feet per hour, and we call 6 2/5 a mixed number. A mixed number is a number that contains a whole number, called the whole part, along with a fraction, called the fraction part.

In general, the mixed number a b/c is the same as a + b/c. For instance, we can write our mixed number, 6 2/5 as 6 + 2/5. Now, if we were to pull out a calculator and plug this in, it would spit out that 6 + 2/5 = 6.4, so we have that 6 2/5 = 6.4.

When we rewrite 6 2/5 as 6.4, we are actually converting it to a different type of number, and that type of number is called a decimal number. A decimal number is simply a number with a decimal point in it. We call the number to the left of the decimal point the whole part, and we call the rest of the number (to the right of the decimal point) the decimal part.

The relationship between mixed numbers and decimal numbers is that both types of numbers are equivalent to their whole parts added to the fraction part or decimal part. This is why we can convert a mixed number to its equivalent decimal number by adding the whole part to the fractional part. This is the same as rewriting the fractional part as a decimal.

Similarly, we can convert a decimal number to a mixed number by just writing the decimal part as a fraction. For instance, consider our number again. To convert 6.4 back to a mixed number, we just write the decimal part as a fraction. To do this, make the decimal part the numerator (top number) and make the place value that the decimal part ends on the denominator (bottom number).

• 6.4 = 6 4/10. This can be simplified to 6 2/5.

Pretty neat, wouldn't you say?

Equations Involving Decimals and Mixed Numbers

So far so good! We are now familiar with both mixed numbers and decimal numbers and how to convert between the two. Now, let's consider equations involving these two types of numbers. Consider Speedy again. Let's say that during his training session, he wants to go 10 feet total. Based on this, we can set up the following equation:

• 6 2/5 t = 10

In this equation, t is the duration of the training session. We could also write this using the decimal number as follows:

• 6.4t = 10

We can use either equation to figure out how long Speedy's training session will be by solving for t. But how do we solve equations involving mixed numbers and decimal numbers? Well, solving these equations is the exact same process as solving equations without these types of numbers! It's just a matter of isolating the variable by using the following techniques:

1. Adding or subtracting the same term from both sides of the equation
2. Multiplying or dividing the same term, excluding zero, from both sides of the equation
3. Simplifying both sides of the equation
4. Interchanging sides of the equation

Also, we can convert numbers within an equation if we prefer working with one type of number (decimals or mixed) over the other.

For example, to figure out the duration of Speedy's training session, we can solve either of the equations we came up with. In this case, the equation with the decimal number is easier to deal with. Notice that, if we try to solve the equation with the mixed number, we would divide both sides by 6 2/5 to isolate t.

It looks like we would end up converting 6 2/5 to decimal form anyway in order to solve for t, so it's easier to just solve the equation with the decimal number in the first place.

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