The 43rd Term of Ap 26 Find The first tom of the progress in given that is common difference is 1/2 Two Aps have same first &last term
The first term of the arithmetic progression is 5
What is an arithmetic sequence?An arithmetic sequence can simply be defined as a sequence of numbers or terms in which the differences between successive or consecutive terms are always equal.
The formula for the nth term of an arithmetic sequence is expressed as;
Tn = a + (n-1)d
Where;
Tn is the nth term of the sequencea is the first term of the sequencen is the number of termsd is the common difference between successive termsGiven that the first term of the sequence is a, the 43rd term is 26 and the common difference is 1/2
Substitute the values into the formula
26 = a + (43 -1)1/2
expand the bracket
26 = a + (42)1/2
Multiply through
26 = a + 21
collect like terms
a = 26 - 21
a = 5
Thus, the value is 5
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what is the line that passes through the point -1, four and is parallel to the line 2
X plus Y equals one
The line that passes through the point -1, four and is parallel to the line 2 X plus Y equals one is y + 2x = 2.
What is a slope?Slope is measure of steepness of a line.
The Equation of line is y = mx + c
where,
m is slope
c is y intercept
To find equation of the parallel line we must first find the slope of the original line
The original line is 2x + y = 1
We must first write equation in the general form above
That's
y = - 2x + 1
Comparing with general equation above
Slope = - 2
Since lines are parallel their slope are also same
Slope of parallel line = - 2
So the equation of line using point
( - 1 , 4) and the slope - 2 is
y-4= -2(x+1)
y-4= 2x-2
y+2x= -2+4
We have final answer as
y + 2x = 2
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Which among the given set is a function A = [(4,6), (A, B), (A, 6)]C = {(6,9), (6,6), (6,9)B = [(1, Z), ( 2, Y), (3, X)]
The set which is a function is B = [(1, Z), ( 2, Y), (3, X)]
What is a function ?
Each element of X receives exactly one element of Y when a function from one set to the other is used. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
Since, we know that function does not repeat any x coordinate
Which means, the first coordinate of every pair does not repeat
Therefore the first coordinates which is repeated is not a function
Now, let's check out the given sets and check whether its a function or not
A = [(4,6), (A, B), (A, 6)], is not a function
Since, its first coordinate is repeated
C = {(6,9), (6,6), (6,9), is not a function
Since, its first coordinate is repeated
B = [(1, Z), ( 2, Y), (3, X)], is a function
Because, its first coordinate is not repeated
Hence, B = [(1, Z), ( 2, Y), (3, X)], is a function
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the hypotenuse of a right triangle measures 16 cm and one of its legs measures 5 cm. Find the measure of the other leg.
Step-by-step explanation:
Use the Pythagorean Theorem.
a^2+b^2=c^2
a^2 + 5^2 = 16^2
a^2 + 25 = 256
a^2 = 231
a = square root of 231
a= 15.20
Answer:15.20
Step-by-step explanation:
Write an algebraic expression to represent each statement.
the number, b, divided by thirty-six
Write an algebraic expression to represent each statement.
the number, b, divided by thirty-six
b/36
Drag the tiles to the correct boxes. Not all tiles will be used.What are the domain and the range of function ?3-5f)2 - 3 - 18=(-00, -3) U(-3,0)(-o, -3U-3, 6) U6, có)(-0, 6) U (6,00)(-ao, -5) U(-5, 3) U3, có)(co, 0) U6, 3) Uề, có)(-0, 0) U(0, 0)rangedomai
The Solution:
Given the function:
Required:
To determine the domain and the range of the function.
Below is the graph of the function:
Step 1:
The domain of the function is:
[tex](-\infty,-3)\cup(-3,\infty)[/tex]Step 2:
The range of the function is:
[tex](-\infty,0)\cup(0,\infty)[/tex]Therefore, the correct answer is :
[tex]undefined[/tex]Determine whether the graph represents a function.
Multiplicative inverses or ______, is a number that when multiplied by the ___________ gives the product as 1.
Answer: Multiplicative inverses or reciprocal, is a number that when multiplied by the given number gives the product as 1.
Step-by-step explanation:
Use an integer or a fraction to describe the slope of a perpendicular To this line.
Answer:
slope of perpendicular line = - [tex]\frac{2}{3}[/tex]
Step-by-step explanation:
calculate the slope m of the given line using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (1, 0) and (x₂, y₂ ) = (3, 3) ← 2 points on the line
m = [tex]\frac{3-0}{3-1}[/tex] = [tex]\frac{3}{2}[/tex]
given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{3}{2} }[/tex] = - [tex]\frac{2}{3}[/tex]
What is the sum of the 8th square
number and the 5th cube number?
8 square = 64
5 cube = 125
Step-by-step explanation:
=8² + 5³
=64 + 125
=189
What are square and cube ?
The square of a number is the multiplication by itself. That means, for any integer, we can obtain the square by multiplying the integer itself. For example, the square of an integer 4 will be: 4 × 4 = 16, i.e., the square of 4 is equal to 16.
The cube of a number is the multiplication by itself thrice. That means, for any integer, we can obtain the cube by multiplying the integer by its square. For example, the cube of an integer 5 will be: 52 × 5 or 5 × 5 × 5, i.e., the cube of 5 is equal to 125.
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a pizza shop delivered 8 pepperoni pizzas to a college on the first night of final exams. the total cost of the pizzas was $56. a small pizza costs $3 and contains 4 ounces of pepperoni. a medium pizza costs $5 and contains 9 ounces of pepperoni. a large pizza cost $11 and contains 12 ounces of pepperoni. the owner of the pizza shop used 4 pounds 12 ounces of pepperoni in making the pizzas. how many pizzas of each size were delivered to the college?
From the word problem given, the number of pizzas each that were delivered are:
1 small7 medium; and2 large.Step I
Let a = number of small pizzas delivered
Let b = number of medium pizzas delivered
Let c = number of large pizzas delivered
Step 2 - Creating relevant expressions based on number and type ordered, we have:
a + b + c = 8.............................1
3a + 5b + 11c = 56..................2
4a + 9b + 12C = 4 * 8 + 12.....3a
We can rewrite (3) as
4a + 9b + 12C = 44 ....3b
Step 3
Multiply both sides of (1) by 4
We have
4a +4b +4c = 32
If we remove (1) from (3) we have-
4a +4b +4c = 32
5b +8C = 12......................(4)
Multiply both sides of (1) by 3 and subtract from (2), and we have:
3a + 3b + 3c = 24, removing this from (2) we have
3a + 5b + 11c = 56
-
3a + 3b + 3c = 24
2b + 8c = 32.....................(5)
Step 4 - Subtract (4) from (5)
2b + 8c = 32
-
5b + 8c = 12
-3b = 20
b = - 20/3
= - 6.67
Since we cannot have negative values, we work with absolute value to get:
b [tex]\approx[/tex] 7
Since b = 7
2(7) + 8c = 32
8c = 32 - 14
c = 18/8
c [tex]\approx[/tex] 2
Plug in that back into a we have
a + b + c = 8
a + 7 + 2 = 8
a = 8-9
a = -1 (recall we must have absolute values)
a therefore = 1.
Hence the number of pizzas that were ordered are:
1 small
7 medium; and
2 large.
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estimate using a rate per 100 24% of 289
The estimate of 24% of 289 is 69.36
Estimating percentagesFrom the question, we are to determine the estimate the given rate
The given rate is
24% of 289
This can be expressed as
24% × 289
Now, we will evaluate the percentage
To evaluate the percentage, we will divide the number by 100.
That is,
24% = 24/100
Thus,
The expression can further be expressed as
24/100 × 289
= 0.24 × 289
= 69.36
Hence, the estimate is 69.36
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Determina el valor de un número negativo tal que la suma de
ese número y su cuadrado es 42.
The value of a negative number such that the sum of that number and its square is 42 is -7.
What are negative numbers?A negative number is one that always has a value lower than zero and is denoted by the minus (-) symbol. On a number line, negative numbers are displayed to the left of zero. Negative numbers include -6 and -15 as examples.Let the negative number be -x.
According to the question, the sum of the negative number and its square is 42.
So, we form an equation
- x + (-x)² = 42
x² - x - 42 = 0
Solving the quadratic equation,
x² - 7x + 6x - 42 = 0
x(x - 7) + 6(x - 7) = 0
(x - 7)(x + 6) = 0
x = -6 and x = 7
We have taken the number -x to be negative.
So, -x = -6 and -x = 7
x = 6 and x = -7
Hence, the value of the negative number is -7.
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The correct question is: "Determine the value of a negative number such that the sum of that number and its square is 42."
A line has a slope of – 3 and includes the points (5, – 6) and (s, – 3). What is the value of s?
Answer:
[tex]{ \tt{slope = \frac{y _{2} - y _{1} }{x _{2} - x _{1} } }} \\ \\ { \tt{ - 3 = \frac{ - 3 - ( - 6)}{s - 5} }} \\ \\ { \tt{ - 3(s - 5) = 3}} \\ \\ { \tt{s - 5 = - 1}} \\ \\ { \tt{s = 4}}[/tex]
6.105 + 10.4 + 3.075 ?
[ this is middle school math, i dont get why it says highschool :/ ]
Answer:
Step-by-step explanation:
6.105 + 10.4 + 3.075 ? :19.58
The director will share out 18000 as follows
9 office workers will each get £500
The 4 managers will each get at least twice the bonus of an office worker
The directory will get at least 3 times the bonus of an office worker.
Answer:
Total = £18,000
9 Office workers = £500
4 Managers = 500 × 2 (twice the bonus of an office worker) = £1,000
Directory = 500 × 3 (3 times the bonus of an office worker) = £1,500
7:8:9=__:12:__:
please help me
Answer:
10.5:12:13.5
Step-by-step explanation:
The only two values you know and can compare are 8 and 12, so you have to find the relationship between them. 12/8 = 1.5. To get the two unknown values, you multiply 1.5 by the ones you do know.
7 * 1.5 = 10.5
9 * 1.5 = 13.5
So, you get 10.5:12:13.5
Use the figure below to answer questions #1-3.
What is the special angle pair relationship between 1 and 23?
m
2/7
3/6
4/5
at
1/8
The special angle pair relationship between the given angles are:
1. Corresponding angles
2. Alternate exterior angles
3. Consecutive
What are Some Special Angle Pair Relationship?The following are some special angle pair relationships:
Corresponding Angles: These are pairs of angles that lie on relative position to each other on the two parallel lines that are intercepted by a transversal. They are congruent to each other.Alternate Exterior Angles: These are exterior angles that are on each parallel lines that are crossed by a transversal and lie alternate to each other. They are congruent to each other also.Consecutive Interior Angles: These are angles on the same side of a transversal but lie each on the parallel lines that the transversal cuts across. They are supplementary.1. Angles 1 and 3 form a special angle pair relationship known as: corresponding angles.
2. Angles 8 and 4 form a special angle pair relationship known as: alternate exterior angles.
3. Angles 2 and 3 both lie on the same side of the given transversal, therefore, the special angle pair relationship between angle 2 and angle 3 is: consecutive interior angles.
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need help with this, thanks
Answer:
[tex]x=\boxed{13}\\\\y=\boxed{20}[/tex]
Step-by-step explanation:
Vertical Angle Theorem
When two straight lines intersect, they form two pairs of angles. The vertically opposite (non-adjacent) angles are congruent.
Solving for x:
[tex]\boxed{\begin{aligned}(6x-26)^{\circ}&=4x^{\circ}\\6x-26&=4x\\6x-26+26&=4x+26\\6x&=4x+26\\6x-4x&=4x+26-4x\\2x&=26\\2x \div 2&=26 \div 2\\x&=13\\\end{aligned}}[/tex]
Solving for y:
[tex]\boxed{\begin{aligned}(7y-12)^{\circ}&=(6y+8)^{\circ}\\7y-12&=6y+8\\7y-12-6y&=6y+8-6y\\y-12&=8\\y-12+12&=8+12\\y&=20\end{aligned}}[/tex]
Therefore:
[tex]x=\boxed{13}\\\\y=\boxed{20}[/tex]
Help ! Pleaseeeeeeee asap
The obtained decimal number is 0.155.
To solve the question [tex]\frac{3.1 * 10}{2.0 * 10^{2}}[/tex].
Canceling out 10 from the denominator and numerator
= [tex]\frac{3.1 }{2.0 * 10}[/tex]
= 3.1 / 20
Remove the decimal point from the numerator and add one zero in the denominator
= 31/ 200
As we divide the numbers using the long division method we get the decimal number as a result,
= 0.155
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Question 8 of 10
Which choices are equivalent to the expression below? Check all that apply.
√5 √5
●
A. √5
B. √5.5
C. 5
☐D. 10
E. 25
F. √25
expression :
[tex]\sqrt{5} \sqrt{5}[/tex] = [tex]\sqrt{5X5} = \sqrt{25} = 5[/tex].
What is a square root?In mathematics, the square root is a factor that, when multiplied by itself, equals the original integer.The radical symbol used to denote a number's root is "[tex]\sqrt{x}[/tex]" The square of the positive number is represented by multiplying it by itself. The original number is obtained by taking the square root of the square of a positive number.It is easy to determine a number's square. To determine the given number's square, we must multiply it by itself. [tex]x^{2}[/tex] integer is always used to denote the square term.To learn more about : Square and Root
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a family decides to have children until it has three chil- dren of the same gender. assuming p(b) 5 p(g) 5 .5, what is the pmf of x 5 the number of children in the family?
one is the pmf or probability mass function of X = number of children in the family
The Probability Mass Function (PMF) is also called a probability function or frequency function which characterizes the distribution of a discrete random variable.
X takes values: { 3, 4, 5 }
where: P ( X = 3 ) + P ( X = 4 ) + P ( X = 5 ) = 1
For X = 3 we have : BBB or GGG.
P ( X = 3 ) = ( 1/2 * 1/2 * 1/2 ) * 2 = 1/8 * 2 = 2 / 8 = 1/4
P ( X = 4 ) = ( 1/2 * 1/2 * 1/2 * 1/2 ) * 6 = 6 /16 = 3/8
P ( X = 5 ) = ( 1/2 * 1/2 * 1/2 * 1/2 * 1/2 ) * 12 = 12 / 32 = 3/8
1 / 4 + 3 / 8 + 3 / 8 = 8 / 8 = 1
the complete question is
A family decides to have children until it has three kids of the same gender. Assuming P(B) = P(G) = 0.5, what is the pmf of X = number of children in the family?
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Simplify
(x + 3)
(x + 3)
Answer:
1. X+3x8=24
2.X+3x5=15
Step-by-step explanation:
Hello! So...
We are given the following to simplify:
[tex]\frac{(x+3)^8}{(x+3)^5}[/tex]
_______________________________
1. Apply the exponent rule ([tex]a^b^+^c=a^b * a^c[/tex]).
[tex](x+3)^8=(x+3)^5(x+3)^3[/tex]
=[tex]\frac{(x+3)^5(x+3)^3}{(x+3)^5}[/tex]
________________
2. Cancel out the common factor, given us our solution: [tex](x+3)^5[/tex].
________________________________________________________
Hope this helps! If so, please lmk! Thanks and good luck!
25) The line through x-intercept 3 and y-intercept -1 in slope-intercept form.
this dude has a farm and 1/8 were pigs, 2/3 were horses, and the rest were chickens. there are 10 chickens, how many animals are on the farm?
The number of animals that are on the farm is 48.
How many animals are on the farm?A fraction is a non-integer that is made up of a numerator and a denominator. An example of a fraction is 2/3.
The first step is to determine the fraction of the animals that are chickens.
Fraction that are chickens = 1 - fraction that are pigs and horses.
[tex]\frac{1}{8} + \frac{2}{3}[/tex]
[tex]\frac{3 + 16}{24}[/tex] = [tex]\frac{19}{24}[/tex]
Fraction that are chickens = 1 - 19/24 = 5/24
The next step is to divide the number of chickens by the fraction of the chickens.
Number of animals = number of chickens ÷ fraction of chickens
10 ÷ 5/24
10 x 24/ 5 = 48
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Please help me! I will give 50 points! I have a test tomorrow and don't understand how to do this! Please explain step by step
1) [tex]\angle A \cong \angle D, \angle C \cong \angle E, \overline{AB} \cong \overline{DF}[/tex] (given)
2) [tex]\triangle ABC \cong \triangle DFE[/tex] (AAS)
3) [tex]\angle ACB \cong \angle DEF[/tex] (CPCTC)
In 1787 the French scientist Jacques Charles observed that when he plotted the graph of volume of a fixed amount of air versus the temperature of the air, the points lay along a straight line. Therefore, he concluded that volume of a gas varies linearly with temperature. Suppose that at 27º C a certain amount of air occupies a volume of 500 cm3. When it is warmed to 90º C, it occupies 605 cm3.
a) Write the particular equation in slope-intercept form expressing volume in terms of temperature.
b) Predict the volume at 60º C.
c) The process of predicting a value between two data points is called “interpolation”.
What is the origin of the word?
d) Predict the volume at 300º C.
e) The process of predicting a value beyond any given data points is called
“extrapolation”. What is the origin of this word?
f) Extrapolate your mathematical model back to the point where the volume is zero.
That is, find the temperature-intercept.
g) What is the special name given to the temperature in part f)?
The relationship between the volume, V, and temperature, T, of a given mass of gas, found by the French scientist Jacques Charles, gives;
a) The slope–intercept form of the equation that expresses the volume in terms of the temperature of the gas is; [tex] \displaystyle{ V = 1. \bar 6 \cdot T + 455}[/tex]
b) The volume at 60 °C is 555 cm³
c) Interpolation originates from the verb interpolare
d) The temperature at 300 °Cis 955 cm³
e) Extrapolation originates from 'extra' (meaning outside) and the shortened form of Interpolation
f) The temperature–intercept is -273 °C
g) The special name is the absolute zero temperature
What is Charles law for gases?Charles law states that the volume of a given mass of gas is directly proportional to its temperature given in Kelvin.
Mathematically, Charles Law states that; V = K•T
Where;
V = The volume of the gas
T = The gas temperature
K = The proportionality constant
The volume occupied by the gas at 27 °C = 500 cm³
The volume occupied at 90 °C = 605 cm³
a) To write the equation in slope–intercept form, we have;
[tex] \displaystyle{ Slope = \frac{605 - 500}{90 - 27} =1. \bar{6} =1. \bar{6}}[/tex]
The equation of the line in point and slope form is therefore;
Point–slope form; [tex] \displaystyle{ V - 500 = 1. \bar{6} \times (T - 27) }[/tex]
[tex] \displaystyle{ V = 1. \bar{6} \cdot T -45 + 500 = 1. \bar{6} \cdot T + 455}[/tex]
The equation in slope–intercept form of expressing the volume, V, in terms of the temperature, T, is therefore;
Volume equation; [tex] \displaystyle{ V = 1. \bar{6} \cdot T + 455}[/tex]
b) The volume at 60 °C is given from the equation that gives the volume, V, in terms of time, T as follows;
[tex] \displaystyle{ V(T) = 1. \bar{6} \cdot T + 455}[/tex]
When T = 60 °C
[tex] \displaystyle{ V(60) = 1. \bar{6} \times 60 + 455=555}[/tex]
The volume occupied by the gas at 60 °C is 555 cm³c) The word interpolate originated from the Latin word interpolare which means to alter or refurbish
d) The volume occupied by the gas at 300 °C is given by the function for the volume, V, in terms of temperature, T, equation as follows;
When T = 300 °C
[tex] \displaystyle{ V(300) = 1. \bar{6} \times 300 + 455 = 955}[/tex]
The volume of the gas at 300 °C is 955 cm³e) Extrapolation originate from the word 'extra' which refers to the outside of an area. Extra is combined with a short form of interpolate to give the word, extrapolation
f) When the volume is zero, the volume function equation can be used to find the temperature as follows;
Volume function; [tex] \displaystyle{ V = 1. \bar{6} \cdot T + 455}[/tex]
When V = 0
[tex] \displaystyle{ V = 0 =1. \bar{6} \cdot T + 455}[/tex]
[tex] \displaystyle{ -455 = 1. \bar{6} \cdot T }[/tex]
[tex] \displaystyle{T = \frac{3 \times (-455)}{5}= -273 }[/tex]
The temperature at which the volume of the gas is zero is -273 °Cg) The special name given to -273 °C of the gas is the absolute zero temperature.
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Help me I’m so confused I don’t understand how to do this problem…
a) The problem is in the second step, the square root can't be distributed under sum.
b) 10*√41
How to simplify the numeric expression?a) The incorrect step is the second one, the square root can't be distributed like that.
Here we have the numeric expression:
5*√164
And we want to simplify it.
We can write:
5 = √25
Then:
5*√164 = √25*√164 = √(164*25) = √4,010
Now we can decompose this as a product of square numbers:
4,010 = 4*1,025 = 4*25*41
Then we rewrite:
√4,010 = √4*√25*√41 = 2*5*√41 = 10*√41
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Find the values of p and q if x-1 is a common factor of f(x)=x⁴-px³ +7qx+1, and g(x)=x²-4x² + px² +qx-3,
The values of p and q if x-1 is a common factor of f(x)=x⁴-px³ +7qx+1, and g(x)=x²-4x² + px² +qx-3,are 13/2 and -1/2 respectively.
What is a factor?In Mathematics, factors are positive integers that can divide a number evenly. Suppose we multiply two numbers to get product. The number that is multiplied are the factors of the product. Each number is factor of itself. Factors have many real-life examples, such as arranging sweets in box, arranging numbers in pattern, distributing chocolates among children, etc. To find the factors of number, we need to use the multiplication or division method.
Factors are numbers that can divide a number exactly. Hence, after the division, there is no remainder left. Factors are numbers you multiply together to get the another number. Thus, factor is the divisor of another number.
f(x)=[tex]x^{4}-px^{3}+7qx+1[/tex]
= 1-p+7q+1
= p-7q=2---------1
g(x)= [tex]-3x^{2} +px^{2} +qx-3[/tex]
= -3+p+q-3
=p+q= 6----------2
p-7q=2
p+q=6
-----------------
By subtracting 2 from 1
-8q=4
q=-1/2
putting value of q in equation 2 we get,
p+(-1/2)=6
p=6+[tex]\frac{1}{2}[/tex]
p=13/2
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There are only green pens and blue pens in a box.
There are three more blue pens than green pens in the box.
There are more than 12 pens in the box.
Simon is going to take at random two pens from the box.
The probability that Simon will take two pens of the same colour is
27/55
Work out the number of green pens in the box.
Answer:
Green Pens : 21
Step-by-step explanation:
Let the number of
green pens = x
Three more blue pens than green pens translates to
=> number of blue pens = number of green pens + 3
=> blue pens = x + 3
Total number of pens = blues + green = x + 3 + x = 2x + 3
Probability of taking 2 green pens:
The first time a green pen is taken out, the probability
P( 1s green pen) = Number of green pens/Total number of pens
= [tex]\dfrac{x}{2x + 3}[/tex]
After the first green pen is removed, the number of green pens will be one less and so will the total number of pens
Number of green pens = x - 1
Total number of pens = 2x + 3 - 1 = 2x + 2
P(second green pen | first green pen)
[tex]=\dfrac{x-1}{2x + 2}[/tex]
P(2 green pens) = P(first green pen) x P(second green pen)
[tex]= \dfrac{x}{2x + 3} \times \dfrac{x-1}{2x + 2}\\\\=\dfrac{x(x -1)}{(2x +3)(2x+2)}[/tex] [1]
Probability of 2 blue pens
P( 1st blue pen) = number of blue pens/total number of pens
[tex]=\dfrac{x + 3}{2x + 3}[/tex]
P(second blue pen = (number of pens remaining if first was blue) ÷ total number of pens - 1
If the first pen taken was blue then there will be 2x + 3 - 1 blue pens out of a total of 2x + 3 - 1 pens
P(2nd blue pen | first blue pen)
[tex]=\dfrac{x + 2}{2x + 2}[/tex]
[tex]\text{P(2 blue pens)} =\dfrac{x + 3}{2x + 3} \times \dfrac{x + 2}{2x + 2}[/tex][tex]=\dfrac{(x + 3)(x+2)}{(2x + 3)({2x + 2})}[/tex] [2]
The probability of either 2 green pens or 2 blue pens is the sum of the two probability expressions [1] and [2]
[tex]= \dfrac{x(x -1)}{(2x +3)(2x+2)} + \dfrac{(x + 3)(x+2)}{(2x + 3)({2x + 2})}[/tex]
[tex]= \dfrac{x(x - 1)+ (x + 3) (x +2)}{(2x + 3)(2x + 2)}\\\\\\= \dfrac{2x^2+4x+6}{(2x + 3)(2x + 2)}\\\\\\\\[/tex]
Numerator: factor out 2 to get 2(x² + 2x + 3)
denominator = 4x²+10x+6 = 2 (2x² + 5x + 3)
2 of numerator and 2 of denominator cancel out to give the final expression:
[tex]\dfrac{x^2\:+\:2x\:+\:3}{2x^2+\:5x\:+\:3}[/tex]
But we are given that the probability of picking 2 pens of the same color = 27/55
This will give us:
[tex]\dfrac{x^2\:+\:2x\:+\:3}{2x^2+\:5x\:+\:3} = \dfrac{27}{55}[/tex]
Cross-multiplying we get
[tex]55\left(x^2+\:2x\:+\:3\right)=27\left(2x^2+\:5x\:+\:3\right)[/tex]
=> [tex]55x^2+110x+165 = 54x^2+135x+81[/tex]
Grouping like terms and simplifying gives
[tex]x^2-25x+84=0[/tex]
This is a quadratic equation which can be solved using a scientific calculator (as I did)
The solution set is
x = 21, x = 4
Remember x = number of green pens. We can ignore x = 4 because the total number of pens given by the expression 2x + 3 will be less than 12 and we are told there are more than 12 pens total
So the solution is x = 21 (number of green pens)
Number of blue pens = x + 3 = 21 + 3 = 24
Answer: Green Pens : 21 and Blue Pens: 24