Unit 2; Part 2: Adding and Subtracting Fractions

First, lets talk about reducing fractions to the lowest terms. Also known as simplifying, we can use Prime Factorization to reduce fractions.

How to Reduce Fractions Using Prime Factorization

The beauty of using the prime factorization method is that you can be sure that the fraction’s reduction possibilities are exhausted — that is, you can be certain that you haven’t missed any factors that the numerator and denominator may have in common.

I will reduce the following fraction using prime factorization:

$\frac{16}{64}$ $\frac{16}{64}$

As we saw in the previous section we need to break each number into its prime numbers:

16 = 2x2x2x2 and 64 = 2x2x2x2x2x2

Write: $\frac{2\cdot2\cdot2\cdot2}{2\cdot2\cdot2\cdot2\cdot2\cdot2}$ $\frac{2\cdot2\cdot2\cdot2}{2\cdot2\cdot2\cdot2\cdot2\cdot2}$ and cancel out: you're left with: $\frac{1}{4}$ $\frac{1}{4}$

So: $\frac{16}{64}\:=\frac{1}{4}$ $\frac{16}{64}\:=\frac{1}{4}$

See Khan Academy on Reducing Fractions to Lowest Terms Links to an external site.

2.2.A Adding Fractions

Like fractions are fractions with the same denominator. You can add and subtract like fractions easily - simply add or subtract the numerators and write the sum over the common denominator.

$\frac{4}{9}\:+\:\frac{3}{9}\:=\:\frac{7}{9}$ $\frac{4}{9}\:+\:\frac{3}{9}\:=\:\frac{7}{9}$

The tricky part comes when you add or subtract fractions that have different denominators. To do this, you need to know how to find the least common denominator.

Before you can add or subtract fractions with different denominators, you must first find equivalent fractions with the same denominator, like this:

Find the smallest multiple (LCM) of both numbers.
Rewrite the fractions as equivalent fractions with the LCM as the denominator.

When working with fractions, the LCM is called the least common denominator (LCD).

Lets just practice with adding fractions now.

Example 1: Add the following fractions

$\frac{2}{3}\:+\:\frac{1}{4}$ $\frac{2}{3}\:+\:\frac{1}{4}$

Find the lowest common denominator: 3 is a prime number, 4 = 2 x 2, so 3 x 2 x 2 = 12
The lowest common multiple of 3 and 4 is their product, 12.

We will convert each fraction to an equivalent fraction with denominator 12.

To make the denominator 12 in $\frac{2}{3}$ $\frac{2}{3}$ we need to multiply 3 by 4, what ever we do to the denominator we MUST do to the numerator.

$\frac{2\cdot4}{3\cdot4}\:=\:\frac{8}{12}$ $\frac{2\cdot4}{3\cdot4}\:=\:\frac{8}{12}$

To make the denominator 12 in $\frac{1}{4}$ $\frac{1}{4}$ we need to multiply 4 by 3, what ever we do to the denominator we MUST do to the numerator.

$\frac{1\cdot3}{4\cdot3}\:=\:\frac{3}{12}$ $\frac{1\cdot3}{4\cdot3}\:=\:\frac{3}{12}$

Our new addition problem, with common denominators will look like this:

$\frac{8}{12}\:+\:\frac{3}{12}\:=\:\frac{11}{12}$ $\frac{8}{12}\:+\:\frac{3}{12}\:=\:\frac{11}{12}$

You only need to add across the top (numerators), and keep the bottom the same (denominator). If necessary, you can reduce to lowest terms.

Sometime you will end up with improper fractions. You will need to convert improper fractions to mixed numbers.

As we discussed earlier, an improper fractions is one that has a numerator higher than it's denominator. A mixed number is a whole number and a fraction (part of a number).

improper fraction: $\frac{7}{3}$ $\frac{7}{3}$ mixed number: $2\frac{1}{3}$ $2\frac{1}{3}$

All improper fractions should be turned into mixed numbers. To turn improper fractions into mixed numbers you will need to do some division. Lets convert the improper fraction $\frac{7}{3}$ $\frac{7}{3}$ into a mixed number:

$convertfrac2.gif$

When you are adding, subtracting, multiplying or dividing fractions with mixed numbers it is ALWAYS a good idea to convert mixed numbers into improper fractions before conducting operations, this way you will always get it right! Especially with subtraction, this also prevents you from forgetting about the whole numbers.

This may seem like unnecessary extra steps, but it's always a good idea to do it the same all the time to ensure the right answer.

Now, Lets convert a mixed number into an improper fraction, lets convert $4\frac{1}{2}$ $4\frac{1}{2}$ into an improper fraction:

$convertfrac1.gif$

Now that you understand how to deal with mixed numbers, lets practice.

See Khan Academy for adding and subtracting fractions Links to an external site.

Example 2: Add the following mixed fractions

$3\frac{1}{3}\:+\:2\frac{1}{4}$ $3\frac{1}{3}\:+\:2\frac{1}{4}$

To convert the mixed numbers into an improper fractions I need to start with the $3\frac{1}{3}$ $3\frac{1}{3}$ , $3\times3+1=10$ $3\times3+1=10$ ; I keep the same denominator and my improper fraction is: $\frac{10}{3}$ $\frac{10}{3}$ . Now, for $2\frac{1}{4}$ $2\frac{1}{4}$ , $4\times2+1=9$ $4\times2+1=9$ ; I keep the same denominator and my improper fractions is: $\frac{9}{4}$ $\frac{9}{4}$ . Now you can find common denominators.

To find common denominators, I use prime factoring: 3 is a prime number; 4 = 2 x 2; so, 3 x 2 x 2 = 12; my common denominator will be 12.

$\frac{10\cdot4}{3\cdot4}\:=\frac{40}{12}$ $\frac{10\cdot4}{3\cdot4}\:=\frac{40}{12}$ and $\frac{9\cdot3}{4\cdot3}=\frac{27}{12}$ $\frac{9\cdot3}{4\cdot3}=\frac{27}{12}$ . Now that we have changed our mixed numbers into improper fractions and found our common denominators, we can now add straight across the top.

$\frac{40}{12}+\frac{27}{12}=\frac{67}{12}$ $\frac{40}{12}+\frac{27}{12}=\frac{67}{12}$

But, still not done. I need to now convert back to a mixed number and reduce to lowest terms.

To change $\frac{67}{12}$ $\frac{67}{12}$ to a mixed number, we divide: $67\div12\:=\:5\:r7$ $67\div12\:=\:5\:r7$ , so, 5 is the whole number, the remainder(7) is the numerator and the denominator is the same (12).

$5\frac{7}{12}$ $5\frac{7}{12}$ , we cannot reduce, so the answer is $5\frac{7}{12}$ $5\frac{7}{12}$ .

Lets see one more example without all the explanations.

Example 3: Add the following fractions and/or mixed numbers

$5\frac{1}{7}+\frac{5}{9}$ $5\frac{1}{7}+\frac{5}{9}$

7 and 9 are prime numbers, 7 x 9 = 63

$5\frac{1\cdot9}{7\cdot9}=5\frac{9}{63}$ $5\frac{1\cdot9}{7\cdot9}=5\frac{9}{63}$ $\frac{5\cdot7}{9\cdot7}=\frac{35}{63}$ $\frac{5\cdot7}{9\cdot7}=\frac{35}{63}$

$5\frac{9}{63}+\frac{35}{63}=5\frac{44}{63}$ $5\frac{9}{63}+\frac{35}{63}=5\frac{44}{63}$

Can this be reduced? No, it is the smallest form possible.

Notice I did not turn this number into an improper fraction, if you are ADDING, and there is only ONE whole number, it is not necessary to conduct this step, but, if you did, it would still come out he same.

Your turn.

Practice 2.2.A

Add the following fractions and mixed numbers. Convert all improper fractions to mixed numbers and reduce to lowest terms.

1. $2\frac{3}{4}+2\frac{8}{10}$ $2\frac{3}{4}+2\frac{8}{10}$

2. $\frac{1}{12}+\frac{6}{19}$ $\frac{1}{12}+\frac{6}{19}$

3. $10\frac{4}{10}+1\frac{2}{5}$ $10\frac{4}{10}+1\frac{2}{5}$

4. $1\frac{1}{4}+\frac{16}{20}$ $1\frac{1}{4}+\frac{16}{20}$

5. $\frac{10}{11}+\frac{12}{15}$ $\frac{10}{11}+\frac{12}{15}$

2.2.B Subtracting Fractions

Subtracting fractions is the same as we have learned about adding, except, we now subtract and not add. When subtracting mixed numbers it is very important to always convert mixed numbers into improper fractions to get the correct answer.

Lets show you a couple examples, then you practice a few.

Example 1: Subtract the following fractions and/or mixed numbers.

$6\frac{1}{4}-3\frac{6}{5}$ $6\frac{1}{4}-3\frac{6}{5}$

First, convert to improper fractions: $4\times6+1=25\:\:\:\frac{25}{4}\:\:\:and\:\:\:\:5\times3+6=21\:\:\:\frac{21}{5}$ $4\times6+1=25\:\:\:\frac{25}{4}\:\:\:and\:\:\:\:5\times3+6=21\:\:\:\frac{21}{5}$

Find common denominators: 4 = 2x2 and 5 is a prime number, so the common denominator is: 2x2x5=20

$\frac{25\cdot5}{4\cdot5}=\frac{125}{20}$ $\frac{25\cdot5}{4\cdot5}=\frac{125}{20}$ and $\frac{21\cdot4}{5\cdot4}=\frac{84}{20}$ $\frac{21\cdot4}{5\cdot4}=\frac{84}{20}$ , so: $\frac{125}{20}-\frac{84}{20}=\frac{41}{20}$ $\frac{125}{20}-\frac{84}{20}=\frac{41}{20}$

Now, convert the improper fraction to a mixed number:

$41\div20=2\:r1$ $41\div20=2\:r1$

So, 2 is the whole number, the remainder(1) is the numerator and the denominator is the same (20).

$2\frac{1}{20}$ $2\frac{1}{20}$ (cannot be reduced)

Example 2: Subtract the following fractions and/or mixed numbers

$1\frac{3}{4}-\frac{10}{19}$ $1\frac{3}{4}-\frac{10}{19}$

Convert mixed numbers into improper fractions.

$4\times1+3=7\:\:\:\:\frac{7}{4}$ $4\times1+3=7\:\:\:\:\frac{7}{4}$

Find common denominators.

4 = 2 x 2 and 19 is a prime number, so 2 x 2 x 19 = 76

$\frac{7\cdot19}{4\cdot19}=\frac{133}{76}\:\:\:\:\:\:and\:\:\:\:\:\:\:\frac{10\cdot1}{19\cdot1}=\frac{10}{19}$ $\frac{7\cdot19}{4\cdot19}=\frac{133}{76}\:\:\:\:\:\:and\:\:\:\:\:\:\:\frac{10\cdot1}{19\cdot1}=\frac{10}{19}$ so, $\frac{133}{76}-\frac{10}{76}=\frac{123}{76}$ $\frac{133}{76}-\frac{10}{76}=\frac{123}{76}$

Now, convert to improper fraction.

$123\div76=1\:r47$ $123\div76=1\:r47$

So, 1 is the whole number, the remainder(47) is the numerator and the denominator is the same (76).

$1\frac{47}{76}$ $1\frac{47}{76}$ (cannot be reduced)

Example 3: Subtract the following fractions and/or mixed numbers

$14\:-\frac{15}{19}$ $14\:-\frac{15}{19}$

Since we have a whole number and we need to subtract a fraction, we need to turn our whole number into a fraction. This is done simply by putting the whole number over 1.

$\frac{14}{1}-\frac{15}{19}$ $\frac{14}{1}-\frac{15}{19}$

Find common denominators: 1 and 19 are a prime numbers, so the common denominator is: 1 x 19 = 19.

$\frac{14\cdot19}{1\cdot19}=\frac{266}{19}\:\:\:\:\:and\:\:\:\:\:\frac{15\cdot1}{19\cdot1}=\frac{15}{19}$ $\frac{14\cdot19}{1\cdot19}=\frac{266}{19}\:\:\:\:\:and\:\:\:\:\:\frac{15\cdot1}{19\cdot1}=\frac{15}{19}$ so, $\frac{266}{19}-\frac{15}{19}=\frac{251}{19}$ $\frac{266}{19}-\frac{15}{19}=\frac{251}{19}$