to view answer.To see more practice with fractions, follow the links:
Simplifying Fractions
Multiplying and Dividing Fractions
The most important thing to remember about fractions is that they represent a part of a whole. When adding fractions, you are adding up parts of a whole.
Example 1:
Add the following fractions:
$${\Large\frac{3}{14}}+{\Large\frac{7}{4}}$$
In this example, we can imagine a whole pie sliced into 14 equal sized pieces (into fourtheenth‘s). The fraction, $${\Large\frac{3}{14}}$$, indicates 3 of the 14 pieces in the whole pie. We also have some other pies each sliced into 4 pieces (into fourths). The fraction, $${\Large\frac{7}{4}}$$, indicates 7 such pieces from pies cut into fourths.
When adding $${\Large\frac{3}{14}}$$ and $${\Large\frac{7}{4}}$$, we are adding different sizes of pie pieces. Thus, we cannot just add 3 and 7 to get 10 pieces, because some of the pieces are fourths and some of the pieces are fourteenths. We need to have the same sized pieces to add them. This is why we look for a common denominator when adding fractions. The common denominator is the number of pieces that we can cut the fourths and fourteenths into so that we are adding pieces of the same size.
To find the least common denominator for the two fractions, we find the smallest number that is a multiple of the two denominators (in other words, the least common multiple). So, in this case, 4 cannot be multiplied by any number to get 14, but $$4 \cdot 7 = 28$$ which is also $$14 \cdot 2$$, so the least common multiple of 4 and 14 is 28.
Thus, we slice each of our fourteenths in half to get twenty-eighths of a pie. In this case, we have 3 fourteenths which are each cut in half to get 6 twenty-eighths of a pie. Numerically, this is the same as multiplying the numerator and the denominator each by 2 to get a denominator of 28.
$${\Large\frac{3 \cdot 2}{14 \cdot 2}} ={\Large \frac{6}{28}}$$
Similarly, if we cut each of the 7 fourths into 7 pieces, then we get a total of $$7 \cdot 7 = 49$$ twenty-eighths of a pie.
This is the same as multiplying the numerator and the denominator of $${\Large\frac{7}{4}}$$ by 7 to get a denominator of 28.
$${\Large\frac{7 \cdot 7}{4 \cdot 7}} ={\Large \frac{49}{28}}$$
Now the fractions have common denominators, so we are adding pieces of pie that are the same size. So, we can just add the numerators to get the total number of twenty-eighths of pie that we have:
$${\Large\frac{3}{14}}+{\Large\frac{7}{4}} ={\Large \frac{6}{28}}+{\Large\frac{49}{28}} = {\Large\frac{6+49}{28}}={\Large\frac{55}{28}}$$
Since nothing divides both 55 and 28, this fraction is simplified and $${\Large\frac{55}{28}}$$ is the final answer.
See Simplifying Fractions for more help on this.
Example 2:
Subtract the following fractions and simplify your answer:
$${\Large\frac{5}{6}}-{\Large\frac{13}{15}}$$
1. So, first, we need to have common denominators in order to add or subtract the fractions. Otherwise, we are trying to add different sizes of pieces of the whole.
To find the common denominator, we find a number that is a multiple of both 6 and 15. Preferably, we would like to find the smallest multiple of both, or the “least common multiple”. In this case, 15 is not a multiple of 6. A good strategy, then, is to look for the lowest multiple of 15 that is a multiple of 6. We have: $$15 \cdot 2=30$$. The number 30 is also a multiple of 6, since $$6 \cdot 5 = 30$$. Thus, 30 is the least common multiple.
2. Now, we need to make the denominator of both fractions 30. Do this by multiplying the top and bottom of each fraction by the same number – the number that will make the denominator 30.
$${\Large\frac{5 \cdot 5}{6 \cdot 5}}={\Large\frac{25}{30}}$$ and:
$${\Large\frac{13 \cdot 2}{15 \cdot 2}} ={\Large \frac{26}{30}}$$
3. Now the problem has changed to include fractions with common denominators, but is equivalent to our original problem:
$${\Large\frac{5}{6}}-{\Large\frac{13}{15}}={\Large\frac{25}{30}}-{\Large\frac{26}{30}}$$
Then, with a common denominator, we perform the subtraction on the numerators:
$${\Large\frac{25}{30}}-{\Large\frac{26}{30}} = {\Large\frac{1}{30}}$$
4. Then, since there is no number that divides both 1 and 30 (except 1), we cannot simplify the fraction any further, and so the final answer is $$-{\Large\frac{1}{30}}$$.
Some practice problems to check your skills:
1. Find the sum and simplify your answer: $$
{\Large{1 \over 2} + {7 \over {10}}}
Think about it for a moment and then access this link to view answer.
2. Find the sum and simplify your answer: $$
{\Large{{24} \over {35}} + {3 \over 5} + {1 \over 7}}
$$
Think about it for a moment and then access this link to view answer.
3. Find the sum and simplify your answer: $$
{\Large{{2300} \over {1200}} + {{300} \over {800}}}
$$
Think about it for a moment and then access this link to view answer.
4. Find the difference and simplify your answer: $$
{\Large{3 \over {18}} - {2 \over {27}}}
$$
Think about it for a moment and then access this link to view answer.
5. Find the total and simplify your answer: $$
{\Large{3 \over 8} + {{11} \over {12}} - {5 \over {48}}}
$$
Think about it for a moment and then access this link to view answer.