# Subtracting Mixed Number Calculator

Mixed Fraction A
Mixed Fraction B
A - B  =  -23/35
GENERATE WORK

## Mixed Number Subtraction - work with steps

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

Objective :
Find the difference when subtracting a mixed number from mixed number?

Solution :
5 8/7 - 5 8/7 = ?

Convert mixed numbers into equivalent frations and do the subtraction

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)

Subtract two numerator of the fraction
(215)/(35) - (238)/(35) = (215 - 238)/35 = -23/35

5 8/7 + 5 8/7 = -23/35

Subtracting mixed number calculatoruses 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 difference between $A\frac{a}{b}$ and $B\frac{c}{d}$. It is an online algebra tool for finding the mixed numbers difference in the simplest form between the first mixed number and the second mixed number and give the stepwise procedure for subtracting a mixed number from mixed number.

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 subtraction calculator will give the mixed numbers difference between the first number and the second number.
Input: Two mixed numbers;
Output: A fraction in the simplest form or mixed number 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 Subtraction Rule: The mixed numbers difference between $A\frac{a}{b}$ 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 Subtract a Mixed Number from Another?

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, $$10\frac 35=\frac{10\times5+3}{5}=\frac{53}5$$ The result in the subtraction of numbers is a difference. A difference of two numbers depends on their order, i.e. the subtraction is non-commutative operation. When we deal with mixed numbers, there are two types of subtracting a mixed number from a mixed number:
• When denominators of proper fractions of mixed numbers are equal
When denominators of proper fractions of mixed numbers are equal, then the mixed numbers difference between the first number $A\frac ab$ and the second number $B\frac cb$ 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 subtract one number from another, 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. Subtract the second numerator from the firs one;
5. The result is the difference 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 $$A\frac{a}{b}-B\frac{c}{d}=\frac{A\times b+a}{b}-\frac{B\times d+c}{d}=\frac{(A\times b+a)\times d-(B\times d+c)\times b}{b\times d},\quad \mbox{for}\;b,d\ne0$$ For example, let us use subtracting mixed numbers step by step calculation to find the mixed numbers difference between $5\frac 37$ 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 $$5\frac 37-6\frac 45=\frac {38\times 5-34\times 7}{5\times 7}=\frac {-48}{35}$$ To write the result in simplest form, find the GCF of the numerator and denominator of the difference. Because $48$ and $35$ are relatively prime numbers, the final result is $-\frac{48}{35}$. To write the difference as a mixed number, use the above mentioned conversion from an improper fraction to a mixed number: $$-\frac{48}{35}=-1\frac{13}{35}$$ The similar consideration can be applied in subtraction of algebraic expressions.
The mixed number subtraction work with steps shows the complete step-by-step calculation for finding the mixed numbers difference of two mixed numbers $5\frac{3}{7}$ and $6\frac{4}{5}$ using the mixed numbers subtraction 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 number subtraction calculator to generate the work, verify the results of subtracting numbers derived by hand, or do their homework problems efficiently.

### Real World Problems Using Mixed Numbers Subtraction

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, $1\; {\rm inch}=2\frac{54}{100}\; \rm{cm}$. Subtracting of a mixed number from another is often occurs in practise problems.

### Mixed Numbers Subtraction Practice Problems

Practice Problem 1 :
There are $36\frac 14$ boxes of tomatoes in a truck. Farmer sold $21\frac 37$ boxes of tomatoes. How many boxes of tomatoes left in the truck?

Practice Problem 2 :
Mitchell's ice cream recipe calls for $3 \frac 35$ caps of sugar and Ann's recipe calls for $1 \frac38$ caps of sugar. How many more caps of sugar are used in Mitchell's recipe than in Ann's recipe?

The mixed number subtraction 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 subtraction of two or more numbers represented as mixed numbers. Using this concept they can be able to solve complex algebraic problems and equations.