# 1.10 Fractions: order of operations

 Page 1 / 1
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses combinations of operations with fractions. By the end of the module students should gain a further understanding of the order of operations.

## Section overview

• The Order of Operations

## The order of operations

To determine the value of a quantity such as

$\frac{1}{2}+\frac{5}{8}\cdot \frac{2}{\text{15}}$

where we have a combination of operations (more than one operation occurs), we must use the accepted order of operations.

## The order of operations:

1. In the order (2), (3), (4) described below, perform all operations inside group­ing symbols: ( ), [ ], ( ), . Work from the innermost set to the outermost set.
2. Perform exponential and root operations.
3. Perform all multiplications and divisions moving left to right.
4. Perform all additions and subtractions moving left to right.

## Sample set a

Determine the value of each of the following quantities.

$\frac{1}{4}+\frac{5}{8}\cdot \frac{2}{\text{15}}$

1. Multiply first.

$\frac{1}{4}+\frac{\stackrel{1}{\overline{)5}}}{\underset{4}{\overline{)8}}}\cdot \frac{\stackrel{1}{\overline{)2}}}{\underset{3}{\overline{)\text{15}}}}=\frac{1}{4}+\frac{1\cdot 1}{4\cdot 3}=\frac{1}{4}+\frac{1}{\text{12}}$

2. Now perform this addition. Find the LCD.

$\left(\begin{array}{}4={2}^{2}\\ \text{12}={2}^{2}\cdot 3\end{array}}\text{The LCD}={2}^{2}\cdot 3=\text{12}\text{.}$

$\begin{array}{ccc}\hfill \frac{1}{4}+\frac{1}{\text{12}}& =& \frac{1\cdot 3}{\text{12}}+\frac{1}{\text{12}}=\frac{3}{\text{12}}+\frac{1}{\text{12}}\hfill \\ & =& \frac{3+1}{\text{12}}=\frac{4}{\text{12}}=\frac{1}{3}\hfill \end{array}$

Thus, $\frac{1}{4}+\frac{5}{8}\cdot \frac{2}{\text{15}}=\frac{1}{3}$

$\frac{3}{5}+\frac{9}{\text{44}}\left(\frac{5}{9}-\frac{1}{4}\right)$

1. Operate within the parentheses first, $\left(\frac{5}{9}-\frac{1}{4}\right)$ .

$\left(\begin{array}{}9={3}^{2}\\ 4={2}^{2}\end{array}}\text{The LCD}={2}^{2}\cdot {3}^{2}=4\cdot 9=\text{36}\text{.}$

$\frac{5\cdot 4}{\text{36}}-\frac{1\cdot 9}{\text{36}}=\frac{\text{20}}{\text{36}}-\frac{9}{\text{36}}=\frac{\text{20}-9}{\text{36}}=\frac{\text{11}}{\text{36}}$

Now we have

$\frac{3}{5}+\frac{9}{\text{44}}\left(\frac{\text{11}}{\text{36}}\right)$

2. Perform the multiplication.

$\frac{3}{5}+\frac{\stackrel{1}{\overline{)9}}}{\underset{4}{\overline{)\text{44}}}}\cdot \frac{\stackrel{1}{\overline{)\text{11}}}}{\underset{4}{\overline{)\text{36}}}}=\frac{3}{5}+\frac{1\cdot 1}{4\cdot 4}=\frac{3}{5}+\frac{1}{\text{16}}$

3. Now perform the addition. The LCD=80.

$\frac{3}{5}+\frac{1}{\text{16}}=\frac{3\cdot \text{16}}{\text{80}}+\frac{1\cdot 5}{\text{80}}=\frac{\text{48}}{\text{80}}+\frac{5}{\text{80}}=\frac{\text{48}+5}{\text{80}}=\frac{\text{53}}{\text{80}}$

Thus, $\frac{3}{5}+\frac{9}{\text{44}}\left(\frac{5}{9}-\frac{1}{4}\right)=\frac{\text{53}}{\text{80}}$

$8-\frac{\text{15}}{\text{426}}\left(2-1\frac{4}{\text{15}}\right)\left(3\frac{1}{5}+2\frac{1}{8}\right)$

1. Work within each set of parentheses individually.

$\begin{array}{ccc}2-1\frac{4}{\text{15}}& =& 2\frac{1\cdot \text{15}+4}{\text{15}}=2-\frac{\text{19}}{\text{15}}\hfill \\ & =& \frac{\text{30}}{\text{15}}-\frac{\text{19}}{\text{15}}=\frac{\text{30}-\text{19}}{\text{15}}=\frac{\text{11}}{\text{15}}\hfill \\ 3\frac{1}{5}+2\frac{1}{8}& =& \frac{3\cdot 5+1}{5}+\frac{2\cdot 8+1}{8}\hfill \\ & =& \frac{\text{16}}{5}+\frac{\text{17}}{8}\text{LCD}=\text{40}\hfill \\ & =& \frac{\text{16}\cdot 8}{\text{40}}+\frac{\text{17}\cdot 5}{\text{40}}\hfill \\ & =& \frac{\text{128}}{\text{40}}+\frac{\text{85}}{\text{40}}\hfill \\ & =& \frac{\text{128}+\text{85}}{\text{40}}\hfill \\ & =& \frac{\text{213}}{\text{40}}\hfill \end{array}$

Now we have

$8-\frac{\text{15}}{\text{426}}\left(\frac{\text{11}}{\text{15}}\right)\left(\frac{\text{213}}{\text{40}}\right)$

2. Now multiply.

$8-\frac{\stackrel{1}{\overline{)\text{15}}}}{\underset{2}{\overline{)\text{426}}}}\cdot \frac{\text{11}}{\underset{1}{\overline{)\text{15}}}}\cdot \frac{\stackrel{1}{\overline{)\text{213}}}}{\text{40}}=8-\frac{1\cdot \text{11}\cdot 1}{2\cdot 1\cdot \text{40}}=8-\frac{\text{11}}{\text{80}}$

3. Now subtract.

$8-\frac{\text{11}}{\text{80}}=\frac{\text{80}\cdot 8}{\text{80}}-\frac{\text{11}}{\text{80}}=\frac{\text{640}}{\text{80}}-\frac{\text{11}}{\text{80}}=\frac{\text{640}-\text{11}}{\text{80}}=\frac{\text{629}}{\text{80}}\phantom{\rule{4px}{0ex}}\text{or}\phantom{\rule{4px}{0ex}}7\frac{\text{69}}{\text{80}}$

Thus, $8-\frac{15}{426}\left(2-1\frac{4}{15}\right)\left(3\frac{1}{5}+2\frac{1}{8}\right)=7\frac{69}{80}$

${\left(\frac{3}{4}\right)}^{2}\cdot \frac{8}{9}-\frac{5}{\text{12}}$

1. Square $\frac{3}{4}$ .

${\left(\frac{3}{4}\right)}^{2}=\frac{3}{4}\cdot \frac{3}{4}=\frac{3\cdot 3}{4\cdot 4}=\frac{9}{\text{16}}$

Now we have

$\frac{9}{\text{16}}\cdot \frac{8}{9}-\frac{5}{\text{12}}$

2. Perform the multiplication.

$\frac{\stackrel{1}{\overline{)9}}}{\underset{2}{\overline{)\text{16}}}}\cdot \frac{\stackrel{1}{\overline{)8}}}{\underset{1}{\overline{)9}}}-\frac{5}{\text{12}}=\frac{1\cdot 1}{2\cdot 1}-\frac{5}{\text{12}}=\frac{1}{2}-\frac{5}{\text{12}}$

3. Now perform the subtraction.

$\frac{1}{2}-\frac{5}{\text{12}}=\frac{6}{\text{12}}-\frac{5}{\text{12}}=\frac{6-5}{\text{12}}=\frac{1}{\text{12}}$

Thus, ${\left(\frac{4}{3}\right)}^{2}\cdot \frac{8}{9}-\frac{5}{\text{12}}=\frac{1}{\text{12}}$

$2\frac{7}{8}+\sqrt{\frac{\text{25}}{\text{36}}}÷\left(2\frac{1}{2}-1\frac{1}{3}\right)$

1. Begin by operating inside the parentheses.

$\begin{array}{ccc}2\frac{1}{2}-1\frac{1}{3}& =& \frac{2\cdot 2+1}{2}-\frac{1\cdot 3+1}{3}=\frac{5}{2}-\frac{4}{3}\hfill \\ & =& \frac{\text{15}}{6}-\frac{8}{6}=\frac{\text{15}-8}{6}=\frac{7}{6}\hfill \end{array}$

2. Now simplify the square root.

$\sqrt{\frac{\text{25}}{\text{36}}}=\frac{5}{6}\left(\text{since}{\left(\frac{5}{6}\right)}^{2}=\frac{\text{25}}{\text{36}}\right)$

Now we have

$2\frac{7}{8}+\frac{5}{6}÷\frac{7}{6}$

3. Perform the division.

$2\frac{7}{8}+\frac{5}{\underset{1}{\overline{)6}}}\cdot \frac{\stackrel{1}{\overline{)6}}}{7}=2\frac{7}{8}+\frac{5\cdot 1}{1\cdot 7}=2\frac{7}{8}+\frac{5}{7}$

Thus, $2\frac{7}{8}+\sqrt{\frac{\text{25}}{\text{36}}}÷\left(2\frac{1}{2}-1\frac{1}{3}\right)=3\frac{\text{33}}{\text{56}}$

## Practice set a

Find the value of each of the following quantities.

$\frac{5}{\text{16}}\cdot \frac{1}{\text{10}}-\frac{1}{\text{32}}$

0

$\frac{6}{7}\cdot \frac{\text{21}}{\text{40}}÷\frac{9}{\text{10}}+5\frac{1}{3}$

$\frac{\text{35}}{6}$ or $5\frac{5}{\text{6}}$

$8\frac{7}{\text{10}}-2\left(4\frac{1}{2}-3\frac{2}{3}\right)$

$\frac{\text{211}}{\text{30}}$ or $7\frac{1}{\text{30}}$

$\frac{\text{17}}{\text{18}}-\frac{\text{58}}{\text{30}}\left(\frac{1}{4}-\frac{3}{\text{32}}\right)\left(1-\frac{\text{13}}{\text{29}}\right)$

$\frac{7}{9}$

$\left(\frac{1}{\text{10}}+1\frac{1}{2}\right)÷\left(1\frac{4}{5}-1\frac{6}{\text{25}}\right)$

$2\frac{6}{7}$

$\frac{\frac{2}{3}-\frac{3}{8}\cdot \frac{4}{9}}{\frac{7}{\text{16}}\cdot 1\frac{1}{3}+1\frac{1}{4}}$

$\frac{3}{\text{11}}$

${\left(\frac{3}{8}\right)}^{2}+\frac{3}{4}\cdot \frac{1}{8}$

$\frac{\text{15}}{\text{64}}$

$\frac{2}{3}\cdot 2\frac{1}{4}-\sqrt{\frac{4}{\text{25}}}$

$\frac{\text{11}}{\text{10}}$

## Exercises

Find each value.

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

$\frac{5}{4}$

$\frac{7}{9}-\frac{4}{5}\cdot \frac{5}{\text{36}}$

$2\frac{2}{7}+\frac{5}{8}÷\frac{5}{\text{16}}$

$4\frac{2}{7}$

$\frac{3}{\text{16}}÷\frac{9}{\text{14}}\cdot \frac{\text{12}}{\text{21}}+\frac{5}{6}$

$\frac{4}{\text{25}}÷\frac{8}{\text{15}}-\frac{7}{\text{20}}÷2\frac{1}{\text{10}}$

$\frac{2}{\text{15}}$

$\frac{2}{5}\cdot \left(\frac{1}{\text{19}}+\frac{3}{\text{38}}\right)$

$\frac{3}{7}\cdot \left(\frac{3}{\text{10}}-\frac{1}{\text{15}}\right)$

$\frac{1}{\text{10}}$

$\frac{\text{10}}{\text{11}}\cdot \left(\frac{8}{9}-\frac{2}{5}\right)+\frac{3}{\text{25}}\cdot \left(\frac{5}{3}+\frac{1}{4}\right)$

$\frac{2}{7}\cdot \left(\frac{6}{7}-\frac{3}{\text{28}}\right)+5\frac{1}{3}\cdot \left(1\frac{1}{4}-\frac{1}{8}\right)$

$6\frac{3}{\text{14}}$

$\frac{\left(\frac{6}{\text{11}}-\frac{1}{3}\right)\cdot \left(\frac{1}{\text{21}}+2\frac{\text{13}}{\text{42}}\right)}{1\frac{1}{5}+\frac{7}{\text{40}}}$

${\left(\frac{1}{2}\right)}^{2}+\frac{1}{8}$

$\frac{3}{8}$

${\left(\frac{3}{5}\right)}^{2}-\frac{3}{\text{10}}$

$\sqrt{\frac{\text{36}}{\text{81}}}+\frac{1}{3}\cdot \frac{2}{9}$

$\frac{\text{20}}{\text{27}}$

$\sqrt{\frac{\text{49}}{\text{64}}}-\sqrt{\frac{9}{4}}$

$\frac{2}{3}\cdot \sqrt{\frac{9}{4}}-\frac{\text{15}}{4}\cdot \sqrt{\frac{\text{16}}{\text{225}}}$

0

${\left(\frac{3}{4}\right)}^{2}+\sqrt{\frac{\text{25}}{\text{16}}}$

${\left(\frac{1}{3}\right)}^{2}\cdot \sqrt{\frac{\text{81}}{\text{25}}}+\frac{1}{\text{40}}÷\frac{1}{8}$

$\frac{2}{5}$

${\left(\sqrt{\frac{4}{\text{49}}}\right)}^{2}+\frac{3}{7}÷1\frac{3}{4}$

${\left(\sqrt{\frac{\text{100}}{\text{121}}}\right)}^{2}+\frac{\text{21}}{{\left(\text{11}\right)}^{2}}$

1

$\sqrt{\frac{3}{8}+\frac{1}{\text{64}}}-\frac{1}{2}÷1\frac{1}{3}$

$\sqrt{\frac{1}{4}}\cdot {\left(\frac{5}{6}\right)}^{2}+\frac{9}{\text{14}}\cdot 2\frac{1}{3}-\sqrt{\frac{1}{\text{81}}}$

$\frac{\text{125}}{\text{72}}$

$\sqrt{\frac{1}{9}}\cdot \sqrt{\frac{6\frac{3}{8}+2\frac{5}{8}}{\text{16}}}+7\frac{7}{\text{10}}$

$\frac{3\frac{3}{4}+\frac{4}{5}\cdot {\left(\frac{1}{2}\right)}^{3}}{\frac{\text{67}}{\text{240}}+{\left(\frac{1}{3}\right)}^{4}\cdot \left(\frac{9}{\text{10}}\right)}$

$\frac{\text{252}}{\text{19}}$

$\sqrt{\sqrt{\frac{\text{16}}{\text{81}}}}+\frac{1}{4}\cdot 6$

$\sqrt{\sqrt{\frac{\text{81}}{\text{256}}}}-\frac{3}{\text{32}}\cdot 1\frac{1}{8}$

$\frac{\text{165}}{\text{256}}$

## Exercises for review

( [link] ) True or false: Our number system, the Hindu-Arabic number system, is a positional number system with base ten.

( [link] ) The fact that 1 times any whole number = that particular whole number illustrates which property of multiplication?

multiplicative identity

( [link] ) Convert $8\frac{6}{7}$ to an improper fraction.

( [link] ) Find the sum. $\frac{3}{8}+\frac{4}{5}+\frac{5}{6}$ .

$\frac{\text{241}}{\text{120}}$ or $2\frac{1}{\text{120}}$

( [link] ) Simplify $\frac{6+\frac{1}{8}}{6-\frac{1}{8}}$ .

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!