How to Add and Multiply Fractions
Look at the denominators., Find the lowest common denominator (LCD) of the two fractions., Find the new numerator of the first fraction., Find the new numerator of the second fraction., Add the numerators of the two fractions, and place the sum over...
Step-by-Step Guide
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Step 1: Look at the denominators.
The denominator is the number below the fraction bar.
In order to add fractions, the denominators of the fractions have to be the same.
If they are not the same, you will need to find the lowest common denominator.
The denominator tells you how many pieces make one whole.If the two fractions do not use the same denominator, then you are not adding same-sized pieces, and your answer will be incorrect.
For example, if you are adding the fractions 64{\displaystyle {\frac {6}{4}}} and 53{\displaystyle {\frac {5}{3}}}, you would compare the 4 and the
3.
They are different, and so you cannot add these fractions as is, because you cannot add fourths and thirds together. -
Step 2: Find the lowest common denominator (LCD) of the two fractions.
If the denominators are the same, you can skip this step.
If the denominators are not the same, you must find the LCD.
To find the LCD, you are looking for the lowest common multiple of the two denominators.
A multiple is a number that results from multiplying a number by another number.
One way to find the LCD is to make a list of multiples of the two denominators until you find one that matches.
To learn other ways to find the LCD, read Find the Least Common Denominator.
For example, if your denominators are 4 and 3, first list multiples of 4 (4, 8, 12, 16,
20...) and then list multiples of 3 (3, 6, 9, 12, 15…).
The lowest multiple the two numbers have in common is 12, so 12 is the least common denominator. , The numerator is the number above the fraction bar.
To find the new numerator, compare the original denominator to the LCD.
Determine what factor you have to multiply the original denominator by to get to the LCD.
Then, multiply the numerator by this same factor.
The numerator tells you how many pieces you have.Just remember when preparing fractions for adding that whatever you do to the denominator, you must also do to the numerator.
For example, if the original denominator was 4, and the LCD is 12, you would have to multiply the original denominator by the factor 3, since 4×3=12{\displaystyle 4\times 3=12}.
As such, you would then multiply the numerator by the factor
3.
If the original numerator was 6, then the new numerator would be 18, since 6×3=18{\displaystyle 6\times 3=18}.
So, your new fraction becomes 1812{\displaystyle {\frac {18}{12}}}. , To do this, follow the same process you did to find the new numerator for the first fraction.
For example, if the original denominator was 3, and the LCD is 12, you would have to multiply the original denominator by the factor 4, since 3×4=12{\displaystyle 3\times 4=12}.
As such, you would then multiply the numerator by the factor
4.
If the original numerator was 5, then the new numerator would be 20, since 5×4=20{\displaystyle 5\times 4=20}.
So, your new fraction becomes 2012{\displaystyle {\frac {20}{12}}}. , Now that you have found the lowest common denominator, you are adding individual pieces that are the same size, so you can proceed with adding the numerators.
You do NOT add the denominators, because the size of the pieces you are adding does not change, only how many of the pieces you have.
For example, if you are adding 1812+2012{\displaystyle {\frac {18}{12}}+{\frac {20}{12}}}, you will add the numerators 18 and
20. 18+20=38{\displaystyle 18+20=38}.
Then, you will place 38 over the LCD, which is
12.
So the sum is 3812{\displaystyle {\frac {38}{12}}}. , Usually you will be asked to reduce, or simplify a final answer if it is a fraction.
To simplify, find the highest number that will divide evenly into the numerator and denominator,then divide the numerator and denominators by that number.
For example, if your answer is 3812{\displaystyle {\frac {38}{12}}}, divide the numerator and denominator by
2. 38/2=19{\displaystyle 38/2=19} and 12/2=6{\displaystyle 12/2=6}, so the simplified fraction is 196{\displaystyle {\frac {19}{6}}}.
Since 19 cannot be divided evenly by any other number, you cannot simplify 196{\displaystyle {\frac {19}{6}}}any further. -
Step 3: Find the new numerator of the first fraction.
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Step 4: Find the new numerator of the second fraction.
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Step 5: Add the numerators of the two fractions
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Step 6: and place the sum over the LCD.
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Step 7: Simplify your answer
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Step 8: if possible.
Detailed Guide
The denominator is the number below the fraction bar.
In order to add fractions, the denominators of the fractions have to be the same.
If they are not the same, you will need to find the lowest common denominator.
The denominator tells you how many pieces make one whole.If the two fractions do not use the same denominator, then you are not adding same-sized pieces, and your answer will be incorrect.
For example, if you are adding the fractions 64{\displaystyle {\frac {6}{4}}} and 53{\displaystyle {\frac {5}{3}}}, you would compare the 4 and the
3.
They are different, and so you cannot add these fractions as is, because you cannot add fourths and thirds together.
If the denominators are the same, you can skip this step.
If the denominators are not the same, you must find the LCD.
To find the LCD, you are looking for the lowest common multiple of the two denominators.
A multiple is a number that results from multiplying a number by another number.
One way to find the LCD is to make a list of multiples of the two denominators until you find one that matches.
To learn other ways to find the LCD, read Find the Least Common Denominator.
For example, if your denominators are 4 and 3, first list multiples of 4 (4, 8, 12, 16,
20...) and then list multiples of 3 (3, 6, 9, 12, 15…).
The lowest multiple the two numbers have in common is 12, so 12 is the least common denominator. , The numerator is the number above the fraction bar.
To find the new numerator, compare the original denominator to the LCD.
Determine what factor you have to multiply the original denominator by to get to the LCD.
Then, multiply the numerator by this same factor.
The numerator tells you how many pieces you have.Just remember when preparing fractions for adding that whatever you do to the denominator, you must also do to the numerator.
For example, if the original denominator was 4, and the LCD is 12, you would have to multiply the original denominator by the factor 3, since 4×3=12{\displaystyle 4\times 3=12}.
As such, you would then multiply the numerator by the factor
3.
If the original numerator was 6, then the new numerator would be 18, since 6×3=18{\displaystyle 6\times 3=18}.
So, your new fraction becomes 1812{\displaystyle {\frac {18}{12}}}. , To do this, follow the same process you did to find the new numerator for the first fraction.
For example, if the original denominator was 3, and the LCD is 12, you would have to multiply the original denominator by the factor 4, since 3×4=12{\displaystyle 3\times 4=12}.
As such, you would then multiply the numerator by the factor
4.
If the original numerator was 5, then the new numerator would be 20, since 5×4=20{\displaystyle 5\times 4=20}.
So, your new fraction becomes 2012{\displaystyle {\frac {20}{12}}}. , Now that you have found the lowest common denominator, you are adding individual pieces that are the same size, so you can proceed with adding the numerators.
You do NOT add the denominators, because the size of the pieces you are adding does not change, only how many of the pieces you have.
For example, if you are adding 1812+2012{\displaystyle {\frac {18}{12}}+{\frac {20}{12}}}, you will add the numerators 18 and
20. 18+20=38{\displaystyle 18+20=38}.
Then, you will place 38 over the LCD, which is
12.
So the sum is 3812{\displaystyle {\frac {38}{12}}}. , Usually you will be asked to reduce, or simplify a final answer if it is a fraction.
To simplify, find the highest number that will divide evenly into the numerator and denominator,then divide the numerator and denominators by that number.
For example, if your answer is 3812{\displaystyle {\frac {38}{12}}}, divide the numerator and denominator by
2. 38/2=19{\displaystyle 38/2=19} and 12/2=6{\displaystyle 12/2=6}, so the simplified fraction is 196{\displaystyle {\frac {19}{6}}}.
Since 19 cannot be divided evenly by any other number, you cannot simplify 196{\displaystyle {\frac {19}{6}}}any further.
About the Author
Susan Webb
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