Enter two numbers just as they look (whole-number box plus numerator and denominator boxes), and choose the operation (+ − × ÷) in the drop-down in the middle. Mixed numbers, fractions and whole numbers all work. The answer is shown as a simplified fraction, a mixed number and a decimal, with the steps.
Enter a whole number in each box. Leave unused boxes blank (the numerator and denominator for a whole number only, the whole-number box for a fraction only). For a negative mixed number, put the minus sign in the whole-number box (it applies to the whole number).
Result
Enter two numbers in the fields on the left, choose the operation (+ − × ÷) in the middle, and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formula
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Changing a mixed number to an improper fraction
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Adding mixed numbers (change to improper fractions, use a common denominator, add the numerators)
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Subtracting mixed numbers (change to improper fractions, use a common denominator, subtract the numerators)
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Multiplying mixed numbers (change to improper fractions, multiply numerators and denominators)
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Dividing mixed numbers (change to improper fractions, multiply by the reciprocal)
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Simplifying and changing back to a mixed number
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Add, subtract, multiply and divide mixed numbers such as \(2\,\dfrac{1}{3} \times 1\,\dfrac{1}{2}\) on the spot. Just enter the whole number, numerator and denominator the way they look
- The answer is shown three ways: as a fraction in simplest form, as a mixed number and as a decimal. You can also see the steps in between, such as changing to improper fractions, finding a common denominator and simplifying
- Besides mixed numbers, you can mix in whole numbers and fractions (proper and improper), and use negative numbers
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Enter a mixed number just as it looks, in the whole-number box and the numerator and denominator boxes (for a whole number only or a fraction only, leave the unused boxes blank). For a negative mixed number, put the minus sign in the whole-number box. It applies to the whole number (for example, whole number -2, numerator 3, denominator 4 means \(-2\,\dfrac{3}{4} = -(2 + \dfrac{3}{4}) = -\dfrac{11}{4}\)).
What is this calculation used for?
Scaling a recipe up or down
Making 1.5 times (\(1\,\dfrac{1}{2}\) times) a recipe that calls for \(\dfrac{3}{4}\) cup of butter needs \(\dfrac{3}{4} \times \dfrac{3}{2} = \dfrac{9}{8} = 1\,\dfrac{1}{8}\) cups.
Recipes are written with mixed numbers such as "\(1\,\dfrac{1}{2}\) cups" and "\(2\,\dfrac{3}{4}\) cups", so this calculation comes up every time you adjust a recipe for more or fewer people.
Adding up inch measurements in DIY and woodworking
Sizes of screws, bolts and lumber in inches are traditionally written as mixed numbers, such as "\(2\,\dfrac{3}{4}\) inches". For example, a 2×4 is actually \(1\,\dfrac{1}{2}\) inches thick, so two of them stacked are \(1\,\dfrac{1}{2} + 1\,\dfrac{1}{2} = 3\) inches.
Whenever you add up measurements or figure out how many pieces you can cut from a board, you really need to add, subtract, multiply and divide mixed numbers.
Figuring out fabric and ribbon for sewing and crafts
If one project uses \(1\,\dfrac{3}{4}\) yards and another uses \(2\,\dfrac{1}{2}\) yards, you need \(1\,\dfrac{3}{4} + 2\,\dfrac{1}{2} = 4\,\dfrac{1}{4}\) yards in total.
Fabric stores cut fabric in steps of \(\dfrac{1}{8}\) yard, so you can ask for exactly \(4\,\dfrac{1}{4}\) yards (4.25 yards as a decimal) and avoid buying too much or too little.
Planning study or work time
Studying \(1\,\dfrac{1}{2}\) hours (1 hour 30 minutes) a day, 6 days a week, adds up to \(1\,\dfrac{1}{2} \times 6 = 9\) hours a week.
Time works well with fractions (30 minutes = \(\dfrac{1}{2}\) hour, 15 minutes = \(\dfrac{1}{4}\) hour), so if you can multiply mixed numbers, you can quickly estimate the total time in a plan.
Checking a child's math homework (a core topic in Grades 4–6)
Changing between mixed numbers and improper fractions starts in Grade 4, adding and subtracting fractions with unlike denominators is Grade 5, multiplying fractions is Grade 5, and dividing fractions by fractions is Grade 6. It is a core part of elementary and middle school math.
If an adult can explain the thinking in words (change to improper fractions → find a common denominator → calculate → simplify), checking homework becomes more than marking answers right or wrong. The steps on this page follow that same order.
Formula
Changing a mixed number to an improper fraction
Standard notation
(the usual math form)
\(a\,\dfrac{b}{c}\)
\(=\)
\(\dfrac{a \times c + b}{c}\)
In words
(symbols replaced with words)
①
mixed number (whole part \(a\), fraction part \(\dfrac{b}{c}\))
\(=\)
②
\(\dfrac{\text{whole × denominator + numerator}}{\text{same denominator}}\)
The formula in words
①
For a
mixed number (whole part \(a\), fraction part \(\dfrac{b}{c}\))
multiply the whole part \(a\) by the denominator \(c\) and add the numerator \(b\). Use that as the new numerator, and you get the improper fraction
②
\(\dfrac{\text{whole × denominator + numerator}}{\text{same denominator}}\)
Quick example
Changing the mixed number \(2\,\dfrac{1}{3}\) (two and one third) to an improper fraction gives
mixed number \(2\,\dfrac{1}{3}\)
\(=\)
improper fraction \(\dfrac{2 \times 3 + 1}{3} = \dfrac{7}{3}\)
\(2\,\dfrac{1}{3} = \dfrac{2 \times 3 + 1}{3} = \dfrac{7}{3}\)
Key idea
The mixed number \(2\,\dfrac{1}{3}\) means \(2 + \dfrac{1}{3}\), so you rewrite the whole part 2 as \(\dfrac{6}{3}\) and combine it with the fraction part. That is where "whole × denominator + numerator" comes from.
For a negative mixed number, remember that the minus sign applies to the whole number (\(-2\,\dfrac{3}{4} = -(2 + \dfrac{3}{4}) = -\dfrac{11}{4}\)).
For adding, subtracting, multiplying and dividing mixed numbers, the surest method is always to change them to improper fractions first.
Adding mixed numbers (change to improper fractions, use a common denominator, add the numerators)
Standard notation
(the usual math form)
\(a\,\dfrac{b}{c}\)
\(+\)
\(d\,\dfrac{e}{f}\)
\(=\)
\(\dfrac{a \times c + b}{c}\)
\(+\)
\(\dfrac{d \times f + e}{f}\)
\(=\)
\(\dfrac{(a \times c + b) \times f + (d \times f + e) \times c}{c \times f}\)
In words
(symbols replaced with words)
①
left mixed number \(a\,\dfrac{b}{c}\)
\(+\)
②
right mixed number \(d\,\dfrac{e}{f}\)
\(=\)
③
\(\dfrac{\text{left: whole × denominator + numerator}}{\text{left denominator}}\)
\(+\)
④
\(\dfrac{\text{right: whole × denominator + numerator}}{\text{right denominator}}\)
\(=\)
⑤
\(\dfrac{\text{sum of the numerators after rewriting}}{\text{common denominator (product of the denominators)}}\)
The formula in words
①
Take the
left mixed number \(a\,\dfrac{b}{c}\)
②
and the
right mixed number \(d\,\dfrac{e}{f}\)
and for each, use "whole × denominator + numerator" as the new numerator. This changes them to the
③
left number as an improper fraction \(\dfrac{a \times c + b}{c}\)
④
and the
right number as an improper fraction \(\dfrac{d \times f + e}{f}\)
Rewrite them with a common denominator and add the numerators, and you get
⑤
\(\dfrac{\text{sum of the numerators after rewriting}}{\text{common denominator (product of the denominators)}}\)
Quick example
For \(1\,\dfrac{1}{2} + 2\,\dfrac{1}{3}\), change to the improper fractions \(\dfrac{3}{2}\) and \(\dfrac{7}{3}\), and rewrite them with the least common denominator 6
left mixed number \(1\,\dfrac{1}{2}\)
\(+\)
right mixed number \(2\,\dfrac{1}{3}\)
\(=\)
sum of the two mixed numbers \(3\,\dfrac{5}{6}\)
\(1\,\dfrac{1}{2} + 2\,\dfrac{1}{3} = \dfrac{3}{2} + \dfrac{7}{3} = \dfrac{9}{6} + \dfrac{14}{6} = \dfrac{23}{6}\)
\(\dfrac{23}{6} = 3\,\dfrac{5}{6}\ \ (\approx 3.833)\)
Key idea
Trying to add fraction parts with different denominators while they are still mixed numbers is where people get stuck. Change them to improper fractions first, and the rest is ordinary fraction addition: find a common denominator and add the numerators.
Multiplying the two denominators, \(c \times f\), always gives a common denominator, but using the least common multiple of the denominators keeps the numbers small and makes simplifying easier later (for example, with denominators 4 and 6, use 12 rather than 24).
You can also add the whole parts and the fraction parts separately (\(1 + 2 = 3\), \(\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{5}{6}\), so \(3\,\dfrac{5}{6}\)), and the answer is the same. But when the fraction parts add up to more than 1 (for example, \(\dfrac{1}{2} + \dfrac{2}{3} = \dfrac{7}{6}\)), you have to carry into the whole part, so if in doubt, the improper fraction method is the safe choice.
Subtracting mixed numbers (change to improper fractions, use a common denominator, subtract the numerators)
Standard notation
(the usual math form)
\(a\,\dfrac{b}{c}\)
\(-\)
\(d\,\dfrac{e}{f}\)
\(=\)
\(\dfrac{a \times c + b}{c}\)
\(-\)
\(\dfrac{d \times f + e}{f}\)
\(=\)
\(\dfrac{(a \times c + b) \times f - (d \times f + e) \times c}{c \times f}\)
In words
(symbols replaced with words)
①
left mixed number \(a\,\dfrac{b}{c}\)
\(-\)
②
right mixed number \(d\,\dfrac{e}{f}\)
\(=\)
③
\(\dfrac{\text{left: whole × denominator + numerator}}{\text{left denominator}}\)
\(-\)
④
\(\dfrac{\text{right: whole × denominator + numerator}}{\text{right denominator}}\)
\(=\)
⑤
\(\dfrac{\text{difference of the numerators after rewriting}}{\text{common denominator (product of the denominators)}}\)
The formula in words
①
Take the
left mixed number \(a\,\dfrac{b}{c}\)
②
and the
right mixed number \(d\,\dfrac{e}{f}\)
and for each, use "whole × denominator + numerator" as the new numerator. This changes them to the
③
left number as an improper fraction \(\dfrac{a \times c + b}{c}\)
④
and the
right number as an improper fraction \(\dfrac{d \times f + e}{f}\)
Rewrite them with a common denominator and subtract the right numerator from the left, and you get
⑤
\(\dfrac{\text{difference of the numerators after rewriting}}{\text{common denominator (product of the denominators)}}\)
Quick example
For \(1\,\dfrac{1}{2} - \dfrac{3}{4}\), change the mixed number to the improper fraction \(\dfrac{3}{2}\), and rewrite with the least common denominator 4
left mixed number \(1\,\dfrac{1}{2}\)
\(-\)
right fraction \(\dfrac{3}{4}\)
\(=\)
left minus right \(\dfrac{3}{4}\)
\(1\,\dfrac{1}{2} - \dfrac{3}{4} = \dfrac{3}{2} - \dfrac{3}{4} = \dfrac{6}{4} - \dfrac{3}{4} = \dfrac{3}{4}\)
Key idea
The idea is exactly the same as addition: change to improper fractions, find a common denominator, then subtract the numerators.
Subtracting the whole parts and the fraction parts separately needs regrouping (borrowing 1 from the whole part) when the fraction part cannot be subtracted (for example, the fraction parts of \(2\,\dfrac{1}{4} - 1\,\dfrac{1}{2}\), \(\dfrac{1}{4} - \dfrac{1}{2}\)). That is easy to get wrong, so the improper fraction method is the safe choice.
Subtracting a larger number from a smaller one gives a negative fraction (for example, \(\dfrac{1}{4} - \dfrac{3}{4} = -\dfrac{2}{4} = -\dfrac{1}{2}\)). This calculator shows negative answers as they are.
Multiplying mixed numbers (change to improper fractions, multiply numerators and denominators)
Standard notation
(the usual math form)
\(a\,\dfrac{b}{c}\)
\(\times\)
\(d\,\dfrac{e}{f}\)
\(=\)
\(\dfrac{a \times c + b}{c}\)
\(\times\)
\(\dfrac{d \times f + e}{f}\)
\(=\)
\(\dfrac{(a \times c + b) \times (d \times f + e)}{c \times f}\)
In words
(symbols replaced with words)
①
left mixed number \(a\,\dfrac{b}{c}\)
\(\times\)
②
right mixed number \(d\,\dfrac{e}{f}\)
\(=\)
③
\(\dfrac{\text{left: whole × denominator + numerator}}{\text{left denominator}}\)
\(\times\)
④
\(\dfrac{\text{right: whole × denominator + numerator}}{\text{right denominator}}\)
\(=\)
⑤
\(\dfrac{\text{product of the improper fractions' numerators}}{\text{product of the denominators}}\)
The formula in words
①
Change the
left mixed number \(a\,\dfrac{b}{c}\)
②
and the
right mixed number \(d\,\dfrac{e}{f}\)
to improper fractions, the
③
left number as an improper fraction \(\dfrac{a \times c + b}{c}\)
④
and the
right number as an improper fraction \(\dfrac{d \times f + e}{f}\)
Multiply the numerators together and the denominators together, and you get
⑤
\(\dfrac{\text{product of the improper fractions' numerators}}{\text{product of the denominators}}\)
Quick example
For \(2\,\dfrac{1}{3} \times 1\,\dfrac{1}{2}\), change to the improper fractions \(\dfrac{7}{3}\) and \(\dfrac{3}{2}\), then multiply
left mixed number \(2\,\dfrac{1}{3}\)
\(\times\)
right mixed number \(1\,\dfrac{1}{2}\)
\(=\)
product of the two mixed numbers \(3\,\dfrac{1}{2}\)
\(2\,\dfrac{1}{3} \times 1\,\dfrac{1}{2} = \dfrac{7}{3} \times \dfrac{3}{2} = \dfrac{7 \times 3}{3 \times 2} = \dfrac{21}{6} = \dfrac{7}{2} = 3\,\dfrac{1}{2}\)
Key idea
Unlike addition and subtraction, multiplication and division do not need a common denominator. On the other hand, you cannot calculate with the mixed numbers as they are. Multiplying the whole parts together and the fraction parts together is a common mistake. For example, the correct answer to \(1\,\dfrac{1}{2} \times 1\,\dfrac{1}{2}\) is \(\dfrac{3}{2} \times \dfrac{3}{2} = \dfrac{9}{4} = 2\,\dfrac{1}{4}\), but putting the whole parts (\(1 \times 1 = 1\)) next to the fraction parts (\(\dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4}\)) gives \(1\,\dfrac{1}{4}\), which is wrong. Always change to improper fractions before you multiply.
Dividing mixed numbers (change to improper fractions, multiply by the reciprocal)
Standard notation
(the usual math form)
\(a\,\dfrac{b}{c}\)
\(\div\)
\(d\,\dfrac{e}{f}\)
\(=\)
\(\dfrac{a \times c + b}{c}\)
\(\times\)
\(\dfrac{f}{d \times f + e}\)
\(=\)
\(\dfrac{(a \times c + b) \times f}{c \times (d \times f + e)}\)
In words
(symbols replaced with words)
①
mixed number being divided \(a\,\dfrac{b}{c}\)
\(\div\)
②
mixed number you divide by \(d\,\dfrac{e}{f}\)
\(=\)
③
\(\dfrac{\text{left: whole × denominator + numerator}}{\text{left denominator}}\)
\(\times\)
④
\(\dfrac{\text{divisor's denominator}}{\text{divisor: whole × denominator + numerator}}\)
\(=\)
⑤
\(\dfrac{\text{product of the numerators (left × reciprocal)}}{\text{product of the denominators (left × reciprocal)}}\)
The formula in words
①
Change the
mixed number being divided \(a\,\dfrac{b}{c}\)
②
and the
mixed number you divide by \(d\,\dfrac{e}{f}\)
to improper fractions. Multiply the
③
number being divided as an improper fraction \(\dfrac{a \times c + b}{c}\)
by the
④
reciprocal of the divisor \(\dfrac{f}{d \times f + e}\)
(the divisor with its numerator and denominator swapped), and you get
⑤
\(\dfrac{\text{product of the numerators (left × reciprocal)}}{\text{product of the denominators (left × reciprocal)}}\)
Quick example
Dividing the whole number 4 by the mixed number \(1\,\dfrac{1}{3}\) (\(\dfrac{4}{3}\) as an improper fraction) means multiplying by the reciprocal \(\dfrac{3}{4}\)
number being divided, 4
\(\div\)
mixed number you divide by \(1\,\dfrac{1}{3}\)
\(=\)
4 divided by \(1\,\dfrac{1}{3}\), 3
\(4 \div 1\,\dfrac{1}{3} = \dfrac{4}{1} \div \dfrac{4}{3} = \dfrac{4}{1} \times \dfrac{3}{4} = \dfrac{12}{4} = 3\)
Key idea
Dividing by \(\dfrac{1}{2}\) is the same as doubling. In the same way, dividing by a fraction is the same as multiplying by its reciprocal (the fraction flipped upside down). Think of "divide by X" as "count how many Xs fit". Two halves fit in 1, so dividing by \(\dfrac{1}{2}\) doubles the answer.
When you divide by a whole number, you can use the same formula by treating it as a fraction such as \(\dfrac{4}{1}\). But you cannot divide by 0.
Simplifying and changing back to a mixed number
Standard notation
(the usual math form)
\(\dfrac{p}{q}\)
\(=\)
\(\dfrac{p \div g}{q \div g}\)
\(\dfrac{p}{q}\)
\(=\)
\(w\,\dfrac{r}{q}\)
In words
(symbols replaced with words)
①
fraction before simplifying \(\dfrac{p}{q}\)
\(=\)
②
\(\dfrac{\text{numerator divided by the GCF}}{\text{denominator divided by the GCF}}\)
③
answer that is an improper fraction (numerator ≥ denominator)
\(=\)
④
mixed number (whole part = quotient \(w\) of numerator ÷ denominator, numerator = remainder \(r\))
The formula in words
①
Divide the numerator and denominator of the
fraction before simplifying \(\dfrac{p}{q}\)
by the largest number that divides both (the greatest common factor \(g\)), and you get
②
\(\dfrac{\text{numerator divided by the GCF}}{\text{denominator divided by the GCF}}\)
(a fraction that cannot be simplified further)
③
An
answer that is an improper fraction (numerator ≥ denominator)
can be changed back: use the quotient \(w\) of numerator ÷ denominator as the whole part and the remainder \(r\) as the new numerator, to get the
④
mixed number (whole part = quotient \(w\) of numerator ÷ denominator, numerator = remainder \(r\))
Quick example
Simplifying \(\dfrac{21}{6}\) from the multiplication example (the GCF of 21 and 6 is 3) gives
fraction before simplifying \(\dfrac{21}{6}\)
\(=\)
numerator and denominator divided by 3, \(\dfrac{7}{2}\) (as a mixed number, \(3\,\dfrac{1}{2}\))
\(\dfrac{21}{6} = \dfrac{21 \div 3}{6 \div 3} = \dfrac{7}{2} = 3\,\dfrac{1}{2}\)
Key idea
The rule is to simplify a fraction answer at the end, so that it cannot be simplified any further (simplest form).
If the simplified result is an improper fraction (numerator ≥ denominator), you can change it to a mixed number. Divide the numerator by the denominator: the quotient is the whole part, and the remainder is the new numerator (for example, for \(\dfrac{7}{2}\), \(7 \div 2 = 3\) remainder \(1\), so \(3\,\dfrac{1}{2}\)). This calculator shows the answer both as an improper fraction and as a mixed number.
Every calculation with mixed numbers starts by changing them to improper fractions. For addition and subtraction, find a common denominator and work with the numerators. For multiplication, multiply the numerators together and the denominators together. For division, multiply by the reciprocal. Finally, simplify with the greatest common factor, and if the answer is an improper fraction, change it back to a mixed number.
Symbols and terms
Symbols
| \(\dfrac{a}{b}\) | a over b | A fraction. The top \(a\) is the numerator and the bottom \(b\) is the denominator. It has the same value as \(a \div b\). The denominator \(b\) cannot be 0. |
| \(a\,\dfrac{b}{c}\) | a and b over c (mixed number) | The left mixed number. It is the whole part \(a\) written next to the proper fraction \(\dfrac{b}{c}\), and it means \(a + \dfrac{b}{c}\). In this calculator, you enter it in separate whole-number, numerator and denominator boxes. |
| \(d\,\dfrac{e}{f}\) | d and e over f (mixed number) | The right mixed number. Different letters are used to tell it apart from the left one in the formulas. It means \(d + \dfrac{e}{f}\), in the same way. |
| \(g\) | g | The greatest common factor used in the simplifying formula. It is the largest number that divides both the numerator and the denominator (for example, 3 for 21 and 6). |
| \(\div\) | divided by | The division sign. In fraction division, you replace it with multiplication by the reciprocal of the divisor. |
Terms
| mixed number | A number written as a whole number followed by a proper fraction, such as \(3\,\dfrac{1}{2}\) ("three and one half"). It means "whole number + fraction", and its strength is that you can see its size at a glance. It suits everyday amounts, such as "three and a half cups". |
| improper fraction | A fraction whose numerator is equal to or greater than its denominator, such as \(\dfrac{7}{2}\). It is better than a mixed number for calculating, especially for multiplication and division. The quotient and remainder of numerator ÷ denominator turn it into a mixed number. |
| proper fraction | A fraction whose numerator is smaller than its denominator, such as \(\dfrac{1}{2}\) or \(\dfrac{3}{4}\). Its value is always less than 1. The fraction part of a mixed number is a proper fraction. |
| numerator | The top number of a fraction. The denominator tells how many equal parts 1 is split into, and the numerator tells how many of those parts there are. |
| denominator | The bottom number of a fraction. It tells how many equal parts 1 is split into. It cannot be 0 (that would mean dividing by 0). |
| common denominator | A denominator shared by two or more fractions. Rewriting fractions with different denominators so that they have the same one usually uses the least common multiple of the denominators (the least common denominator, LCD). It is needed before adding or subtracting, but not for multiplying or dividing. |
| simplify | To divide the numerator and denominator of a fraction by a common factor to make it simpler (also called reducing). Dividing by the greatest common factor does it in one step (for example, \(\dfrac{21}{6} = \dfrac{7}{2}\)). |
| least common multiple (LCM) | The smallest multiple that two numbers share. Example - the least common multiple of 4 and 7 is 28. It is used as the common denominator. |
| greatest common factor (GCF, GCD) | The largest number that divides both of two numbers. Example - the greatest common factor of 21 and 6 is 3. It is used to simplify. It is also called the greatest common divisor (GCD). |
| reciprocal | A fraction with its numerator and denominator swapped (for example, the reciprocal of \(\dfrac{4}{3}\) is \(\dfrac{3}{4}\)). A number times its reciprocal is 1. Dividing by a fraction can be replaced by multiplying by its reciprocal. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.
| What a fraction means (Grade 3) |
|
| Mixed numbers and improper fractions (Grade 4) |
|
| Simplifying, common denominators, multiples and factors (Grades 4–6) |
|
| Adding and subtracting fractions (Grade 5) |
|
| Multiplying and dividing fractions (Grades 5–6) |
|
| Positive and negative numbers (Grade 7) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to change a mixed number to an improper fraction
| Whole part of the mixed number a | 2 |
| Numerator of the mixed number b | 1 |
| Denominator of the mixed number c | 3 |
| Improper fraction numerator (a×c+b) | =B1*B3+B2 |
| Improper fraction denominator (c) | =B3 |
Table to add fractions
| Left numerator a | 3 |
| Left denominator b | 2 |
| Right numerator c | 1 |
| Right denominator d | 3 |
| Answer numerator (a×d+c×b) | =B1*B4+B3*B2 |
| Answer denominator (b×d) | =B2*B4 |
| As a decimal | =B5/B6 |
Table to subtract fractions
| Left numerator a | 3 |
| Left denominator b | 2 |
| Right numerator c | 3 |
| Right denominator d | 4 |
| Answer numerator (a×d−c×b) | =B1*B4-B3*B2 |
| Answer denominator (b×d) | =B2*B4 |
| As a decimal | =B5/B6 |
Table to multiply fractions
| Left numerator a | 7 |
| Left denominator b | 3 |
| Right numerator c | 3 |
| Right denominator d | 2 |
| Answer numerator (a×c) | =B1*B3 |
| Answer denominator (b×d) | =B2*B4 |
| As a decimal | =B5/B6 |
Table to divide fractions
| Numerator of the fraction being divided a | 4 |
| Denominator of the fraction being divided b | 1 |
| Numerator of the divisor c | 4 |
| Denominator of the divisor d | 3 |
| Answer numerator (a×d) | =B1*B4 |
| Answer denominator (b×c) | =B2*B3 |
| As a decimal | =B5/B6 |
Table to simplify and change back to a mixed number
| Numerator before simplifying p | 21 |
| Denominator before simplifying q | 6 |
| Simplified numerator (p÷GCF) | =B1/GCD(B1,B2) |
| Simplified denominator (q÷GCF) | =B2/GCD(B1,B2) |
| Whole part of the mixed number (quotient) | =QUOTIENT(B3,B4) |
| Numerator of the mixed number (remainder) | =MOD(B3,B4) |
An Excel cell cannot hold a mixed number as it is, so enter the whole part, numerator and denominator separately. In each table, the upper rows are your inputs, and the formulas in bold green are calculated automatically.
To calculate with mixed numbers, first change them to improper fractions with the first table, put those numerators and denominators into tables 2 to 5, then simplify the answer's numerator and denominator with the sixth table.
For example, the fourth table, for multiplication (\(\dfrac{7}{3} \times \dfrac{3}{2}\)), gives an answer numerator of 21 and denominator of 6. Simplifying with the sixth table gives \(\dfrac{7}{2}\), and as a mixed number, whole part 3 and numerator 1 (= \(3\,\dfrac{1}{2}\)).
GCD finds the greatest common factor (greatest common divisor), QUOTIENT the whole-number part of a division, and MOD the remainder.
To calculate with mixed numbers, first change them to improper fractions with the first table, put those numerators and denominators into tables 2 to 5, then simplify the answer's numerator and denominator with the sixth table.
For example, the fourth table, for multiplication (\(\dfrac{7}{3} \times \dfrac{3}{2}\)), gives an answer numerator of 21 and denominator of 6. Simplifying with the sixth table gives \(\dfrac{7}{2}\), and as a mixed number, whole part 3 and numerator 1 (= \(3\,\dfrac{1}{2}\)).
GCD finds the greatest common factor (greatest common divisor), QUOTIENT the whole-number part of a division, and MOD the remainder.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to change a mixed number to an improper fraction
| Whole part of the mixed number a | 2 |
| Numerator of the mixed number b | 1 |
| Denominator of the mixed number c | 3 |
| Improper fraction numerator (a×c+b) | =B1*B3+B2 |
| Improper fraction denominator (c) | =B3 |
Table to add fractions
| Left numerator a | 3 |
| Left denominator b | 2 |
| Right numerator c | 1 |
| Right denominator d | 3 |
| Answer numerator (a×d+c×b) | =B1*B4+B3*B2 |
| Answer denominator (b×d) | =B2*B4 |
| As a decimal | =B5/B6 |
Table to subtract fractions
| Left numerator a | 3 |
| Left denominator b | 2 |
| Right numerator c | 3 |
| Right denominator d | 4 |
| Answer numerator (a×d−c×b) | =B1*B4-B3*B2 |
| Answer denominator (b×d) | =B2*B4 |
| As a decimal | =B5/B6 |
Table to multiply fractions
| Left numerator a | 7 |
| Left denominator b | 3 |
| Right numerator c | 3 |
| Right denominator d | 2 |
| Answer numerator (a×c) | =B1*B3 |
| Answer denominator (b×d) | =B2*B4 |
| As a decimal | =B5/B6 |
Table to divide fractions
| Numerator of the fraction being divided a | 4 |
| Denominator of the fraction being divided b | 1 |
| Numerator of the divisor c | 4 |
| Denominator of the divisor d | 3 |
| Answer numerator (a×d) | =B1*B4 |
| Answer denominator (b×c) | =B2*B3 |
| As a decimal | =B5/B6 |
Table to simplify and change back to a mixed number
| Numerator before simplifying p | 21 |
| Denominator before simplifying q | 6 |
| Simplified numerator (p÷GCF) | =B1/GCD(B1,B2) |
| Simplified denominator (q÷GCF) | =B2/GCD(B1,B2) |
| Whole part of the mixed number (quotient) | =QUOTIENT(B3,B4) |
| Numerator of the mixed number (remainder) | =MOD(B3,B4) |
The same formulas as in Excel work as is (Google Sheets also has GCD, QUOTIENT and MOD). Copy the whole table, paste it into cell A1, and replace the input numbers with your own fractions.
How to calculate it in Python
from fractions import Fraction
# Write a mixed number as "whole part + fraction part"
# (a negative mixed number -2 3/4 gets a minus sign on the whole thing: -(Fraction(2) + Fraction(3, 4)))
left = Fraction(2) + Fraction(1, 3) # mixed number 2 1/3 → improper fraction 7/3
right = Fraction(1) + Fraction(1, 2) # mixed number 1 1/2 → improper fraction 3/2
answer = left * right # change to + to add, - to subtract, / to divide
print(f"Simplified fraction: {answer}") # 7/2 (simplified automatically)
print(f"Decimal: {float(answer)}") # 3.5
# Change the improper fraction to a mixed number (e.g. 7/2 → 3 1/2)
whole = abs(answer.numerator) // answer.denominator # whole part (quotient of numerator ÷ denominator)
part = abs(answer.numerator) % answer.denominator # new numerator (remainder)
sign = "-" if answer < 0 else ""
print(f"Mixed number: {sign}{whole} {part}/{answer.denominator}")
Runs with only the fractions module from the standard library. Fraction handles fractions exactly and finds common denominators and simplifies for you. Change left and right at the top to your own numbers and run it.
How to write it in LaTeX and other math languages (copy and paste)
Changing a mixed number to an improper fraction
a b/c = (a×c + b)/c
a\dfrac{b}{c} = \dfrac{a \times c + b}{c}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>a</mi>
<mfrac><mi>b</mi><mi>c</mi></mfrac>
<mo>=</mo>
<mfrac>
<mrow><mi>a</mi><mo>×</mo><mi>c</mi><mo>+</mo><mi>b</mi></mrow>
<mi>c</mi>
</mfrac>
</mrow>
</math>
a b/c = (a xx c + b)/c
a + b/c
improper := (a*c + b)/c;
improper = (a*c + b)/c;
a b/c = (a×c + b)/c
Adding mixed numbers (change to improper fractions, use a common denominator, add the numerators)
a b/c + d e/f = (a×c + b)/c + (d×f + e)/f
a\dfrac{b}{c} + d\dfrac{e}{f} = \dfrac{a \times c + b}{c} + \dfrac{d \times f + e}{f}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>a</mi><mfrac><mi>b</mi><mi>c</mi></mfrac>
<mo>+</mo>
<mi>d</mi><mfrac><mi>e</mi><mi>f</mi></mfrac>
<mo>=</mo>
<mfrac>
<mrow><mi>a</mi><mo>×</mo><mi>c</mi><mo>+</mo><mi>b</mi></mrow>
<mi>c</mi>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow><mi>d</mi><mo>×</mo><mi>f</mi><mo>+</mo><mi>e</mi></mrow>
<mi>f</mi>
</mfrac>
</mrow>
</math>
a b/c + d e/f = (a xx c + b)/c + (d xx f + e)/f
(a + b/c) + (d + e/f) // Together
answer := normal((a + b/c) + (d + e/f));
answer = simplifyFraction((a + b/c) + (d + e/f));
a b/c + d e/f = (a×c + b)/c + (d×f + e)/f
Subtracting mixed numbers (change to improper fractions, use a common denominator, subtract the numerators)
a b/c − d e/f = (a×c + b)/c − (d×f + e)/f
a\dfrac{b}{c} - d\dfrac{e}{f} = \dfrac{a \times c + b}{c} - \dfrac{d \times f + e}{f}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>a</mi><mfrac><mi>b</mi><mi>c</mi></mfrac>
<mo>−</mo>
<mi>d</mi><mfrac><mi>e</mi><mi>f</mi></mfrac>
<mo>=</mo>
<mfrac>
<mrow><mi>a</mi><mo>×</mo><mi>c</mi><mo>+</mo><mi>b</mi></mrow>
<mi>c</mi>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow><mi>d</mi><mo>×</mo><mi>f</mi><mo>+</mo><mi>e</mi></mrow>
<mi>f</mi>
</mfrac>
</mrow>
</math>
a b/c - d e/f = (a xx c + b)/c - (d xx f + e)/f
(a + b/c) - (d + e/f) // Together
answer := normal((a + b/c) - (d + e/f));
answer = simplifyFraction((a + b/c) - (d + e/f));
a b/c - d e/f = (a×c + b)/c - (d×f + e)/f
Multiplying mixed numbers (change to improper fractions, multiply numerators and denominators)
a b/c × d e/f = ((a×c + b) × (d×f + e))/(c×f)
a\dfrac{b}{c} \times d\dfrac{e}{f} = \dfrac{(a \times c + b) \times (d \times f + e)}{c \times f}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>a</mi><mfrac><mi>b</mi><mi>c</mi></mfrac>
<mo>×</mo>
<mi>d</mi><mfrac><mi>e</mi><mi>f</mi></mfrac>
<mo>=</mo>
<mfrac>
<mrow><mo>(</mo><mi>a</mi><mo>×</mo><mi>c</mi><mo>+</mo><mi>b</mi><mo>)</mo><mo>×</mo><mo>(</mo><mi>d</mi><mo>×</mo><mi>f</mi><mo>+</mo><mi>e</mi><mo>)</mo></mrow>
<mrow><mi>c</mi><mo>×</mo><mi>f</mi></mrow>
</mfrac>
</mrow>
</math>
a b/c xx d e/f = ((a xx c + b) xx (d xx f + e))/(c xx f)
(a + b/c)*(d + e/f) // Together
answer := normal((a + b/c)*(d + e/f));
answer = simplifyFraction((a + b/c)*(d + e/f));
a b/c × d e/f = ((a×c + b) × (d×f + e))/(c×f)
Dividing mixed numbers (change to improper fractions, multiply by the reciprocal)
a b/c ÷ d e/f = (a×c + b)/c × f/(d×f + e)
a\dfrac{b}{c} \div d\dfrac{e}{f} = \dfrac{a \times c + b}{c} \times \dfrac{f}{d \times f + e}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>a</mi><mfrac><mi>b</mi><mi>c</mi></mfrac>
<mo>÷</mo>
<mi>d</mi><mfrac><mi>e</mi><mi>f</mi></mfrac>
<mo>=</mo>
<mfrac>
<mrow><mi>a</mi><mo>×</mo><mi>c</mi><mo>+</mo><mi>b</mi></mrow>
<mi>c</mi>
</mfrac>
<mo>×</mo>
<mfrac>
<mi>f</mi>
<mrow><mi>d</mi><mo>×</mo><mi>f</mi><mo>+</mo><mi>e</mi></mrow>
</mfrac>
</mrow>
</math>
a b/c -: d e/f = (a xx c + b)/c xx f/(d xx f + e)
(a + b/c)/(d + e/f) // Together
answer := normal((a + b/c)/(d + e/f));
answer = simplifyFraction((a + b/c)/(d + e/f));
a b/c ÷ d e/f = (a×c + b)/c × f/(d×f + e)
Simplifying and changing back to a mixed number
p/q = (p ÷ g)/(q ÷ g)
\dfrac{p}{q} = \dfrac{p \div g}{q \div g}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfrac><mi>p</mi><mi>q</mi></mfrac>
<mo>=</mo>
<mfrac>
<mrow><mi>p</mi><mo>÷</mo><mi>g</mi></mrow>
<mrow><mi>q</mi><mo>÷</mo><mi>g</mi></mrow>
</mfrac>
</mrow>
</math>
p/q = (p -: g)/(q -: g)
Simplify[p/q]
answer := simplify(p/q);
answer = simplifyFraction(sym(p)/sym(q));
p/q = (p÷g)/(q÷g)
How to have ChatGPT do the calculation
You are a fraction calculation assistant. Do the following calculation by actually running Python code (the fractions module is recommended), and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). For the mixed numbers 2 1/3 (two and one third) and 1 1/2 (one and one half), calculate the following four: 1. Addition (2 1/3 + 1 1/2) 2. Subtraction (2 1/3 − 1 1/2) 3. Multiplication (2 1/3 × 1 1/2) 4. Division (2 1/3 ÷ 1 1/2) Give each answer three ways: as a fraction in simplest form, as a mixed number and as a decimal. Show the formulas you used and the numbers from the execution result in a table.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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