Complete Guide to Adding and Subtracting Rational Expressions

Numbers	Multiples of Numbers	Common Multiples	Least Common Multiple
$2$ and $3$	$Multiples of 2 : Multiples of 3 : 2, 4, 6, 8, 10, 12, \dots 3, 6, 9, 12, 15, \dots$	$\underline{6}, 12, 18, 24, \dots$	$LCM (2, 3) = 6$
$8$ and $12$	$Multiples of 8 : Multiples of 12 : 8, 16, 24, 32, 40, 48, \dots 12, 24, 36, 48, \dots$	$\underline{24}, 48, 72, 96, \dots$	$LCM (8, 12) = 24$

Numbers

Multiples of Numbers

Common Multiples

Least Common Multiple

2

and

3

Multiples of 2 : Multiples of 3 : 2, 4, 6, 8, 10, 12, \dots 3, 6, 9, 12, 15, \dots

\underline{6}, 12, 18, 24, \dots

LCM (2, 3) = 6

8

and

12

Multiples of 8 : Multiples of 12 : 8, 16, 24, 32, 40, 48, \dots 12, 24, 36, 48, \dots

\underline{24}, 48, 72, 96, \dots

LCM (8, 12) = 24

Coprimes	LCM
$3$ and $5$	$15$
$5$ and $4$	$20$
$4$ and $9$	$36$

Coprimes

LCM

3

and

5

15

5

and

4

20

4

and

9

36

Polynomials	Factor	LCM	Explanation
$4 x^{3} and 6 x y^{2}$	$2^{2} \cdot x^{3} and 2 \cdot 3 \cdot x \cdot y^{2}$	$12 x^{3} y^{2}$	$12 x^{3} y^{2}$ is the smallest expression that is divisible by both $4 x^{3}$ and $6 x y^{2} .$
$3 x^{3} + 3 x^{2} and x^{2} + 5 x + 4$	$3 \cdot x^{2} \cdot (x + 1) and (x + 1) (x + 4)$	$3 x^{2} (x + 1) (x + 4)$	$3 x^{2} (x + 1) (x + 4)$ is the smallest expression that is divisible by both $3 x^{3} + 3 x^{2}$ and $x^{2} + 5 x + 4 .$

Polynomials

Factor

LCM

Explanation

4 x^{3} and 6 x y^{2}

2^{2} \cdot x^{3} and 2 \cdot 3 \cdot x \cdot y^{2}

12 x^{3} y^{2}

12 x^{3} y^{2}

is the smallest expression that is divisible by both

4 x^{3}

and

6 x y^{2} .

3 x^{3} + 3 x^{2} and x^{2} + 5 x + 4

3 \cdot x^{2} \cdot (x + 1) and (x + 1) (x + 4)

3 x^{2} (x + 1) (x + 4)

3 x^{2} (x + 1) (x + 4)

is the smallest expression that is divisible by both

3 x^{3} + 3 x^{2}

and

x^{2} + 5 x + 4 .

	Standard Form	Factored Form	All Related Factors
Polynomail I	$12 x^{2} y + 48 x y + 48 y$	$2^{2} \cdot 3 \cdot y \cdot (x + 2)^{2}$	$2^{2} \cdot 3^{1} \cdot x^{0} \cdot y^{1} \cdot (x + 2)^{2} \cdot (x - 8)^{1}$
Polynomail II	$3 x^{3} y - 18 x^{2} y - 48 x y$	$3 \cdot x \cdot y \cdot (x + 2) \cdot (x - 8)$	$2^{0} \cdot 3^{1} \cdot x^{1} \cdot y^{1} \cdot (x + 2)^{1} \cdot (x - 8)^{1}$

Standard Form

Factored Form

All Related Factors

Polynomail I

12 x^{2} y + 48 x y + 48 y

2^{2} \cdot 3 \cdot y \cdot (x + 2)^{2}

2^{2} \cdot 3^{1} \cdot x^{0} \cdot y^{1} \cdot (x + 2)^{2} \cdot (x - 8)^{1}

Polynomail II

3 x^{3} y - 18 x^{2} y - 48 x y

3 \cdot x \cdot y \cdot (x + 2) \cdot (x - 8)

2^{0} \cdot 3^{1} \cdot x^{1} \cdot y^{1} \cdot (x + 2)^{1} \cdot (x - 8)^{1}

	Given Form	Factored Form	All Related Factors
Heichi	$6 x^{2} y - 6 x$	$2 \cdot 3 \cdot x \cdot (x y - 1)$	$2^{1} \cdot 3^{1} \cdot x^{1} \cdot (x y - 1)^{1}$
Ramsha	$3 x^{3} y^{2} - 6 x^{2} y + 3 x$	$3 \cdot x \cdot (x y - 1)^{2}$	$2^{0} \cdot 3^{1} \cdot x^{1} \cdot (x y - 1)^{2}$

Given Form

Factored Form

All Related Factors

Heichi

6 x^{2} y - 6 x

2 \cdot 3 \cdot x \cdot (x y - 1)

2^{1} \cdot 3^{1} \cdot x^{1} \cdot (x y - 1)^{1}

Ramsha

3 x^{3} y^{2} - 6 x^{2} y + 3 x

3 \cdot x \cdot (x y - 1)^{2}

2^{0} \cdot 3^{1} \cdot x^{1} \cdot (x y - 1)^{2}

Fractions	Denominators	Multiples of Denominators	Common Denominators	LCM of Denominators (LCD)
$\frac{2}{3}$ and $\frac{1}{2}$	$3$ and $2$	$Multiples of 3 : Multiples of 2 : 3, 6, 9, 12, 15, \dots 2, 4, 6, 8, 10, 12, \dots$	$6,$ $12$	$6$
$\frac{5}{6}$ and $\frac{1}{4}$	$6$ and $4$	$Multiples of 6 : Multiples of 4 : 6, 12, 18, 24, 30, \dots 4, 8, 12, 16, 20, 24, \dots$	$12,$ $24$	$12$
$\frac{1}{4}$ and $\frac{5}{2}$	$4$ and $2$	$Multiples of 4 : Multiples of 2 : 4, 8, 12, \dots 2, 4, 6, 8, 10, 12, \dots$	$4,$ $8,$ $12$	$4$

Fractions

Denominators

Multiples of Denominators

Common Denominators

LCM of Denominators (LCD)

\frac{2}{3}

and

\frac{1}{2}

3

and

2

Multiples of 3 : Multiples of 2 : 3, 6, 9, 12, 15, \dots 2, 4, 6, 8, 10, 12, \dots

6,

12

6

\frac{5}{6}

and

\frac{1}{4}

6

and

4

Multiples of 6 : Multiples of 4 : 6, 12, 18, 24, 30, \dots 4, 8, 12, 16, 20, 24, \dots

12,

24

12

\frac{1}{4}

and

\frac{5}{2}

4

and

2

Multiples of 4 : Multiples of 2 : 4, 8, 12, \dots 2, 4, 6, 8, 10, 12, \dots

4,

8,

12

4

Denominator	Factored Form
$2 x - 2$	$2 (x - 1)$
$x^{2} - 4 x + 3$	$(x - 1) (x - 3)$

Denominator

Factored Form

2 x - 2

2 (x - 1)

x^{2} - 4 x + 3

(x - 1) (x - 3)

Pipe	Time Needed to Fill the Pool (hours)
Pipe I	$t$
Pipe II	$2 \cdot t = 2 t$
Pipe III	$1.5 \cdot 2 t = 3 t$
All of Them	$6$

Pipe

Time Needed to Fill the Pool (hours)

Pipe I

t

Pipe II

2 \cdot t = 2 t

Pipe III

1.5 \cdot 2 t = 3 t

All of Them

6

Pipe	Time Needed to Fill the Pool (hours)	Filling Rate (per hour)
Pipe I	$t$	$\frac{1}{t}$
Pipe II	$2 t$	$\frac{1}{2 t}$
Pipe III	$3 t$	$\frac{1}{3 t}$
All of Them	$6$	$\frac{1}{6}$

Pipe

Time Needed to Fill the Pool (hours)

Filling Rate (per hour)

Pipe I

t

\frac{1}{t}

Pipe II

2 t

\frac{1}{2 t}

Pipe III

3 t

\frac{1}{3 t}

All of Them

6

\frac{1}{6}

Denominator	Factored Form
$t$	$t$
$2 t$	$2 \cdot t$
$3 t$	$3 \cdot t$
$6$	$2 \cdot 3$

Denominator

Factored Form

t

t

2 t

2 \cdot t

3 t

3 \cdot t

6

2 \cdot 3

a There are many strategies for writing the given fraction as a sum of unit fractions. One way to write

\frac{2}{7}

as a sum of unit fractions would be to find the greatest unit fraction less than

\frac{2}{7} .

Note that

\frac{1}{4}

is less than

\frac{2}{7} .

\frac{1}{4} < \frac{2}{7} because \frac{7}{2 8} < \frac{8}{2 8}

Subtracting

\frac{1}{4}

from

\frac{2}{7}

gives

\frac{1}{2 8} .

Therefore, the given fraction is the sum of

\frac{1}{4}

and

\frac{1}{2 8} .

\frac{2}{7} = \frac{1}{4} + \frac{1}{2 8}

This is an Egyptian fraction representation of

\frac{2}{7} .

Note that there are different representations of the given fraction. Here, only one of them is shown.

b In order to find the sum of the rational expressions, both must have a common denominator.

\frac{1}{x + 1} + \frac{1}{x ( x + 1 )}

Looking at the denominators, the numerator and denominator of the first expression should be multiplied by

x .

\frac{1}{x + 1} + \frac{1}{x ( x + 1 )}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot x}{b \cdot x}$

\frac{x}{x ( x + 1 )} + \frac{1}{x ( x + 1 )}

AddFrac

Add fractions

\frac{x + 1}{x ( x + 1 )}

CancelCommonFac

Cancel out common factors

\frac{x + 1}{x ( x + 1 )}

SimpQuot

Simplify quotient

\frac{1}{x}

The sum is equal to

\frac{1}{x} .

\frac{1}{x + 1} + \frac{1}{x ( x + 1 )} = \frac{1}{x}

c In Part B, the sum of the rational expressions was found to be

\frac{1}{x} .

The reverse of this identity can be applied to find Egyptian fractions of the form

\frac{1}{N},

where

N > 2 .

\frac{1}{N} = \frac{1}{N + 1} + \frac{1}{N ( N + 1 )}

The expressions of the right hand side can also be rewritten. An equivalent expression for

\frac{1}{N + 1}

is obtained by replacing

N

with

N + 1

in the identity.

\frac{1}{x} = \frac{1}{x + 1} + \frac{1}{x ( x + 1 )}

SubstituteExpressions

Substitute expressions

\frac{1}{N + 1} = \frac{1}{N + 1 + 1} + \frac{1}{( N + 1 ) ( N + 1 + 1 )}

AddTerms

Add terms

\frac{1}{N + 1} = \frac{1}{N + 2} + \frac{1}{( N + 1 ) ( N + 2 )}

This equation can be used to rewrite the right-hand side of the first equation.

\frac{1}{N} \frac{1}{N} = \frac{1}{N + 1} + \frac{1}{N ( N + 1 )} ⇓ = \frac{1}{N + 2} + \frac{1}{( N + 1 ) ( N + 2 )} + \frac{1}{N ( N + 1 )}

As it is shown in the equations above, each unit fraction

\frac{1}{N},

with

N \geq 2,

can be written as a sum of two or more different unit fractions.

d Positive rational numbers of the form

\frac{p}{q},

where

p > 0

and

q > 2,

can be written as a sum of unit fractions

\frac{1}{q} .

\frac{p}{q} = p \frac{1}{q} + \frac{1}{q} + \dots + \frac{1}{q}

p = 1,

then it is already an Egyptian fraction. If

p = 2,

then the first

\frac{1}{q}

is kept and then the second one is replaced with

\frac{1}{q + 1} + \frac{1}{q ( q + 1 )} .

\frac{2}{q} = \frac{1}{q} + \frac{1}{q} = \frac{1}{q} + \frac{1}{q + 1} + \frac{1}{q ( q + 1 )}

p > 2,

then the third

\frac{1}{q}

is written as

\frac{1}{q + 1} + \frac{1}{q ( q + 1 )}

and then each of these is replaced using the same idea.

\frac{p}{q} = \frac{1}{q} + \frac{1}{q} + \frac{1}{q} + \dots

▼

Rewrite

Substitute

$\frac{1}{q} = \frac{1}{q + 1} + \frac{1}{q ( q + 1 )}$

\frac{p}{q} = \frac{1}{q} + \frac{1}{q + 1} + \frac{1}{q ( q + 1 )} + \frac{1}{q} + \dots

Substitute

$\frac{1}{q} = \frac{1}{q + 1} + \frac{1}{q ( q + 1 )}$

\frac{p}{q} = \frac{1}{q} + \frac{1}{q + 1} + \frac{1}{q ( q + 1 )} + \frac{1}{q + 1} + \frac{1}{q ( q + 1 )} + \dots

Substitute

$\frac{1}{q + 1} = \frac{1}{q + 2} + \frac{1}{( q + 1 ) ( q + 2 )}$

\frac{p}{q} = \frac{1}{q} + \frac{1}{q + 1} + \frac{1}{q ( q + 1 )} + \frac{1}{q + 2} + \frac{1}{( q + 1 ) ( q + 2 )} + \frac{1}{q ( q + 1 )} + \dots

Substitute

$\frac{1}{q ( q + 1 )} = \frac{1}{q ( q + 1 ) + 1} + \frac{1}{q ( q + 1 ) ( q ( q + 1 ) + 1 )}$

\frac{p}{q} = \frac{1}{q} + \frac{1}{q + 1} + \frac{1}{q ( q + 1 )} + \frac{1}{q + 2} + \frac{1}{( q + 1 ) ( q + 2 )} + \frac{1}{q ( q + 1 ) + 1} + \frac{1}{q ( q + 1 ) ( q ( q + 1 ) + 1 )} + \dots

By repeating this procedure, replacing every duplicate unit fraction with two unit fractions with larger denominators, all repeating unit fractions will eventually disappear. Therefore, any positive rational number $\frac{p}{q}$ can be written as an Egyptian fraction.

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Catch-Up and Review

Egyptian Fractions

Least Common Multiple of Polynomials and How to Find It

Least Common Multiple

Finding the Least Common Multiple of Polynomials

Writing an Expression for the Meeting Time

Hint

Solution

Using the Least Common Denominator to Add or Subtract Rational Expressions

Least Common Denominator

Adding and Subtracting Rational Expressions

Adding and Subtracting With Like Denominators

Adding and Subtracting With Unlike Denominators

Acidity of the Mouth

Hint

Solution

Finding the Time It Takes for a Pipe to Fill a Pool

Hint

Solution

Finding the Total Time of Izabella's Trip

Hint

Solution

Finding Monthly Payment

Hint

Solution

Egyptian Fractions

Answer

Hint

Solution

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