GENERATE WORK

GENERATE WORK

**Input Data :**

Mixed Fraction A = `7 1/6`

Mixed Fraction B = `2 1/3`

**Objective :**

Find the average of two given mixed numbers?

**Formula :**

Average = `(A + B)/2`

**Solution :**

`7 1/6\divide7 1/6 = ?`

Convert mixed numbers into equivalent frations

`7 1/6 = ((7\times6)+1)/6`
= `(42 + 1)/6`
` = 43/6`

`2 1/3 = ((2\times3)+1)/3`
= `(6 + 1)/3`
` = 7/3`

denominator of the two fraction is different. Therefore, find lcm for two denominators (6, 3) = 6

Multiply lcm with both numerator & denominator

`43/6 + 7/3 = (43\times6)/(6\times6) + (7\times6)/(3\times6)`

= `(43\times1)/(6) + (7\times2)/(6)`

= `(43)/(6) + (14)/(6)`

Add two numerator of the fraction

`(43)/(6) + (14)/(6) = (43 + 14)/6 = 57/6`

Average = `((57/6))/2`

Average = `57/12`

Simplify above fraction `57/12`

Common divisor of (57, 12) is 3

Divide both numerator & denominator by gcd value 3

`57/12 = frac{57\divide3}{12\divide3} = 19/4`

Average = `19/4`

** 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:

- 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.
- Press the "GENERATE WORK" button to make the computation;
- Average of mixed fractions calculator will give the average of two numbers represented as mixed numbers.

- 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 $d$.

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 unlike

- Convert mixed numbers to corresponding improper fractions;
- Find the LCM of denominators of derived improper fractions;
- Rewrite these fractions over the LCM;
- Add new numerators;
- Put the sum of numerators over the LCM;
- Divide this fraction by the number of fractions, $n$;
- Simplify the result if needed.

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:
$$ \sqrt{\frac{x_1^2+\ldots+x_n^2}{n}}\geq\frac{x_1+\ldots+x_n}{n}\geq\sqrt[n]{x_1\cdot x_2\ldots\cdot x_n}\geq \frac{n}{\frac{1}{x_1}+\ldots+\frac{1}{x_n}}$$

**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.