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Add and Subtract Mixed Numbers

Model addition of mixed numbers with a common denominator

Model Addition of Mixed Numbers with a Common Denominator

So far, we’ve added and subtracted proper and improper fractions, but not mixed numbers. Let’s begin by thinking about addition of mixed numbers using money.

If Ron has $1$ dollar and $1$ quarter, he has $1\frac{1}{4}$ dollars.

If Don has $2$ dollars and $1$ quarter, he has $2\frac{1}{4}$ dollars.

What if Ron and Don put their money together? They would have $3$ dollars and $2$ quarters. They add the dollars and add the quarters. This makes $3\frac{2}{4}$ dollars. Because two quarters is half a dollar, they would have $3$ and a half dollars, or $3\frac{1}{2}$ dollars.

\[\begin{array}{l} \\ \\ \\ 1\frac{1}{4} \\ +2\frac{1}{4} \\ \text{________} \\ \\ \\ 3\frac{2}{4}=3\frac{1}{2}\end{array}\]

When you added the dollars and then added the quarters, you were adding the whole numbers and then adding the fractions.

\[1\frac{1}{4}+2\frac{1}{4}\]

We can use fraction circles to model this same example:

		$1\frac{1}{4}+2\frac{1}{4}$
Start with $1\frac{1}{4}$.	one whole and one $\frac{1}{4}$ pieces
Add $2\frac{1}{4}$ more.	two wholes and one $\frac{1}{4}$ pieces
The sum is:	three wholes and two $\frac{1}{4}$'s

Example

Try it.

Model $2\frac{1}{3}+1\frac{2}{3}$ and give the sum.

Solution

We will use fraction circles, whole circles for the whole numbers and $\frac{1}{3}$ pieces for the fractions.

two wholes and one $\frac{1}{3}$
plus one whole and two $\frac{1}{3}$s
sum is three wholes and three $\frac{1}{3}$s

This is the same as $4$ wholes. So, $2\frac{1}{3}+1\frac{2}{3}=4.$

Example

Try it.

Model $1\frac{3}{5}+2\frac{3}{5}$ and give the sum as a mixed number.

Solution

We will use fraction circles, whole circles for the whole numbers and $\frac{1}{5}$ pieces for the fractions.

one whole and three $\frac{1}{5}$s
plus two wholes and three $\frac{1}{5}$s.
sum is three wholes and six $\frac{1}{5}$s

Adding the whole circles and fifth pieces, we got a sum of $3\frac{6}{5}.$ We can see that $\frac{6}{5}$ is equivalent to $1\frac{1}{5},$ so we add that to the $3$ to get $4\frac{1}{5}.$

Add Mixed Numbers

Modeling with fraction circles helps illustrate the process for adding mixed numbers: We add the whole numbers and add the fractions, and then we simplify the result, if possible.

Example

Try it.

Add: $3\frac{4}{9}+2\frac{2}{9}.$

Solution

	$3\frac{4}{9}+2\frac{2}{9}$
Add the whole numbers.
Add the fractions.
Simplify the fraction.

In , the sum of the fractions was a proper fraction. Now we will work through an example where the sum is an improper fraction.

Example

Try it.

Find the sum: $9\frac{5}{9}+5\frac{7}{9}.$

Solution

	$9\frac{5}{9}+5\frac{7}{9}$
Add the whole numbers and then add the fractions.	$\begin{array}{l} \\ 9\frac{5}{9} \\ \underset{\text{_____}}{+\ 5\frac{7}{9}} \\ 14\frac{12}{9}\end{array}$
Rewrite $\frac{12}{9}$ as a mixed number.	$14+1\frac{3}{9}$
Add.	$15\frac{3}{9}$
Simplify.	$15\frac{1}{3}$

An alternate method for adding mixed numbers is to convert the mixed numbers to improper fractions and then add the improper fractions. This method is usually written horizontally.

Example

Try it.

Add by converting the mixed numbers to improper fractions: $3\frac{7}{8}+4\frac{3}{8}.$

Solution

	$3\frac{7}{8}+4\frac{3}{8}$
Convert to improper fractions.	$\frac{31}{8}+\frac{35}{8}$
Add the fractions.	$\frac{31+35}{8}$
Simplify the numerator.	$\frac{66}{8}$
Rewrite as a mixed number.	$8\frac{2}{8}$
Simplify the fraction.	$8\frac{1}{4}$

Since the problem was given in mixed number form, we will write the sum as a mixed number.

compares the two methods of addition, using the expression $3\frac{2}{5}+6\frac{4}{5}$ as an example. Which way do you prefer?

Mixed Numbers	Improper Fractions
$\begin{array}{l} \\ \\ 3\frac{2}{5} \\ \frac{+6\frac{4}{5}}{\ 9\frac{6}{5}} \\ 9+\frac{6}{5} \\ 9+1\frac{1}{5} \\ 10\frac{1}{5}\end{array}$	$\begin{array}{l} \\ 3\frac{2}{5}+6\frac{4}{5} \\ \frac{17}{5}+\frac{34}{5} \\ \frac{51}{5} \\ 10\frac{1}{5}\end{array}$

Model Subtraction of Mixed Numbers

Let’s think of pizzas again to model subtraction of mixed numbers with a common denominator. Suppose you just baked a whole pizza and want to give your brother half of the pizza. What do you have to do to the pizza to give him half? You have to cut it into at least two pieces. Then you can give him half.

We will use fraction circles (pizzas!) to help us visualize the process.

Start with one whole.

Algebraically, you would write:

Example

Try it.

Use a model to subtract: $1-\frac{1}{3}.$

Solution

What if we start with more than one whole? Let’s find out.

Example

Try it.

Use a model to subtract: $2-\frac{3}{4}.$

Solution

In the next example, we’ll subtract more than one whole.

Example

Try it.

Use a model to subtract: $2-1\frac{2}{5}.$

Solution

What if you start with a mixed number and need to subtract a fraction? Think about this situation: You need to put three quarters in a parking meter, but you have only a $\$1$ bill and one quarter. What could you do? You could change the dollar bill into $4$ quarters. The value of $4$ quarters is the same as one dollar bill, but the $4$ quarters are more useful for the parking meter. Now, instead of having a $\$1$ bill and one quarter, you have $5$ quarters and can put $3$ quarters in the meter.

Example

Try it.

Use a model to subtract: $1\frac{1}{4}-\frac{3}{4}$

Solution

Rewrite vertically. Start with one whole and one fourth.
Since the fractions have denominator 4, cut the whole into 4 pieces. You now have $\frac{4}{4}$ and $\frac{1}{4}$ which is $\frac{5}{4}$.
Take away $\frac{3}{4}$. There is $\frac{1}{2}$ left.

Condensed — the full section is in OpenStax Prealgebra 2e.

Subtract Mixed Numbers with a Common Denominator

Now we will subtract mixed numbers without using a model. But it may help to picture the model in your mind as you read the steps.

Example

Try it.

Find the difference: $5\frac{3}{5}-2\frac{4}{5}.$

Solution


Rewrite the problem in vertical form.
Since $\frac{3}{5}$ is less than $\frac{4}{5}$, take 1 from the 5 and add it to the $\frac{3}{5}:(\frac{5}{5}+\frac{3}{5}=\frac{8}{5})$
Subtract the fractions.
Subtract the whole parts. The result is in simplest form.

Since the problem was given with mixed numbers, we leave the result as mixed numbers.

Just as we did with addition, we could subtract mixed numbers by converting them first to improper fractions. We should write the answer in the form it was given, so if we are given mixed numbers to subtract we will write the answer as a mixed number.

Example

Try it.

Find the difference by converting to improper fractions:

$9\frac{6}{11}-7\frac{10}{11}.$

Solution

	$9\frac{6}{11}-7\frac{10}{11}$
Rewrite as improper fractions.	$\frac{105}{11}-\frac{87}{11}$
Subtract the numerators.	$\frac{18}{11}$
Rewrite as a mixed number.	$1\frac{7}{11}$

Add and Subtract Mixed Numbers with Different Denominators

To add or subtract mixed numbers with different denominators, we first convert the fractions to equivalent fractions with the LCD. Then we can follow all the steps we used above for adding or subtracting fractions with like denominators.

Example

Try it.

Add: $2\frac{1}{2}+5\frac{2}{3}.$

Solution

Since the denominators are different, we rewrite the fractions as equivalent fractions with the LCD, $6.$ Then we will add and simplify.

We write the answer as a mixed number because we were given mixed numbers in the problem.

Example

Try it.

Subtract: $4\frac{3}{4}-2\frac{7}{8}.$

Solution

Since the denominators of the fractions are different, we will rewrite them as equivalent fractions with the LCD $8.$ Once in that form, we will subtract. But we will need to borrow $1$ first.

We were given mixed numbers, so we leave the answer as a mixed number.

Example

Try it.

Subtract: $3\frac{5}{11}-4\frac{3}{4}.$

Solution

We can see the answer will be negative since we are subtracting $4$ from $3.$ Generally, when we know the answer will be negative it is easier to subtract with improper fractions rather than mixed numbers.

	$3\frac{5}{11}-4\frac{3}{4}$
Change to equivalent fractions with the LCD.	$3\frac{5\cdot 4}{11\cdot 4}-4\frac{3\cdot 11}{4\cdot 11}$ $3\frac{20}{44}-4\frac{33}{44}$
Rewrite as improper fractions.	$\frac{152}{44}-\frac{209}{44}$
Subtract.	$-\frac{57}{44}$
Rewrite as a mixed number.	$-1\frac{13}{44}$

Key Concepts

Add mixed numbers with a common denominator.
1. Add the whole numbers.
2. Add the fractions.
3. Simplify, if possible.
Subtract mixed numbers with common denominators.
1. Rewrite the problem in vertical form.
2. Compare the two fractions.
  If the top fraction is larger than the bottom fraction, go to Step 3.
  If not, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction.
3. Subtract the fractions.
4. Subtract the whole numbers.
5. Simplify, if possible.
Subtract mixed numbers with common denominators as improper fractions.
1. Rewrite the mixed numbers as improper fractions.
2. Subtract the numerators.
3. Write the answer as a mixed number, simplifying the fraction part, if possible.

Add and Subtract Mixed Numbers

Model Addition of Mixed Numbers

In the following exercises, use a model to find the sum. Draw a picture to illustrate your model.

Try it.

$1\frac{1}{5}+3\frac{1}{5}$

Try it.

$2\frac{1}{3}+1\frac{1}{3}$

Solution

$3\frac{2}{3}$

Try it.

$1\frac{3}{8}+1\frac{7}{8}$

Try it.

$1\frac{5}{6}+1\frac{5}{6}$

Solution

$3\frac{2}{3}$

Add Mixed Numbers with a Common Denominator

In the following exercises, add.

Try it.

$5\frac{1}{3}+6\frac{1}{3}$

Try it.

$2\frac{4}{9}+5\frac{1}{9}$

Solution

$7\frac{5}{9}$

Try it.

$4\frac{5}{8}+9\frac{3}{8}$

Try it.

$7\frac{9}{10}+3\frac{1}{10}$

Solution

Try it.

$3\frac{4}{5}+6\frac{4}{5}$

Try it.

$9\frac{2}{3}+1\frac{2}{3}$

Solution

$11\frac{1}{3}$

Try it.

$6\frac{9}{10}+8\frac{3}{10}$

Try it.

$8\frac{4}{9}+2\frac{8}{9}$

Solution

$11\frac{1}{3}$

Model Subtraction of Mixed Numbers

In the following exercises, use a model to find the difference. Draw a picture to illustrate your model.

Try it.

$1\frac{1}{6}-\frac{5}{6}$

Try it.

$1\frac{1}{8}-\frac{5}{8}$

Solution

$\frac{1}{2}$

Subtract Mixed Numbers with a Common Denominator

In the following exercises, find the difference.

Try it.

$2\frac{7}{8}-1\frac{3}{8}$

Try it.

$2\frac{7}{12}-1\frac{5}{12}$

Solution

$1\frac{1}{6}$

Try it.

$8\frac{17}{20}-4\frac{9}{20}$

Try it.

$19\frac{13}{15}-13\frac{7}{15}$

Solution

$6\frac{2}{5}$

Try it.

$8\frac{3}{7}-4\frac{4}{7}$

Try it.

$5\frac{2}{9}-3\frac{4}{9}$

Solution

$1\frac{7}{9}$

Try it.

$2\frac{5}{8}-1\frac{7}{8}$

Try it.

$2\frac{5}{12}-1\frac{7}{12}$

Solution

$\frac{5}{6}$

Add and Subtract Mixed Numbers with Different Denominators

In the following exercises, write the sum or difference as a mixed number in simplified form.

Try it.

$3\frac{1}{4}+6\frac{1}{3}$

Try it.

$2\frac{1}{6}+5\frac{3}{4}$

Solution

$7\frac{11}{12}$

Try it.

$1\frac{5}{8}+4\frac{1}{2}$

Try it.

$7\frac{2}{3}+8\frac{1}{2}$

Solution

$16\frac{1}{6}$

Try it.

$9\frac{7}{10}-2\frac{1}{3}$

Try it.

$6\frac{4}{5}-1\frac{1}{4}$

Solution

$5\frac{11}{20}$

Try it.

$2\frac{2}{3}-3\frac{1}{2}$

Try it.

$2\frac{7}{8}-4\frac{1}{3}$

Solution

$-1\frac{11}{24}$

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

Try it.

$2\frac{5}{8}\cdot 1\frac{3}{4}$

Try it.

$1\frac{2}{3}\cdot 4\frac{1}{6}$

Solution

$6\frac{17}{18}$

Try it.

$\frac{2}{7}+\frac{4}{7}$

Try it.

$\frac{2}{9}+\frac{5}{9}$

Solution

$\frac{7}{9}$

Try it.

$1\frac{5}{12}\div \frac{1}{12}$

Try it.

$2\frac{3}{10}\div \frac{1}{10}$

Solution

Try it.

$13\frac{5}{12}-9\frac{7}{12}$

Try it.

$15\frac{5}{8}-6\frac{7}{8}$

Solution

$8\frac{3}{4}$

Try it.

$\frac{5}{9}-\frac{4}{9}$

Try it.

$\frac{11}{15}-\frac{7}{15}$

Solution

$\frac{4}{15}$

Try it.

$4-\frac{3}{4}$

Try it.

$6-\frac{2}{5}$

Solution

$5\frac{3}{5}$

Try it.

$\frac{9}{20}\div \frac{3}{4}$

Try it.

$\frac{7}{24}\div \frac{14}{3}$

Solution

$\frac{1}{16}$

Try it.

$9\frac{6}{11}+7\frac{10}{11}$

Try it.

$8\frac{5}{13}+4\frac{9}{13}$

Solution

$13\frac{1}{13}$

Try it.

$3\frac{2}{5}+5\frac{3}{4}$

Try it.

$2\frac{5}{6}+4\frac{1}{5}$

Solution

$7\frac{1}{30}$

Try it.

$\frac{8}{15}\cdot \frac{10}{19}$

Try it.

$\frac{5}{12}\cdot \frac{8}{9}$

Solution

$\frac{10}{27}$

Try it.

$6\frac{7}{8}-2\frac{1}{3}$

Try it.

$6\frac{5}{9}-4\frac{2}{5}$

Solution

$2\frac{7}{45}$

Try it.

$5\frac{2}{9}-4\frac{4}{5}$

Try it.

$4\frac{3}{8}-3\frac{2}{3}$

Solution

$\frac{17}{24}$

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Draw a model of the fraction $\frac{7}{3}.$
If you missed this problem, review .

उत्तर उघडा
Change $\frac{11}{4}$ to a mixed number.
If you missed this problem, review .

उत्तर उघडा

$2\frac{3}{4}$
Change $3\frac{1}{2}$ to an improper fraction.
If you missed this problem, review .

उत्तर उघडा

$\frac{7}{2}$
Model $2\frac{1}{3}+1\frac{2}{3}$ and give the sum.

उत्तर उघडा

We will use fraction circles, whole circles for the whole numbers and $\frac{1}{3}$ pieces for the fractions.
two wholes and one $\frac{1}{3}$
plus one whole and two $\frac{1}{3}$s
sum is three wholes and three $\frac{1}{3}$s
This is the same as $4$ wholes. So, $2\frac{1}{3}+1\frac{2}{3}=4.$
$1\frac{2}{5}+3\frac{3}{5}$

उत्तर उघडा

5
$2\frac{1}{6}+2\frac{5}{6}$

उत्तर उघडा

5
Model $1\frac{3}{5}+2\frac{3}{5}$ and give the sum as a mixed number.

उत्तर उघडा

We will use fraction circles, whole circles for the whole numbers and $\frac{1}{5}$ pieces for the fractions.
one whole and three $\frac{1}{5}$s
plus two wholes and three $\frac{1}{5}$s.
sum is three wholes and six $\frac{1}{5}$s
Adding the whole circles and fifth pieces, we got a sum of $3\frac{6}{5}.$ We can see that $\frac{6}{5}$ is equivalent to $1\frac{1}{5},$ so we add that to the $3$ to get $4\frac{1}{5}.$
Model, and give the sum as a mixed number. Draw a picture to illustrate your model.
$2\frac{5}{6}+1\frac{5}{6}$

उत्तर उघडा

$4\frac{2}{3}$
Model, and give the sum as a mixed number. Draw a picture to illustrate your model.
$1\frac{5}{8}+1\frac{7}{8}$

उत्तर उघडा

$3\frac{1}{2}$
Add: $3\frac{4}{9}+2\frac{2}{9}.$

उत्तर उघडा

$3\frac{4}{9}+2\frac{2}{9}$
Add the whole numbers.
Add the fractions.
Simplify the fraction.
Find the sum: $4\frac{4}{7}+1\frac{2}{7}.$

उत्तर उघडा

$5\frac{6}{7}$
Find the sum: $2\frac{3}{11}+5\frac{6}{11}.$

उत्तर उघडा

$7\frac{9}{11}$

Find the sum: $9\frac{5}{9}+5\frac{7}{9}.$

उत्तर उघडा

	$9\frac{5}{9}+5\frac{7}{9}$
Add the whole numbers and then add the fractions.	$\begin{array}{l} \\ 9\frac{5}{9} \\ \underset{\text{_____}}{+\ 5\frac{7}{9}} \\ 14\frac{12}{9}\end{array}$
Rewrite $\frac{12}{9}$ as a mixed number.	$14+1\frac{3}{9}$
Add.	$15\frac{3}{9}$
Simplify.	$15\frac{1}{3}$

Find the sum: $8\frac{7}{8}+7\frac{5}{8}.$

उत्तर उघडा

$16\frac{1}{2}$
Find the sum: $6\frac{7}{9}+8\frac{5}{9}.$

उत्तर उघडा

$15\frac{1}{3}$

Add by converting the mixed numbers to improper fractions: $3\frac{7}{8}+4\frac{3}{8}.$

उत्तर उघडा

	$3\frac{7}{8}+4\frac{3}{8}$
Convert to improper fractions.	$\frac{31}{8}+\frac{35}{8}$
Add the fractions.	$\frac{31+35}{8}$
Simplify the numerator.	$\frac{66}{8}$
Rewrite as a mixed number.	$8\frac{2}{8}$
Simplify the fraction.	$8\frac{1}{4}$

Since the problem was given in mixed number form, we will write the sum as a mixed number.

Find the sum by converting the mixed numbers to improper fractions:
$5\frac{5}{9}+3\frac{7}{9}.$

उत्तर उघडा

$9\frac{1}{3}$
Find the sum by converting the mixed numbers to improper fractions:
$3\frac{7}{10}+2\frac{9}{10}.$

उत्तर उघडा

$6\frac{3}{5}$
Use a model to subtract: $1-\frac{1}{3}.$

उत्तर उघडा
Use a model to subtract: $1-\frac{1}{4}.$

उत्तर उघडा

$\frac{3}{4}$
Use a model to subtract: $1-\frac{1}{5}.$

उत्तर उघडा

$\frac{4}{5}$
Use a model to subtract: $2-\frac{3}{4}.$

उत्तर उघडा
Use a model to subtract: $2-\frac{1}{5}.$

उत्तर उघडा

$\frac{9}{5}$
Use a model to subtract: $2-\frac{1}{3}.$

उत्तर उघडा

$\frac{5}{3}$
Use a model to subtract: $2-1\frac{2}{5}.$

उत्तर उघडा
Use a model to subtract: $2-1\frac{1}{3}.$

उत्तर उघडा

$\frac{2}{3}$
Use a model to subtract: $2-1\frac{1}{4}.$

उत्तर उघडा

$\frac{3}{4}$

Use a model to subtract: $1\frac{1}{4}-\frac{3}{4}$

उत्तर उघडा

Rewrite vertically. Start with one whole and one fourth.
Since the fractions have denominator 4, cut the whole into 4 pieces. You now have $\frac{4}{4}$ and $\frac{1}{4}$ which is $\frac{5}{4}$.
Take away $\frac{3}{4}$. There is $\frac{1}{2}$ left.

Use a model to subtract. Draw a picture to illustrate your model.
$1\frac{1}{3}-\frac{2}{3}$

उत्तर उघडा

$\frac{2}{3}$
Use a model to subtract. Draw a picture to illustrate your model.
$1\frac{1}{5}-\frac{4}{5}$

उत्तर उघडा

$\frac{2}{5}$

Find the difference: $5\frac{3}{5}-2\frac{4}{5}.$

उत्तर उघडा


Rewrite the problem in vertical form.
Since $\frac{3}{5}$ is less than $\frac{4}{5}$, take 1 from the 5 and add it to the $\frac{3}{5}:(\frac{5}{5}+\frac{3}{5}=\frac{8}{5})$
Subtract the fractions.
Subtract the whole parts. The result is in simplest form.

Since the problem was given with mixed numbers, we leave the result as mixed numbers.

Find the difference: $6\frac{4}{9}-3\frac{7}{9}.$

उत्तर उघडा

$2\frac{2}{3}$
Find the difference: $4\frac{4}{7}-2\frac{6}{7}.$

उत्तर उघडा

$1\frac{5}{7}$
Find the difference by converting to improper fractions:
$9\frac{6}{11}-7\frac{10}{11}.$

उत्तर उघडा

$9\frac{6}{11}-7\frac{10}{11}$
Rewrite as improper fractions. $\frac{105}{11}-\frac{87}{11}$
Subtract the numerators. $\frac{18}{11}$
Rewrite as a mixed number. $1\frac{7}{11}$
Find the difference by converting the mixed numbers to improper fractions:
$6\frac{4}{9}-3\frac{7}{9}.$

उत्तर उघडा

$2\frac{2}{3}$
Find the difference by converting the mixed numbers to improper fractions:
$4\frac{4}{7}-2\frac{6}{7}.$

उत्तर उघडा

$1\frac{5}{7}$
Add: $2\frac{1}{2}+5\frac{2}{3}.$

उत्तर उघडा

Since the denominators are different, we rewrite the fractions as equivalent fractions with the LCD, $6.$ Then we will add and simplify.
We write the answer as a mixed number because we were given mixed numbers in the problem.
Add: $1\frac{5}{6}+4\frac{3}{4}.$

उत्तर उघडा

$6\frac{7}{12}$
Add: $3\frac{4}{5}+8\frac{1}{2}.$

उत्तर उघडा

$12\frac{3}{10}$
Subtract: $4\frac{3}{4}-2\frac{7}{8}.$

उत्तर उघडा

Since the denominators of the fractions are different, we will rewrite them as equivalent fractions with the LCD $8.$ Once in that form, we will subtract. But we will need to borrow $1$ first.
We were given mixed numbers, so we leave the answer as a mixed number.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Add and Subtract Mixed Numbers

Model addition of mixed numbers with a common denominator
Add mixed numbers with a common denominator
Model subtraction of mixed numbers
Subtract mixed numbers with a common denominator
Add and subtract mixed numbers with different denominators
Rewrite the problem in vertical form.
Compare the two fractions.
If the top fraction is larger than the bottom fraction, go to Step 3.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

स्वतःचा प्रयत्न करा

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

अधिक माहिती Arithmetic

Order of operations Fractions Percentages Powers and roots

		\(1\frac{1}{4}+2\frac{1}{4}\)
Start with \(1\frac{1}{4}\).	one whole and one \(\frac{1}{4}\) pieces
Add \(2\frac{1}{4}\) more.	two wholes and one \(\frac{1}{4}\) pieces
The sum is:	three wholes and two \(\frac{1}{4}\)'s

	\(9\frac{5}{9}+5\frac{7}{9}\)
Add the whole numbers and then add the fractions.	\(\begin{array}{l} \\ 9\frac{5}{9} \\ \underset{\text{_____}}{+\ 5\frac{7}{9}} \\ 14\frac{12}{9}\end{array}\)
Rewrite \(\frac{12}{9}\) as a mixed number.	\(14+1\frac{3}{9}\)
Add.	\(15\frac{3}{9}\)
Simplify.	\(15\frac{1}{3}\)

	\(3\frac{7}{8}+4\frac{3}{8}\)
Convert to improper fractions.	\(\frac{31}{8}+\frac{35}{8}\)
Add the fractions.	\(\frac{31+35}{8}\)
Simplify the numerator.	\(\frac{66}{8}\)
Rewrite as a mixed number.	\(8\frac{2}{8}\)
Simplify the fraction.	\(8\frac{1}{4}\)

	\(9\frac{6}{11}-7\frac{10}{11}\)
Rewrite as improper fractions.	\(\frac{105}{11}-\frac{87}{11}\)
Subtract the numerators.	\(\frac{18}{11}\)
Rewrite as a mixed number.	\(1\frac{7}{11}\)

	\(3\frac{5}{11}-4\frac{3}{4}\)
Change to equivalent fractions with the LCD.	\(3\frac{5\cdot 4}{11\cdot 4}-4\frac{3\cdot 11}{4\cdot 11}\) \(3\frac{20}{44}-4\frac{33}{44}\)
Rewrite as improper fractions.	\(\frac{152}{44}-\frac{209}{44}\)
Subtract.	\(-\frac{57}{44}\)
Rewrite as a mixed number.	\(-1\frac{13}{44}\)

two wholes and one \(\frac{1}{3}\)
plus one whole and two \(\frac{1}{3}\)s
sum is three wholes and three \(\frac{1}{3}\)s

one whole and three \(\frac{1}{5}\)s
plus two wholes and three \(\frac{1}{5}\)s.
sum is three wholes and six \(\frac{1}{5}\)s

	\(3\frac{4}{9}+2\frac{2}{9}\)
Add the whole numbers.
Add the fractions.
Simplify the fraction.

Mixed Numbers	Improper Fractions
\(\begin{array}{l} \\ \\ 3\frac{2}{5} \\ \frac{+6\frac{4}{5}}{\ 9\frac{6}{5}} \\ 9+\frac{6}{5} \\ 9+1\frac{1}{5} \\ 10\frac{1}{5}\end{array}\)	\(\begin{array}{l} \\ 3\frac{2}{5}+6\frac{4}{5} \\ \frac{17}{5}+\frac{34}{5} \\ \frac{51}{5} \\ 10\frac{1}{5}\end{array}\)

Rewrite vertically. Start with one whole and one fourth.
Since the fractions have denominator 4, cut the whole into 4 pieces. You now have \(\frac{4}{4}\) and \(\frac{1}{4}\) which is \(\frac{5}{4}\).
Take away \(\frac{3}{4}\). There is \(\frac{1}{2}\) left.


Rewrite the problem in vertical form.
Since \(\frac{3}{5}\) is less than \(\frac{4}{5}\), take 1 from the 5 and add it to the \(\frac{3}{5}:(\frac{5}{5}+\frac{3}{5}=\frac{8}{5})\)
Subtract the fractions.
Subtract the whole parts. The result is in simplest form.