Mixed Numbers Average Calculator

Mixed numbers average 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 the halfway, mean or average of two mixed numbers by adding them and dividing by $2$. It is an online algebra tool for finding the average in the simplest form of 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. Average of mixed fractions calculator will give the average of two numbers represented as mixed numbers.

Input: Two mixed numbers;
Output: A fraction in the simplest form.

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 Average Rule: The average of two numbers $A\frac{a}{b}$ and $B\frac{c}{d}$ is determined by the following formula

• If proper fractions of mixed numbers are like, $b=d$:
$$\mbox{Average}=\frac{A\frac{a}{b}+B\frac{c}{b}}{2}=\frac{\frac{A\times b+a}{b}+\frac{B\times b+c}{b}}{2}=\frac{A+B}{2}+\frac{a+c}{2\times b},\quad \mbox{for}\;b\ne0$$
• If proper fractions of mixed numbers are unlike, $b\ne d$:
$$\mbox{Average}=\frac{A\frac{a}{b}+B\frac{c}{d}}{2}=\frac{\frac{A\times b+a}{b}+\frac{B\times d+c}{d}}{2}=\frac{(A\times b+a)\times d+(B\times d+c)\times b}{2\times b\times d},\quad \mbox{for}\;b,d\ne0$$ or equivalently, $$\mbox{Average}=\frac{A\frac{a}{b}+B\frac{c}{d}}{2}=\frac{(A\times b+a)\times \frac{LCM(b,d)}{b}+(B\times d+c)\times \frac{LCM(b,d)}{d}}{2\times LCM(b,d)},\quad \mbox{for}\;b,d\ne0$$ where $LCM(b,d)$ is the least common multiple of $b$ and $d$.

How to Find Average of Two Mixed Numbers?

To find the average of a set of numbers, find the sum of the numbers and then divide by the number in the set. Average of one number is the number itself. In other words, a mixed numbers average is the midpoint or balanced point of these numbers. Since a mixed number can be converted to a proper fraction by the corresponding rule, then the average of mixed numbers is the same as the average of corresponding improper fractions.
When we deal with mixed numbers, there are two formulas for finding the mixed numbers average:

• When proper fractions of mixed numbers are like

When proper fractions of mixed numbers are like, $b=d$, then the average of two mixed numbers can be expressed by the following formula: $$\mbox{Average}=\frac{A\frac{a}{b}+B\frac{c}{b}}{2}=\frac{\frac{A\times b+a}{b}+\frac{B\times b+c}{b}}{2}=\frac{A+B}{2}+\frac{a+c}{2\times b},\quad \mbox{for}\;b\ne0$$

• When proper fractions of mixed numbers are unlike

When proper fractions of mixed numbers are unlike, $b\ne d$, to find the mixed numbers average of two or more 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;
4. Add new numerators;
5. Put the sum of numerators over the LCM;
6. Divide this fraction by the number of fractions, $n$;
7. Simplify the result if needed.

This method can be expressed algebraically: $$\mbox{Average}=\frac{A\frac{a}{b}+B\frac{c}{d}}{2}=\frac{(A\times b+a)\times \frac{LCM(b,d)}{b}+(B\times d+c)\times \frac{LCM(b,d)}{d}}{2\times LCM(b,d)},\quad \mbox{for}\;b,d\ne0$$ If $LCM(b,d)=b\times d$, then the previous formula becomes $$\mbox{Average}=\frac{A\frac{a}{b}+B\frac{c}{d}}{2}=\frac{\frac{A\times b+a}{b}+\frac{B\times d+c}{d}}{2}=\frac{(A\times b+a)\times d+(B\times d+c)\times b}{2\times b\times d},\quad \mbox{for}\;b,d\ne0$$ For example, let us find the average of mixed numbers $7\frac 16$ and $2\frac 13$. After converting these numbers to improper fractions, we obtain $$7\frac 16=\frac{7\times 6+1}6=\frac{43}6\quad\mbox{and}\quad 2\frac13=\frac{2\times 3+1}{3}=\frac 73$$ Since $LCM(6,3)=6$, then $$\mbox{Average}=\frac {43\times 1+7\times 2}{2\times 6}=\frac {57}{12}$$ To write the mixed numbers average in simplest form, find the GCF of the numerator and denominator of the result. Because $GCF(57,12)=3$, the average of mixed numbers $7\frac 16$ and $2\frac 13$ is $$\frac{57\div 3}{12\div 3}=\frac{19}4$$ The Average of Mixed Numbers work with steps shows the complete step-by-step calculation for finding the mixed numbers average for $7\frac 16$ and $2\frac 13$ using the mixed numbers average 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 mixed numbers average calculator to generate the work, understand the methods of finding averages, or do their homework problems efficiently.

Real World Problems using Average of Mixed Numbers

Average describes a number which represents the general characteristics of a larger group of unequal objects.The average is a common measure of central tendency among a set of numbers. The arithmetic mean can be visualized as a balancing point on a scale.
There are three main types of averages: arithmetic mean, weighted average and average speed. The weighted arithmetic mean is similar to an arithmetic mean, except that instead of each of the number contributing equally to the final average, some numbers contribute more than others. In other words, we multiply each number by its weighting factor. The average speed of a body in time is the distance traveled by the body divided by the duration of the interval. In statistics, there are several types of "averages". For example, mean, median and mode. In algebra, here are four kinds of means of numbers: arithmetic mean, root mean, harmonic mean and root mean. We use the arithmetic mean when numbers add, geometric mean when numbers multiply, harmonic mean when the reciprocals of the numbers add, quadratic mean when the squares of the numbers add. The inequality between these means of a set of positive real numbers $x_1,\ldots,x_n$ says: $$ \sqrt{\frac{x_1^2+\ldots+x_n^2}{n}}\geq\frac{x_1+\ldots+x_n}{n}\geq\sqrt[n]{x_1\cdot x_2\ldots\cdot x_n}\geq \frac{n}{\frac{1}{x_1}+\ldots+\frac{1}{x_n}}$$

Mixed Numbers Average Practice Problems

Practice Problem 1 :
Find the arithmetic mean of $9\frac 18$ and $3\frac {3}{7}$.

Practice Problem 2 :
Ann washed a car for $4$ hours and earned $\$6\frac12$ per hour. Then she washed windows for $2$ hours and earned $\$5\frac 13$ per hour. Find Ann's average earnings per hour for this $6$ hours?

The mixed numbers average 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 arithmetic mean of two or more numbers represented as mixed numbers. Using this concept they can be able to find the mean of a data set in statistics and extend their knowledge developed in theory of numbers.