GED Mathematical Reasoning: Mixed Numbers

When adding or subtracting, you may keep a mixed number ‘as is.’ OR – you MAY choose to change any mixed number to an improper fraction and proceed as usual, by getting a common denominator and performing the operation.

To perform the addition keeping the mixed numbers as they are,

  • First add the whole number parts together. Keep this sum handy – we’ll need to bring it back around for our final answer.
  • Then, add the fraction parts together.
  • Be sure that the final answer is composed of a whole number and a proper fraction.

To perform the addition by turning the mixed numbers into improper fractions,

  • First change any mixed numbers to improper fractions.
  • Then, add the fractions as usual.
  • Finally, change the result back into a mixed number.

 

Example 1

Perform the indicated operation: 2\frac{1}{3} \div \frac{1}{2}

 

To divide with fractions, we multiply by the reciprocal, or – less formally – “keep change flip.”

But first, let’s change 2\frac{1}{3} into an improper fraction.

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3 times 2 plus 1 equals 7, which will be the numerator of our improper fraction, and we’ll keep the same denominator. So 2\frac{1}{3} as an improper fraction is \frac{7}{3}

To divide \frac{7}{3} \div \frac{1}{2}we need to multiply by the reciprocal. We’ll keep the first fraction the same. Then, change the division sign to multiplication. And then flip the second fraction to read: \frac{2}{1}. Our new calculation, then, will be: \frac{7}{3} \times \frac{2}{1}

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Now that the operation is multiplication, we can consider cross canceling. In this case – there are no factors common to either diagonal so we’ll proceed my multiplying the numerators together and then multiplying the denominators together.

7 times 2 is 14, so the numerator of our answer is 14.

3 times 1 is 3, so the denominator of our answer is 3.

Therefore, the quotient of 2\frac{1}{3} \div \frac{1}{2} is \frac{14}{3}, Let’s change the answer into a mixed number by dividing 14 by 3. 3 goes into 14 four times with a remainder of 2. So our result as a mixed number would be 4\frac{2}{3}.

\frac{7}{3} \times \frac{2}{1} = \frac{7 \times 2}{3 \times 1} = \frac{14}{3} = 4\frac{2}{3}

 

Example 2

Perform the indicated operation: 3\frac{2}{3} + 1\frac{3}{4}.

 

To perform the addition keeping the mixed numbers as they are, first add the whole number parts together.

3  + 1 = 4

Keep this sum handy – we’ll need to bring it back around for our final answer. Then, add the fraction parts together.

\frac{2}{3} + \frac{3}{4}

To add we’ll need a common denominator. The Least Common Multiple of 3 and 4 is 12, therefore our Least Common Denominator will be 12. To rewrite \frac{2}{3} with a denominator of 12, create a new fraction with a 12 in the denominator.

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To go from a denominator of 3 to a denominator of 12, we multiply by 4 so we must multiply the numerator by 4, as well. 2 times 4 is 8, which will be the new numerator.

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To rewrite \frac{3}{4} with a denominator of 12, create another new fraction with a 12 in the denominator.

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To go from a denominator of 4 to a denominator of 12, we multiply by 3 so we must multiply the numerator by 3, as well. 3 times 3 is 9, which will be the new numerator.

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Now we’re ready to add the fraction parts.

To add \frac{2}{3} + \frac{3}{4} we’ll add their equivalents \frac{8}{12} + \frac{9}{12}. 8 plus 9 is 17 and keeping the common denominator of 12 yields a sum of \frac{17}{12}.

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Let’s put everything together and see where we’re at. The sum of the whole number parts is 4. The sum of the fraction parts is \frac{17}{12}. It might be tempting to state our answer as 4\frac{17}{12} and leave it at that. HOWEVER…by definition, a mixed number is composed of a whole number and a proper fraction. As it stands, our answer is composed of a whole number and an improper fraction.

Here’s how we fix this. First, change the improper fraction part to a mixed number.

\frac{17}{12} as a mixed number is 1\frac{5}{12}.

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Now, simply add that 1 to the whole number part of 4, giving us a final answer – in the correct “mixed number” format – of 5\frac{5}{12}.

So to recap, the general strategy for adding with mixed numbers is to add the whole number parts, then add the fraction parts. And be sure that the final answer is composed of a whole number and a proper fraction.

For fun, let’s work the problem again and this time we’ll change the mixed numbers to improper fractions before adding.

When we change the mixed numbers to improper fractions, the original problem of 3\frac{2}{3} + 1\frac{3}{4} becomes \frac{11}{3} + \frac{7}{4}.

To add \frac{11}{3} + \frac{7}{4} we’ll need a common denominator.

The Least Common Multiple of 3 and 4 is 12, therefore our Least Common Denominator will be 12.

To go from a denominator of 3 to a denominator of 12, we multiply by 4 so we must multiply the numerator by 4, as well. 11 times 4 is 44, which will be the new numerator.

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To rewrite \frac{7}{4} with a denominator of 12, create another new fraction with a 12 in the denominator.

To go from a denominator of 4 to a denominator of 12, we multiply by 3 so we must multiply the numerator by 3, as well. 3 times 7 is 21, which will be the new numerator.

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Now we’re ready to add. 44 plus 21 is 65 and keeping the common denominator of 12 yields a sum of \frac{65}{12}.

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Our final task is to change this improper fraction back into a mixed number. 12 goes into 65 five times with a remainder of 5, therefore our result – as we expected – is 5\frac{5}{12}.

So adding mixed numbers by first turning them into improper fractions may seem like a more straightforward process – you simply add the improper fractions as usual and then change the result back into a mixed number.

 

Example 3

Perform the indicated operation: 5\frac{1}{8} - 2\frac{1}{4}.

To perform subtraction with mixed numbers, we’ll start off by following a process similar to adding mixed numbers. First, subtract the whole number parts.

5 – 2 = 3

Then, subtract the fraction parts.

\frac{1}{8} - \frac{1}{4}

To subtract, we’ll need a common denominator. The Least Common Multiple of 8 and 4 is 8, therefore our Least Common Denominator will be 8.

Since the first fraction \frac{1}{8} already has the common denominator, we’ll leave that fraction alone.

As for the second fraction, the fraction equivalent to \frac{1}{4} with a denominator of 8 is \frac{2}{8}.

To subtract \frac{1}{8} - \frac{1}{4} we’ll subtract their equivalents \frac{1}{8} - \frac{2}{8}. But notice that we have a problem. We cannot subtract 2 from 1. This is a common issue when subtracting mixed numbers. To alleviate the problem, we perform a process called “regrouping.” In doing so, we “beef up” the fraction part of the first mixed number so that we are able to perform the subtraction.

To regroup the first fraction, we BORROW a 1 from the whole number of 5 (making it a 4) and GIVE the 1 to the fraction part. This makes 5\frac{1}{8} turn into 4 and 1\frac{1}{8}. Next, turn the mixed number 1\frac{1}{8} into the improper fraction of \frac{9}{8}. So the first mixed number is transformed from 5\frac{1}{8} to 4\frac{9}{8}.

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Now, we may subtract as we first set out to do.

We’ll first subtract the whole number parts.

4 – 2 = 2

Then, subtract the fraction parts.

\frac{9}{8} - \frac{1}{4}

To subtract, we’ll need a common denominator. The Least Common Multiple of 8 and 4 is 8, therefore our Least Common Denominator will be 8.

We’ve already determined that the fraction equivalent to \frac{1}{4}  with a denominator of 8 is \frac{2}{8}.

9 minus 2 is 7 and keeping the common denominator of 8 yields a fraction part of \frac{7}{8}. Combining the whole number part of 2 with the fraction part of  gives us a final answer of 2\frac{7}{8}.

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Now let’s see the process if we change the mixed numbers to improper fractions right off the bat.

When we change the mixed numbers to improper fractions, the original problem of 5\frac{1}{8} - 2\frac{1}{4} becomes \frac{41}{8} - \frac{9}{4}.

The Least Common Multiple of 8 and 4 is 8, therefore our Least Common Denominator will be 8.

We will leave the first fraction \frac{41}{8} unchanged since it already has a denominator of 8.

To rewrite \frac{9}{4} with a denominator of 8, we’ll multiply the numerator and denominator by 2 to get \frac{18}{8}.

Now we’re ready to subtract. 41 minus 18 is 23 and keeping the common denominator of 8 yields a difference of \frac{23}{8}.

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Finally, we’ll change the improper fraction to a mixed number. 8 goes into 23 two times with a remainder of 7. Therefore, the result as a mixed number is 2\frac{7}{8}.

As you can see, we may subtract keeping the mixed numbers ‘as is’ OR change to improper fractions before subtracting. Either way, we arrive at the same answer. so it’s up to you with method you prefer. The nice thing about changing the mixed numbers to improper fractions first, is that we didn’t have to complete the ‘regrouping’ process.

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