Multiple Like and Unlike Fractions Addition Calculator

comma separated input values
Sum  =  `1051/70`

Multiple Fractions Addition - work with steps

Input Data :
`2/7, 6/4, 9/5, 3/7, 11`

Number of Input = `5`

Objective :
Find the sum of multiple like and unlike fractions?

Solution :
Conevert whole number into fraction
For unlike denominators, find LCD for all denominators
LCD(`7, 4, 5, 7, 1`) = 140
Multiply LCD by each numerator & denominator of all fractions
= `(2\times140)/(7\times140)` + `(6\times140)/(4\times140)` + `(9\times140)/(5\times140)` + `(3\times140)/(7\times140)` + `(11\times140)/(1\times140)`

= `(2\times20)/(140)` + `(6\times35)/(140)` + `(9\times28)/(140)` + `(3\times20)/(140)` + `(11\times140)/(140)`

= `(40)/(140)` + `(210)/(140)` + `(252)/(140)` + `(60)/(140)` + `(1540)/(140)`

Add all numerator of the fraction
= `(40 + 210 + 252 + 60 + 1540)/140`
= `(2102)/140`
Simplify above fraction `2102/140`
Common divisor of (2102, 140) is 2
Divide both numerator & denominator by gcd value 2
`2102/140 = frac{2102\divide2}{140\divide2} = 1051/70`

`2/7 + 6/4 + 9/5 + 3/7 + 11 = 1051/70`

Multiple fractions addition calculator uses two or more proper or improper fractions and calculates their sum. It is an online algebra tool for finding the sum in the simplest form of two or more proper or improper fractions.

It is necessary to follow the next steps:

  1. Enter two or more fractions in the box. The denominators of these fractions must be nonzero.
  2. Press the "GENERATE WORK" button to make the computation;
  3. Multiple fractions addition calculator will give the sum of two or more numbers represented as fractions.
Input: Two or more fractions;
Output: A fraction in the simplest form.

Multiple Fractions Addition Rule: The sum of $n$ fractions is determined by the following formula
  • If all fractions are like, $b_1=b_2=\ldots=b_n$:
$$\frac{a_1}{b_1}+\frac{a_2}{b_1}+\ldots+\frac{a_n}{b_1}=\frac{a_1+a_2+\ldots a_n}{b_1},\quad \mbox{for}\;b_1\ne0$$
  • If denominators are different:
$$\begin{align} &\frac{a_1}{b_1}+\frac{a_2}{b_2}+\ldots+\frac{a_n}{b_n}= \frac{a_1\times \frac{LCM(b_1,b_2,\ldots,b_n)}{b_1}+a_2\times\frac{LCM(b_1,b_2,\ldots,b_n)}{b_2}+\ldots+a_n\times\frac{LCM(b_1,b_2,\ldots,b_n)}{b_n}}{LCM(b_1,b_2,\ldots,b_n)}, \\& {for}\;b_1,b_2\ldots, b_n\ne0\end{align}$$ where $LCM(b_1,b_2\ldots,b_n)$ is the least common multiple of $b_1,b_2\ldots,b_n$.

How to Find the Sum of Multiple Like & Unlike Fractions?

A sum of two numbers does not depend on their order. In other words, it satisfies the commutative property. A sum of numbers does not depend on how the numbers are grouped. This property is called the associative property. When we deal with fractions, there are two types of addition:

  • When all fractions are like fractions
When denominators of fractions are equal, then their sum will be the sum of numerators over the common denominator. If necessary, the result may be simplified. This can be expressed algebraically: $$\frac{a_1}{b_1}+\frac{a_2}{b_1}+\ldots+\frac{a_n}{b_1}=\frac{a_1+a_2+\ldots a_n}{b_1},\quad \mbox{for}\;b_1\ne0$$
  • When some of the fractions are unlike fractions
When denominators of fractions are different, to add two or more such fractions, it is necessary to follow the next steps:
  1. Find the LCM of denominators;
  2. Rewrite the fractions over the LCM;
  3. Add new numerators;
  4. The result is the sum of numerators over the LCM;
  5. Simplify the result if needed.
This method can be expressed algebraically: $$\begin{align} &\frac{a_1}{b_1}+\frac{a_2}{b_2}+\ldots+\frac{a_n}{b_n}= \frac{a_1\times \frac{LCM(b_1,b_2,\ldots,b_n)}{b_1}+a_2\times\frac{LCM(b_1,b_2,\ldots,b_n)}{b_2}+\ldots+a_n\times\frac{LCM(b_1,b_2,\ldots,b_n)}{b_n}}{LCM(b_1,b_2,\ldots,b_n)}\end{align}$$ for $b_1,b_2\ldots, b_n\ne0.$

If $LCM(b_1,b_2,\ldots,b_n)=b_1\times b_2\times\ldots\times b_n$, then the previous formula becomes
$$\begin{align} &\frac{a_1}{b_1}+\frac{a_2}{b_2}+\ldots+\frac{a_n}{b_n}= \frac{a_1\times b_2\times\ldots\times b_n+a_2\times b_1\times\ldots\times b_n+\ldots+a_n\times b_1\times b_2\times\ldots\times b_{n-1}}{b_1\times b_2\times\ldots\times b_n} \end{align}$$ for $b_1,b_2\ldots, b_n\ne0.$
For example, let us find the sum for $\frac 27, \frac 64, \frac 85$ and $\frac 87$. Since $LCM(7,4,5,7)=140$, then \begin{align} \frac 27+\frac 64+ \frac 85+\frac 87 & = \frac{2\times 20}{140}+\frac{6\times 35}{140}+\frac{8\times 28}{140}+\frac{8\times 20}{140}\\ & = \frac{40}{140}+\frac{210}{140}+\frac{224}{140}+\frac{160}{140}\\ & = \frac{634}{140} \end{align} To write the sum in simplest form, find the GCF of the numerator and denominator of the sum. Because $GCF(634,140)=2$, the final result is $$\frac{634\div2}{140\div2}=\frac{317}{70}$$ Note that some of fraction may be a negative fraction. The similar consideration can be applied in addition of two or more mixed numbers or two or more algebraic fractions. To add two or more mixed numbers, convert mixed numbers to corresponding improper fractions and apply the procedure above.

The multiple fractions addition work with steps shows the complete step-by-step calculation for finding the sum of four fractions $\frac 27, \frac 64, \frac 85$ and $\frac 87$ using the multiple fractions addition rule. For any other fractions, just supply two or more proper or improper fractions and click on the "GENERATE WORK" button. The grade school students may use this adding multiple like and unlike fractions calculator to generate the work, verify the results of adding two or more numbers derived by hand or do their homework problems efficiently.

Real World Problems Using Fractions Addition

The problem of adding two or more fractions can be found in almost all spheres of life and science.

Multiple Fractions Addition Practice Problems

Practice Problem 1 : Find the number for $\frac 59$ greater than the sum of the numbers $1\frac 3 7$ and $\frac 4{11}$.

Practice Problem 2 : Find the sum of five fractions with unlike or different denominators $\frac35, \frac 27, -\frac 18, \frac 4{11}$ and $\frac 23$.

Practice Problem 3 : Ann needs to read thirty books in the following five weeks. At the first week, he read one-quarter of the books. At the second week, he read one-third of the books. At the third week he read three books. At the fourth week he read one-sixth books. What fraction of the books Brad read?

The multiple fractions addition calculator, formula, example calculation (work with steps), real world problems and practice problems would be very useful for grade school students (K-12 education) to understand the addition of two or more numbers represented as fractions. Using this concept they can be able to solve complex algebraic problems and equations.