# 5.1 Addition and subtraction of fractions with like denominators

 Page 1 / 1
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to add and subtract fractions with like denominators. By the end of the module students should be able to add and subtract fractions with like denominators.

## Section overview

• Addition of Fraction With Like Denominators
• Subtraction of Fractions With Like Denominators

## Addition of fraction with like denominators

Let's examine the following diagram.

2 one-fifths and 1 one fifth is shaded.

It is shown in the shaded regions of the diagram that

(2 one-fifths) + (1 one-fifth) = (3 one-fifths)

That is,

$\frac{2}{5}+\frac{1}{5}=\frac{3}{5}$

From this observation, we can suggest the following rule.

## Method of adding fractions having like denominators

To add two or more fractions that have the same denominators, add the numer­ators and place the resulting sum over the common denominator. Reduce, if necessary.

## Sample set a

Find the following sums.

$\frac{3}{7}+\frac{2}{7}$ . The denominators are the same. Add the numerators and place that sum over 7.

$\frac{3}{7}+\frac{2}{7}=\frac{3+2}{7}=\frac{5}{7}$

$\frac{1}{8}+\frac{3}{8}$ . The denominators are the same. Add the numerators and place the sum over 8. Reduce.

$\frac{1}{8}+\frac{3}{8}=\frac{1+3}{8}=\frac{4}{8}=\frac{1}{2}$

$\frac{4}{9}+\frac{5}{9}$ . The denominators are the same. Add the numerators and place the sum over 9.

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

$\frac{7}{8}+\frac{5}{8}$ . The denominators are the same. Add the numerators and place the sum over 8.

$\frac{7}{8}+\frac{5}{8}=\frac{7+5}{8}=\frac{\text{12}}{8}=\frac{3}{2}$

To see what happens if we mistakenly add the denominators as well as the numerators, let's add

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

$\frac{1}{2}+\frac{1}{2}=\frac{1+1}{2+2}=\frac{2}{4}=\frac{1}{2}$

This means that two $\frac{1}{2}$ 's is the same as one $\frac{1}{2}$ . Preposterous! We do not add denominators .

## Practice set a

Find the following sums.

$\frac{1}{\text{10}}+\frac{3}{\text{10}}$

$\frac{2}{5}$

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

$\frac{1}{2}$

$\frac{7}{\text{11}}+\frac{4}{\text{11}}$

1

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

$\frac{4}{5}$

Show why adding both the numerators and denominators is preposterous by adding $\frac{3}{4}$ and $\frac{3}{4}$ and examining the result.

$\frac{3}{4}+\frac{3}{4}=\frac{3+3}{4+4}=\frac{6}{8}=\frac{3}{4}$ , so two $\frac{3}{4}$ ’s= one $\frac{3}{4}$ which is preposterous.

## Subtraction of fractions with like denominators

We can picture the concept of subtraction of fractions in much the same way we pictured addition.

From this observation, we can suggest the following rule for subtracting fractions having like denominators:

## Subtraction of fractions with like denominators

To subtract two fractions that have like denominators, subtract the numerators and place the resulting difference over the common denominator. Reduce, if possible.

## Sample set b

Find the following differences.

$\frac{3}{5}-\frac{1}{5}$ . The denominators are the same. Subtract the numerators. Place the difference over 5.

$\frac{3}{5}-\frac{1}{5}=\frac{3-1}{5}=\frac{2}{5}$

$\frac{8}{6}-\frac{2}{6}$ . The denominators are the same. Subtract the numerators. Place the difference over 6.

$\frac{8}{6}-\frac{2}{6}=\frac{8-2}{6}=\frac{6}{6}=1$

$\frac{\text{16}}{9}-\frac{2}{9}$ . The denominators are the same. Subtract numerators and place the difference over 9.

$\frac{\text{16}}{9}-\frac{2}{9}=\frac{\text{16}-2}{9}=\frac{\text{14}}{9}$

To see what happens if we mistakenly subtract the denominators, let's consider

$\frac{7}{\text{15}}-\frac{4}{\text{15}}=\frac{7-4}{\text{15}-\text{15}}=\frac{3}{0}$

We get division by zero, which is undefined. We do not subtract denominators.

## Practice set b

Find the following differences.

$\frac{\text{10}}{\text{13}}-\frac{8}{\text{13}}$

$\frac{2}{\text{13}}$

$\frac{5}{\text{12}}-\frac{1}{\text{12}}$

$\frac{1}{3}$

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

0

$\frac{\text{26}}{\text{10}}-\frac{\text{14}}{\text{10}}$

$\frac{6}{5}$

Show why subtracting both the numerators and the denominators is in error by performing the subtraction $\frac{5}{9}-\frac{2}{9}$ .

$\frac{5}{9}-\frac{2}{9}=\frac{5-2}{9-9}=\frac{3}{0}$ , which is undefined

## Exercises

For the following problems, find the sums and differences. Be sure to reduce.

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

$\frac{5}{8}$

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

$\frac{9}{\text{10}}+\frac{1}{\text{10}}$

1

$\frac{3}{\text{11}}+\frac{4}{\text{11}}$

$\frac{9}{\text{15}}+\frac{4}{\text{15}}$

$\frac{\text{13}}{\text{15}}$

$\frac{3}{\text{10}}+\frac{2}{\text{10}}$

$\frac{5}{\text{12}}+\frac{7}{\text{12}}$

1

$\frac{\text{11}}{\text{16}}-\frac{2}{\text{16}}$

$\frac{3}{\text{16}}-\frac{3}{\text{16}}$

0

$\frac{\text{15}}{\text{23}}-\frac{2}{\text{23}}$

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

0

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

$\frac{3}{\text{11}}+\frac{1}{\text{11}}+\frac{5}{\text{11}}$

$\frac{9}{\text{11}}$

$\frac{\text{16}}{\text{20}}+\frac{1}{\text{20}}+\frac{2}{\text{20}}$

$\frac{\text{12}}{8}+\frac{2}{8}+\frac{1}{8}$

$\frac{\text{15}}{8}$

$\frac{1}{\text{15}}+\frac{8}{\text{15}}+\frac{6}{\text{15}}$

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

$\frac{1}{2}$

$\frac{\text{11}}{\text{16}}+\frac{9}{\text{16}}-\frac{5}{\text{16}}$

$\frac{4}{\text{20}}-\frac{1}{\text{20}}+\frac{9}{\text{20}}$

$\frac{3}{5}$

$\frac{7}{\text{10}}-\frac{3}{\text{10}}+\frac{\text{11}}{\text{10}}$

$\frac{\text{16}}{5}-\frac{1}{5}-\frac{2}{5}$

$\frac{\text{13}}{5}$

$\frac{\text{21}}{\text{35}}-\frac{\text{17}}{\text{35}}+\frac{\text{31}}{\text{35}}$

$\frac{5}{2}+\frac{\text{16}}{2}-\frac{1}{2}$

10

$\frac{1}{\text{18}}+\frac{3}{\text{18}}+\frac{1}{\text{18}}+\frac{4}{\text{18}}-\frac{5}{\text{18}}$

$\frac{6}{\text{22}}-\frac{2}{\text{22}}+\frac{4}{\text{22}}-\frac{1}{\text{22}}+\frac{\text{11}}{\text{22}}$

$\frac{9}{\text{11}}$

The following rule for addition and subtraction of two fractions is preposterous. Show why by performing the operations using the rule for the following two problems.

## Preposterous rule

To add or subtract two fractions, simply add or subtract the numerators and place this result over the sum or difference of the denominators.

$\frac{3}{\text{10}}-\frac{3}{\text{10}}$

$\frac{8}{\text{15}}+\frac{8}{\text{15}}$

$\frac{\text{16}}{\text{30}}=\frac{8}{5}$ (using the preposterous rule)

Find the total length of the screw.

Two months ago, a woman paid off $\frac{3}{\text{24}}$ of a loan. One month ago, she paid off $\frac{5}{\text{24}}$ of the total loan. This month she will again pay off $\frac{5}{\text{24}}$ of the total loan. At the end of the month, how much of her total loan will she have paid off?

$\frac{\text{13}}{\text{24}}$

Find the inside diameter of the pipe.

## Exercises for review

( [link] ) Round 2,650 to the nearest hundred.

2700

( [link] ) Use the numbers 2, 4, and 8 to illustrate the associative property of addition.

( [link] ) Find the prime factors of 495.

${3}^{2}\cdot 5\cdot \text{11}$

( [link] ) Find the value of $\frac{3}{4}\cdot \frac{\text{16}}{\text{25}}\cdot \frac{5}{9}$ .

( [link] ) $\frac{8}{3}$ of what number is $1\frac{7}{9}$ ?

$\frac{2}{3}$

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
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
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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
7hours 36 min - 4hours 50 min