Mixed Fraction A
Mixed Fraction B
A + B  =  453/35
GENERATE WORK

## Mixed Numbers Addition - work with steps

Input Data :
Mixed Fraction A = 5 8/7
Mixed Fraction B = 6 4/5

Objective :
Find the sum of two given mixed numbers?

Solution :
5 8/7 + 5 8/7 = ?
Convert Mixed fraction into fration
5 8/7 = ((5\times7)+8)/7 = (35 + 8)/7  = 43/7

6 4/5 = ((6\times5)+4)/5 = (30 + 4)/5  = 34/5
denominator of the two fraction is different. Therefore, find lcm for two denominators (7, 5) = 35
Multiply lcm with both numerator & denominator
43/7 + 34/5 = (43\times35)/(7\times35) + (34\times35)/(5\times35)

= (43\times5)/(35) + (34\times7)/(35)

= (215)/(35) + (238)/(35)

Add two numerator of the fraction
(215)/(35) + (238)/(35) = (215 + 238)/35 = 453/35

5 8/7 + 5 8/7 = 453/35

Mixed numbers addition calculator uses two mixed numbers, i.e. two numbers in terms of a whole numbers and proper fractions, $A\frac{a}{b}$ and $B\frac{c}{d}$ for positive integers $a,b,c$ and $d$, and calculates their sum. It is an online tool for finding the sum in the simplest form of two mixed numbers and give the step by step procedure for adding two mixed numbers.
It is necessary to follow the next steps:

1. Enter two mixed numbers $A\frac{a}{b}$ and $B\frac{c}{d}$ in the box. These numbers must be in terms of whole numbers and proper fractions. The numerators and denominators in the proper fractions must be positive integers.
2. Press the "GENERATE WORK" button to make the computation;
3. Mixed numbers addition calculator will give the sum of two numbers represented as mixed numbers.
Input : Two mixed numbers
Output : A fraction in the simplest form or decimal number

Conversion of Mixed Number to Improper Fraction Rule: The mixed number $A\frac{a}{b}$ for $a,b>0$ can be rewritten as improper fraction by the following formula $$A\frac{a}{b}=\frac{A\times b+a}{b},\quad \mbox{for}\;a,b>0$$ Mixed Numbers Adding Rule: The sum of two mixed numbers $A\frac{a}{b}$ and $B\frac{c}{d}$ is determined by the following formula
• If denominators of proper fractions of mixed numbers are equal, $b=d$: $$A\frac{a}{b}+B\frac{c}{b}=\frac{A\times b+a}{b}+\frac{B\times b+c}{b}=A+B+\frac{a+c}{b},\quad \mbox{for}\;b\ne0$$
• If denominators of proper fractions of mixed numbers are different, $b\ne d$: $$A\frac{a}{b}+B\frac{c}{d}=\frac{A\times b+a}{b}+\frac{B\times d+c}{d}=\frac{(A\times b+a)\times d+(B\times d+c)\times b}{b\times d},\quad \mbox{for}\;b,d\ne0$$ or equivalently, $$A\frac{a}{b}+B\frac{c}{d}=\frac{(A\times b+a)\times \frac{LCM(b,d)}{b}+(B\times d+c)\times \frac{LCM(b,d)}{d}}{LCM(b,d)},\quad \mbox{for}\;b,d\ne0$$ where $LCM(b,d)$ is the least common multiple of $b$ and $d$
• Conversion of Improper Fraction to Mixed Number Rule:The improper fraction $\frac{a}{b}, a>b$ can be rewritten as mixed number by the following formula $$\frac{a}{b}=\Big[\frac a b\Big]\frac{a-\Big[\frac a b\Big]\times b}{b},\quad \mbox{for}\;b\ne0$$ where square brackets $[\; ]$ mean round down to the nearest integer

## How to Add Two Mixed Numbers?

A mixed number $A\frac ab$ or sometimes called a \underline{mixed fraction} represents the sum of a nonzero integer number $A$ and a proper fraction $\frac ab$. The numerator $a$ and denominator $b$ of the proper fraction must be positive integers. In the notation of mixed numbers, the sum does not explicitly use operator plus. For example, two pizza and one-third of another pizza is denoted by $2\frac 13$ instead of $2+\frac 13$. Negative mixed number, for example $-2\frac 13$ represents the sum $-(2+\frac 13)$. Mixed numbers can also be written as decimals, for example, $2\frac 12=2.5$.
Improper fractions are rational numbers where the numerator is greater than the denominator. Improper fractions can be rewritten as a mixed number in the following way:

• Divide the numerator by the denominator;
• The whole part of the quotient is the whole number of the mixed number;
• The reminder is the new numerator of the proper fraction;
• The denominator of the proper fraction is equal to the denominator of the improper fraction.
More precisely, the improper fraction $\frac{a}{b}, a>b,$ can be rewritten as a mixed number in the following way $$\frac{a}{b}=\Big[\frac a b\Big]\frac{a-\Big[\frac a b\Big]\times b}{b},\quad \mbox{for}\;b\ne0,$$ where square brackets $[\; ]$ mean round down to the nearest integer. For example, $\frac 8 5$ is equal to $1\frac 35$.
To rewrite a mixed number to an improper fraction follow the next steps:
• Multiply the denominator of the proper fraction by the whole number in the mixed number and add it to its numerator;
• The denominator of the improper fraction is equal to the denominator of the proper fraction of the mixed number.
This means, a mixed number $A\frac{a}{b}$ for $a,b>0$ can be rewritten as improper fraction in the following way $$A\frac{a}{b}=\frac{A\times b+a}{b},\quad \mbox{for}\;a,b>0$$ For example, $$10\frac 35=\frac{10\times5+3}{5}=\frac{53}5$$ When we deal with mixed numbers, there are two types of addition:
• When denominators of proper fractions of mixed numbers are equal
When denominators of proper fractions of mixed numbers are equal, then the sum of two mixed numbers can be expressed in the following way: $$A\frac{a}{b}+B\frac{c}{b}=\frac{A\times b+a}{b}+\frac{B\times b+c}{b}=A+B+\frac{a+c}{b},\quad \mbox{for}\;b\ne0$$
• When denominators of proper fractions of mixed numbers are different
When denominators of proper fractions of mixed numbers are different, to add two or more mixed numbers, it is necessary to follow the next steps:
1. Convert mixed numbers to corresponding improper fractions;
2. Find the LCM of denominators of derived improper fractions;
3. Rewrite these fractions over the LCM;
5. The result is the sum of numerators over the LCM;
6. Simplify the result if needed.
This method can be expressed algebraically: $$A\frac{a}{b}+B\frac{c}{d}=\frac{(A\times b+a)\times \frac{LCM(b,d)}{b}+(B\times d+c)\times \frac{LCM(b,d)}{d}}{LCM(b,d)},\quad \mbox{for}\;b,d\ne0$$ If $LCM(b,d)=b\times d$, then the previous formula becomes $$A\frac{a}{b}+B\frac{c}{d}=\frac{A\times b+a}{b}+\frac{B\times d+c}{d}=\frac{(A\times b+a)\times d+(B\times d+c)\times b}{b\times d},\quad \mbox{for}\;b,d\ne0$$ To add two or more mixed numbers, convert them to improper fractions then add the fractions. For example, let us find the sum for $5\frac 37$ and $6\frac 45$. After converting these numbers to improper fractions, we obtain $$5\frac 37+6\frac 45=\frac {5\times 7+3}{7}+\frac {6\times 5+4}{5} =\frac {38}7+\frac {34}5$$ Since $LCM(7,5)=7\times 5=35$, then $$5\frac 37+6\frac 45=\frac {38\times 5+34\times 7}{5\times 7}=\frac {428}{35}$$ To write the sum in simplest form, find the GCF of the numerator and denominator of the sum. Because $428$ and $35$ are relatively prime numbers, the final result is $\frac{428}{35}$. To write the sum as a mixed number, we use the above mentioned conversion from an improper fraction to a mixed number: $$\frac{428}{35}=12\frac{8}{35}$$ The similar consideration can be applied in addition of algebraic expressions.
The mixed numbers addition work with steps shows the complete step-by-step calculation for finding the sum of two mixed numbers $5\frac{3}{7}$ and $6\frac{4}{5}$ using the mixed numbers addition rule. For any other mixed numbers, just supply two mixed numbers in terms of a whole number and proper fraction and click on the "GENERATE WORK" button. The grade school students may use this adding mixed numbers calculator to generate the work, verify the results of adding numbers derived by hand, or do their homework problems efficiently.

### Real World Problems Using Mixed Numbers Addition

Mixed numbers are useful in counting whole things and parts of these things together. It is used primarily in measurement. Especially of interest are mixed numbers whose denominator of the fractional part is a power of two. They are commonly used with U.S. customary units such as inches, pounds, etc. For instance, $1\; {\rm inch}=2\frac{54}{100}\; \rm{cm}$.

### Practice Problems for Mixed Numbers Addition

Practice Problem 1:
The biggest watermelon from Joe's farm is $13\frac 13$ kilograms, which is $1\frac 23$ kilograms more than the average weight of watermelon from Ann's. What is the average weight of watermelon from Ann's farm?

Practice Problem 2:
A market opens for $3\frac 13$ hours in the morning and $4\frac 17$ hours in the afternoon. How long has the market opened for the day?

The mixed numbers addition calculator, formula, step by step calculation, 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 mixed numbers. Using this concept they can be able to solve complex algebraic problems and equations.