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Here are a few recommended readings before getting started with this lesson.
Ramsha is studying Egyptian history for class.She learned that ancient Egyptians had an interesting way to represent fractions. They used unit fractions, which are fractions of the form $N1 ,$ to represent all other fractions. The examples of the Egyptian fractions was found in the image of the Eye of Horus. Each part of the eye represents a different fraction.
The least common multiple (LCM) of two whole numbers $a$ and $b$ is the smallest whole number that is a multiple of both $a$ and $b.$ It is denoted as $LCM(a,b).$ The least common multiple of $a$ and $b$ is the smallest whole number that is divisible by both $a$ and $b.$ Some examples can be seen in the table below.
Numbers  Multiples of Numbers  Common Multiples  Least Common Multiple 

$2$ and $3$  $Multiples of2:Multiples of3: 2,4,6,8,10,12,…3,6,9,12,15,… $

$6 ,12,18,24,…$  $LCM(2,3)=6$ 
$8$ and $12$  $Multiples of8:Multiples of12: 8,16,24,32,40,48,…12,24,36,48,… $

$24 ,48,72,96,…$  $LCM(8,12)=24$ 
A special procedure exists for finding the $LCM(a,b)$ of a pair of numeric expressions. The LCM of two relatively prime numbers is always equal to their product.
Coprimes  LCM 

$3$ and $5$  $15$ 
$5$ and $4$  $20$ 
$4$ and $9$  $36$ 
Polynomials can also have a least common multiple. The LCM of two or more polynomials is the smallest multiple of both polynomials. In other words, the LCM is the smallest expression that can be evenly divided by each of the given polynomials.
Polynomials  Factor  LCM  Explanation 

$4x_{3}and6xy_{2} $

$2_{2}⋅x_{3}and2⋅3⋅x⋅y_{2} $

$12x_{3}y_{2}$  $12x_{3}y_{2}$ is the smallest expression that is divisible by both $4x_{3}$ and $6xy_{2}.$ 
$3x_{3}+3x_{2}andx_{2}+5x+4 $

$3⋅x_{2}⋅(x+1)and(x+1)(x+4) $

$3x_{2}(x+1)(x+4)$  $3x_{2}(x+1)(x+4)$ is the smallest expression that is divisible by both $3x_{3}+3x_{2}$ and $x_{2}+5x+4.$ 
Finding the LCM of polynomials requires identifying the factors with the highest power that appear in each polynomial.
Split into factors
Factor out $12y$
$a_{2}+2ab+b_{2}=(a+b)_{2}$
Split into factors
Factor out $3xy$
Both polynomials are now written in factored form. Now, write all the missing related factors to identify the factor with the highest power.
Standard Form  Factored Form  All Related Factors  

Polynomail I  $12x_{2}y+48xy+48y$  $2_{2}⋅3⋅y⋅(x+2)_{2}$  $2_{2}⋅3_{1}⋅x_{0}⋅y_{1}⋅(x+2)_{2}⋅(x−8)_{1}$ 
Polynomail II  $3x_{3}y−18x_{2}y−48xy$  $3⋅x⋅y⋅(x+2)⋅(x−8)$  $2_{0}⋅3_{1}⋅x_{1}⋅y_{1}⋅(x+2)_{1}⋅(x−8)_{1}$ 
What is the least common multiple of the given polynomials?
Split into factors
Factor out $6x$
Split into prime factors
Split into factors
Factor out $3x$
$a_{m}⋅b_{m}=(a⋅b)_{m}$
Split into factors
Write as a power
$a_{2}−2ab+b_{2}=(a−b)_{2}$
Given Form  Factored Form  All Related Factors  

Heichi  $6x_{2}y−6x$  $2⋅3⋅x⋅(xy−1)$  $2_{1}⋅3_{1}⋅x_{1}⋅(xy−1)_{1}$ 
Ramsha  $3x_{3}y_{2}−6x_{2}y+3x$  $3⋅x⋅(xy−1)_{2}$  $2_{0}⋅3_{1}⋅x_{1}⋅(xy−1)_{2}$ 
The least common denominator (LCD) of two fractions is the least common multiple (LCM) of the denominators of the fractions. In other words, the least common denominator is the smallest of all the common denominators. Some examples are provided in the table below.
Fractions  Denominators  Multiples of Denominators  Common Denominators  LCM of Denominators (LCD) 

$32 $ and $21 $  $3$ and $2$  $Multiples of3:Multiples of2: 3,6,9,12,15,…2,4,6,8,10,12,… $

$6,$ $12$  $6$ 
$65 $ and $41 $  $6$ and $4$  $Multiples of6:Multiples of4: 6,12,18,24,30,…4,8,12,16,20,24,… $

$12,$ $24$  $12$ 
$41 $ and $25 $  $4$ and $2$  $Multiples of4:Multiples of2: 4,8,12,…2,4,6,8,10,12,… $

$4,$ $8,$ $12$  $4$ 
When adding and subtracting rational expressions, the same rules apply as when adding and subtracting fractions.
If the rational expressions have a common denominator, the numerators can be added or subtracted directly.
$Q(x)P(x) ±Q(x)R(x) =Q(x)P(x)±R(x) $
Here, $P(x),$ $Q(x),$ and $R(x)$ are polynomials and $Q(x) =0.$
If the denominators are different, the expressions have to be manipulated to find a common denominator before they can be added or subtracted. One way of doing this is to multiply both numerator and denominator of one of the rational expressions with the denominator of the other, and vice versa.
$Q(x)P(x) ±S(x)R(x) =Q(x)S(x)P(x)S(x)±R(x)Q(x) $
Denominator  Factored Form 

$2x−2$  $2(x−1)$ 
$x_{2}−4x+3$  $(x−1)(x−3)$ 
$ba =b⋅(x−3)a⋅(x−3) $
Multiply parentheses
Add terms
$ba =b⋅2a⋅2 $
Distribute $2$
Multiply
Ramsha is very enthusiastic about learning more about Egypt. She finds an Egyptian pen pal, Izabella. Izabella mentions that her favorite dessert is basbousa.
The talk of dessert reminds Ramsha of something she learned in chemistry class. The pH, or acidity, of a person's mouth changes after eating a dessert. The pH level $L$ of a mouth $t$ minutes after eating a dessert is modeled by the following formula.Distribute $6$
Subtract fractions
Commutative Property of Addition
$t=15$
Calculate power
Multiply
Add and subtract terms
Use a calculator
Round to $2$ decimal place(s)
In one of her emails, Izabella tells Ramsha about her father's job. He is in charge of pool maintenance work at a local hotel.
Her father gives Isabella the following set of information.
Let $t$ be the time it takes for the first pipe to fill the pool. Then, the fraction $t1 $ represents the part of the pool that the first pipe fills in one hour.
Let $t$ be the time it takes for Pipe I to fill the pool. Since Pipe II takes 2 times as long to fill the pool as Pipe I, the time needed for Pipe II to fill the pool will be $2$ times $t.$ Similarly, the time needed for Pipe III will be $1.5$ times $2t$ because Pipe III takes $1.5$ times as long to fill the pool as Pipe II.
Pipe  Time Needed to Fill the Pool (hours) 

Pipe I  $t$ 
Pipe II  $2⋅t=2t$ 
Pipe III  $1.5⋅2t=3t$ 
All of Them  $6$ 
Now, think about how much of the pool is filled by each pipe, alone or together, in one hour. For example, it takes all three pipes $6$ hours to fill the pool. Therefore, they would fill $61 $ of the pool in one hour. The filling rates of the other pipes can be determined by this same logic.
Pipe  Time Needed to Fill the Pool (hours)  Filling Rate (per hour) 

Pipe I  $t$  $t1 $ 
Pipe II  $2t$  $2t1 $ 
Pipe III  $3t$  $3t1 $ 
All of Them  $6$  $61 $ 
Denominator  Factored Form 

$t$  $t$ 
$2t$  $2⋅t$ 
$3t$  $3⋅t$ 
$6$  $2⋅3$ 
$ba =b⋅6a⋅6 $
$ba =b⋅3a⋅3 $
$ba =b⋅2a⋅2 $
$ba =b⋅ta⋅t $
Multiply fractions
Add fractions
Izabella is planning a trip that involves a $90$kilometer bus ride and a highspeed train ride. The entire trip is $300$ kilometers.
The average speed of the highspeed train is $30$ kilometers per hour more than twice the average speed of the bus.