According to probability, total number of handshakes will be 36.
How many handshakes are possible?Handshakes equal n * (n - 1)/2. This is the case because a handshake between two people is not counted twice and each of the n people can shake hands with n - 1 people (they wouldn't shake their own hand). Any number of people can be calculated using this formula.
The first person will need to shake eight (8) people's hands. Only seven (7) hands will need to be shook by the second person because the first handshake will not be repeated. Due to the same factors as mentioned above, only six (6) people will require a handshake from the third person. This continues until the final person has no one left to shake hands with because everyone has already done so.
The formula to calculate how many handshakes were exchanged is n = 8 + 7 + 6 +... + 1 = 36.
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will give brainliest please
Answer:
Step-by-step explanation:
a. we konw triangle have 180°
b. A straight line have 180°， so
∠abc=180-5x-12=168-5x,
∠bca=180-10x+37=217-10x
∠cab=180-9x-1=179-9x
then add them all 564 - 24x = 180
c.384=24x, x=16
d.so ∠bca=217-10·16=57.
Answer:
m∠BCA = 57°-------------------------------------------
Sum of exterior angles is 360°:
9x + 1 + 5x + 12 + 10x - 37 = 36024x - 24 = 36024x = 384x = 384/24x = 16The exterior angle at vertex C is:
10x - 37 = 10*16 - 37 = 160 - 37 = 123The ∠BCA is the linear pair with 123°:
m∠BCA = 180° - 123° = 57°How do you tell if a graph is a function and inverse is a function?
If and only if the outcome of reflecting a function's graph about the line y = x is the function's graph, then the function f has an inverse (passes the vertical line test).
What is a function?A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X.
The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
A function, according to a technical definition, is a relationship between a set of inputs and a set of potential outputs, where each input is connected to precisely one output.
The function f has an inverse if and only if the result of reflecting a function's graph about the line y = x is the function's graph (passes the vertical line test).
Therefore, if and only if the outcome of reflecting a function's graph about the line y = x is the function's graph, then the function f has an inverse (passes the vertical line test).
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A cylinder's height equals the diameter of its base. What fraction of the total surface area of the cylinder is the total area of the two bases
If the Cylinder's height equals to the diameter of its base , then the fraction of the total surface area of the cylinder to the total surface area of the two bases is 1/3 .
The curved surface area of the cylinder with height "h" and radius "r" is written as ⇒ [tex]2\pi rh[/tex] ;
the area of the top and bottom of the cylinder is ⇒ [tex]\pi r^{2} + \pi r^{2}[/tex] = [tex]2\pi r^{2}[/tex] ;
the total surface area of the cylinder will be ⇒ [tex]2\pi rh[/tex] [tex]+2\pi r^{2}[/tex] ....equation(1)
it is given that the height of cylinder is equal to the diameter of the base ,
that means ; [tex]h = 2r[/tex]
Substituting the value [tex]h = 2r[/tex] in the equation(1) ,
we get ,
total surface area of cylinder = [tex]4\pi r^{2} +2\pi r^{2}[/tex] = [tex]6\pi r^{2}[/tex]
So , the required fraction is = [tex]\frac{2\pi r^{2} }{6\pi r^{2} }[/tex] ,
On simplifying ,
we get ,
= [tex]\frac{1}{3}[/tex] .
Therefore , the fraction of two bases to the total surface area is [tex]\frac{1}{3}[/tex] .
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(b) If the radius of wheel is 7cm, how far it travels in three rotation
The wheel will travel 131.8cm in 3 rotations.
The radius of the wheel, r = 7cm
Number of rotations, n= 3
distance covered in 1 rotation= circumference of circle=2πr
distance covered in 3 rotation= 2πr x 3
where π=3.14(fixed)
=2 x 3.14 x 7 x 3
Distance=131.88cm
Therefore, the distance covered in 3 rotations is 131.88cm
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Answer:
132cm
Step-by-step explanation:
use formula 22/7 *diameter which is 7*2 getting the answer as 44.For three rotations multiply the answer by three
Denise's cell phone plan is $29.95 per month plus $0.10 per minute of call time. Denise's
cell phone bill is $99.95. For how many minutes was she billed?
Answer:
700 minutes
Step-by-step explanation:
Let y represent the cost and x represent the number of minutes of call time
We have the equation
99.95 = 0.10x + 29.95
Now let's solve
99.95 = 0.10x + 29.95
70 = 0.10x
x = 700 minutes
So, she was billed for 700 minutes
If you can, please give me a Brainliest; thank you!
All the backpacks at sports galore are on sale at 30% off the marked price. Tracy bought a backpack that was marked $60. What was the sale price?
Answer:
Step-by-step explanation:
we know the 30% off the marked price.
so the sale price is 60-60*30%=42
Answer:$42.00
Step-by-step explanation:
13. Which situation can be represented bya true proportion?
A car used 3 gallons of gas to drive 72 miles on a highway and used 4 gallons of gas to drive 72 mileš town.
A store charged $20 for 2 shirts and charged $25 for 3 shirts.
A signal light flashed yellow 7 times in 28 seconds and flashed red 5 times in 35 seconds.
An alarm on a clock beeped 7 times in 6 seconds and beeped 14 times in 12 seconds.
The example of a proportion, is An alarm on a clock beeped 7 times in 6 seconds and beeped 14 times in 12 seconds.
What is a proportion?The term proportionality describes any relationship that is always in the same ratio.
Given are some statements,
a) A car used 3 gallons of gas to drive 72 miles on a highway and used 4 gallons of gas to drive 72 miles in a town.
Solution:- In this case, we see, a car is using 3 and 4 gallons of gas for driving in a highway and in a town for 72 miles respectively.
Since, the distance is same and quantity of gases is different therefore, this can not be represented as a proportion.
b) A store charged $20 for 2 shirts and charged $25 for 3 shirts.
Solution:- In this, let's first find the unit prices for each,
for $20 there are 2 shirts, therefore, for 1 shirt it will be $10, and for 3 shirts it costs $25, therefore, for 1 shirt it will cost $8.3
Since, unit price for both is not equal, therefore, this can not be represented as a proportion.
c) A signal light flashed yellow 7 times in 28 seconds and flashed red 5 times in 35 seconds.
Solution:- As in 2nd option, will here also find unit rate, since, yellow light is flashing 7 times in 28 seconds, therefore, in 4 seconds it will be flashed for 1 time, similarly, red light is flashing 5 times in 35 seconds, therefore, in 7 seconds it will be flashed for 1 time,
Since, both unit rates are not equal, therefore, this can not be represented as a proportion.
d) An alarm on a clock beeped 7 times in 6 seconds and beeped 14 times in 12 seconds.
Solution:- Again we will find unit rate, since, the alarm is beeping 7 times in 6 seconds, therefore, it will beep for 1 time in 6/7 seconds, similarly, it is beeping 14 times in 12 seconds, so for 1 time it will beep in 12/14 = 6/7 seconds.
Here, we see that, both the unit rates are equal, therefore, this is an example of a proportion.
Hence, the last option is the correct example.
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The scatter plot shows a hikers elevation above sea level overtime. The equation of the trend line is shown y equals 8.77x plus 686. To the nearest whole number predicted hikers elevation will be at 145 minutes
The predicted elevation of hikers at 145 minutes, based on the trend line equation, y = 8.77x + 686 is 1,958.
What is a trend line equation?A trend line equation is the equation of a linear relationship, represented as y = mx + b.
In this linear equation, y is the dependent variable, x is the independent variable, m is the slope, and b is the y-intercept.
Equation:y = 8.77x + 686, where x = 145 minutes
Therefore,
y = 8.77(145) + 686
y = 1,271.65 + 686
y = 1,957.65
y = 1,958
Thus, after 145 minutes of hiking, hikers' elevation will be 1,958 units.
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How do you know if a triangle is right acute or obtuse?
if a²+b²>c², the triangle is acute, if a²+b2²=c², the triangle is a right triangle, and if a²+b²<c², the triangle is obtuse.
where a and b are the lengths of the two shorter sides of the triangle and c is the length of the longest side.
A right-angled triangle is a triangle with one of the angles at 90 degrees. A 90-degree angle is called a right angle, and hence the triangle with a right angle is called a right triangle.
An acute triangle has three angles that each measure less than 90°. In an acute-angled triangle, all three internal angles of the triangle are acute, that is, they measure less than 90°.
An obtuse triangle is a triangle with one angle that is greater than 90° degrees, it is not possible for a triangle to have more than one obtuse angle.
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Betsy and her sister have $15 to
spend on lunch. They buy several
mini-hamburgers, which cost $1. 80
cach. If they received change after
buying their food, which inequality
could be used to find h, the number
of mini-hamburgers Betsy and her
sister bought?
A 1. 89 s 15
B 1. 8 > 15
C 1. 82 > 15
D 1. 85 < 15
Therefore, the inequality 1.85 < 15 can be used to find h.
What is inequality?In mathematics, "inequality" refers to a relationship between two expressions or values that is not equal to each other. Therefore, inequality emerges from a lack of balance. For instance, let's say you have 225 and want to purchase a new bicycle that costs $250.
Define expressions or values.When the expression's variables and constants are given values, the calculation it describes yields the expression's value. In an algebraic expression, numbers can be represented by letters.
The inequality 1.85 < 15 can be used to find h, the number of mini-hamburgers Betsy and her sister bought. This inequality represents the constraint that the total cost of the mini-hamburgers (1.85h) must be less than the amount of money they have to spend (15). To solve for h, you would divide both sides of the inequality by 1.85, giving you h < 8.1, so h can be any number less than 8.1.
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Exponential from Two Points (Level 1)
Write an exponential function in the form y = ab* that goes through points (0, 13)
and (2, 325).
a = 13. b^0 = 1, y=a(1), 13 = a 325 = 13(b)^2 b^2 = 325/13 = 25
b = sqr25 = 5 y=13(5)^x is the exponential equation going through the two given points
What is Exponential function?An exponential function is a Mathematical function in the form f (x) = ax, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0. The most commonly used exponential function base is the transcendental number e, which is approximately equal to 2.71828.The graphs of all exponential functions have these characteristics. They all contain the point (0, 1), because a0 = 1. The x-axis is always an asymptote. They are decreasing if 0 < a < 1, and increasing if 1 < a. If your function can be written in the form y = a b x , where and are constants, a ≠ 0 , b > 0 , and b ≠ 1 , then it must be exponential. In quadratic equations, your functions were always to the 2nd power. In exponential functions, the exponent is a variableTo learn more about Exponential function refers to:
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jancie's cheerleading team designed two posters to hang around school to advertise their upcoming competition. the posters were shaped like parallelograms. the smaller of the two posters had a base of 6 1/2 feet and a total area of 16 1/4 feet. the larger poster had a base of 8 1/2 feet and a total area of 34 square feet. how much taller is the larger poster than the smaller poster?
Answer:
1.5 ft
Step-by-step explanation:
The area of a parallelogram is base times height.
So to find the height, you will need to use area/base
Small: 16.25/6.5=2.5 ft
Large: 34/8.5=4ft
4-2.5=1.5ft
How do you reflect over Y =- 4?
a car travels at a constant speed.what does the gradient of a distance-time graph of the car's journey represent?distance the car travelled, the area under the graph,the speed of the car, the acceleration of the car
the speed of the car
=========================================================
Reason:
The term "gradient" is the same as "slope".
Recall that slope = rise/run = (change in y)/(change in x).
The change in y represents how far you travel, as the distance is along the vertical axis. The change in x is the time duration.
Then,
speed = distance/time
which helps show the slope of this distance-time graph is the speed of the car. The steeper the slope, the faster the car.
-------------
An example:
The equation y = 50x has slope 50. The x is the time and y is the distance traveled.
At time x = 0 hours, you have traveled y = 0 miles.
At time x = 1 hour, you have traveled y = 50 miles.
The two points (0,0) and (1,50) are on this line. Use the slope formula to get a slope of 50, which represents the speed of the car in miles per hour (mph). Or note that y = 50x is in the form y = mx+b
m = slope = 50
b = y intercept = 0
At Clara's Beading Boutique, 9 out of the 18 beads on clearance are plastic. What percentage of beads on clearance are plastic?
The percentage of beads on clearance are plastic is 50%.
What is Percentage?Percentage is defined as the parts of a number per fraction of 100. We have to divide a number with it's whole and then multiply with 100 to calculate the percentage of any number.
So the percentage actually means a part per 100.
Total number of beads = 18
Number of beads which are plastic on clearance = 9
Percentage of beads which are plastic on clearance = (9 / 18) × 100
= (1/2) × 100
= 0.5 × 100
= 50%
Hence 50% of beads on clearance are plastic.
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1/4(12x-20)+3(3x+5) I'm extremally confused I'm supposed to simplify
Answer:
12X+10
Step-by-step explanation:
THAT IS THE SIMPLFY 1-4 x (12x-20)+3(3x+5) Facout our 4 from the expression 1-4 x4 (3x-5)+3(3x+5) Simply the expression cancel out the greatest common factor 4 remove unncessary parntheses if so collect the terms 3x 9x 12x and the solutin is 12x+10
How doe etimation help you place the decimal point in a product correctly?Explain
By counting the numbers of figures after the decimal points before you locate the product and shifting the decimal point from the rear to the front in the order you previously counted it after you find the product, an estimation can help you place the decimal point in a product correctly.
What is Estimation?A high number of a certain number is estimated to be the closest whole number or significant number.
When estimating, the use of decimal points aids in separating the fractional portion from the entire figure.
For instance, we have a total of three integers after the decimal point if we want to calculate the product of 4.23 and 2.1.
4.23
× 2.1
4 2 3
________
8 4 6
8 8 8 3
Therefore, we shall move the decimal point three places, working our way from the back to the front.
The result of 4.23 and 2.1 is 8.883 then.
We can therefore say that we completely understand how estimating helps you accurately set the decimal point in a product.
Therefore, by counting the numbers of figures after the decimal points before you locate the product and shifting the decimal point from the rear to the front in the order you previously counted it after you find the product, an estimation can help you place the decimal point in a product correctly.
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Correct question:
How does estimation help you place the decimal Point in a product correctly?
someone help me plsss
x2 - 3x - 5 = 0 has a unique solution of type -11, which is what the equation entails.
what is discriminant ?The "discriminant" is the value that aids in identifying the many types of solutions when using the quadratic formula to solve a problem. It is a unique function of a co-efficient in a mathematical equation (a quadratic equation), and the value it yields provides details about the roots of the equation. Calculating the discriminant involves square rooting the "b" term and subtracting four times the "a" term from the "c" term. Alongside "D," there is a (delta) that represents discrimination. A quadratic equation with two distinct real number solutions is said to have a positive discriminant. A quadratic equation with a discriminant of zero has repeated real number solutions. Both of the solutions are not real numbers, as indicated by a negative discriminant.
given
x2 - 3x - 5 = 0
D=b2−4ac
D= −32−4×1×5
D=92−20
D= −11
x2 - 3x - 5 = 0 has a unique solution of type -11, which is what the equation entails.
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7. Plumber A charges $60 an hour. Plumber B charges $40 to visit your home plus $55 for each
hour. For how many hours will the total cost for each plumber be the same? How much will that
cost be? If a customer thinks they will need a plumber for 5 hours, which plumber should the
customer hire? Explain.
The total cost will be $480.
What is an Equations?
Equations are statements in mathematics that have two algebraic expressions on either side of the equals (=) symbol. It demonstrates how the expressions written on the left and right sides are equivalent. Equations can be solved to find the value of a variable that represents an unknown quantity. If there is no "equal to" symbol in a sentence, it is not an equation. It will be viewed as a phrase.
There should be two equations
For Plumber A = 60x
For Plumber B = 40 + 55x
Now in order to consider the cost to be the same we equate these two equations
60x = 40 + 55x
5x = 40
x = 8 hours
And, the total cost for plumber B is
= 40 + 55(8)
= 480
Hence, The total cost will be $480.
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Give an example of a quadratic function that has a maximum value. How do you know that it has a maximum value?
An example of a quadratic function that has a maximum value is the parabola defined by the equation y = -(x-3)^2 + 4.
What is quadratic function?A quadratic function is a type of polynomial function that can be written in the form y = ax^2 + bx + c, where a, b, and c are constants and x is the variable. The graph of a quadratic function is a parabola, which can either open upward or downward depending on the value of the leading coefficient (a).
What is equation?An equation is a mathematical statement that asserts the equality of two expressions. It is usually written with an "=" sign between the two expressions, such as y = 2x + 3 or x^2 + y^2 = 1. Equations can be used to model real-world situations, and solving an equation means finding the values of the variables that make the equation true.
We can determine that this function has a maximum value by looking at the leading coefficient of the quadratic term (x^2). In this case, the coefficient is -1, which means that the parabola opens downward. Therefore, since the parabola opens downward, the vertex of the parabola is the highest point of the parabola, which means that it is a maximum point.
Also we can find the vertex of the parabola by using the formula x = -b/2a, in this case x= -0 / 2*-1 = 3/2 =1.5, and using this value in the function, we get the y value which is 4, so the vertex is (1.5,4) which confirms that the function has a maximum value at (1.5,4)
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What is the Value of P? 9=p÷9
Answer:
p = 81
Step-by-step explanation:
9 = [tex]\frac{p}{9}[/tex] Multiply both sides by 9
9(9) = [tex]\frac{p}{9}[/tex][tex](\frac{9}{1})[/tex] another name for 9 is [tex]\frac{9}{1}[/tex]
81 = p
Which equations have the same value of x as Three-fifths (30 x minus 15) = 72? Select three options. 18 x minus 15 = 72 50 x minus 25 = 72 18 x minus 9 = 72 3 (6 x minus 3) = 72 x = 4.5
18x - 9 = 72 and x = 4.5 represents the same value as 3/5(30x - 15) =72.
What is equation?Mathematically, an equation can be defined as a statement that supports the equality of two expressions, which are connected by the equals sign “=”. For example, 2x – 5 = 13.
Here,
2x – 5 and 13 are expressions
The sign that connects these two expressions is “=”.
Given
x = 4.5
And equation
3/5(30x - 15) =72
Simplifying
90x/5 - 45/5 = 72
18x - 9 = 72 -----------(a)
18x = 81
x = 81/18
x = 4.5 ------------(b)
Equation (a) and (b) are among the equations given in the options.
Hence, from the above calculation equations 18x - 9 = 72 and x = 4.5 express the same value of x as 3/5(30x - 15) =72 .
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The cost of a certain cell phone plan can be modeled by C(m) = 2m + 15, where C(m) is the total cost for m minutes of cell phone usage. State the meaning of the slope with respect to the cost associated with the cell phone plan.
The interpretation of the slope is that the cost per minute is $2
How to determine the interpretation of the slope?From the question, we have the following parameters that can be used in our computation:
C(m) = 2m + 15
Where
m = number of minutes
C(m) = cost function
The equation of a line can be represented as
y = mx + c
Where
Slope = m
By comparing the equations, we have the following
m = 2
This means that the slope is 2
And it means that the cost per minute is $2
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How do you check if a function has an inverse function?
We can check if a function has an inverse function using horizontal Line Test.
What is Function?
A relation between a collection of inputs and outputs is known as a function. In plain English, a function is an association between inputs in which each input is connect
horizontal Line Test
Make f a function. Any horizontal line that crosses the f graph more than once indicates that f lacks an inverse. F does have an inverse if there are no horizontal lines that cross the graph of f more than once.
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Find the measure of the indicated angle to the nearest degree.
Answer:
57⁰
Step-by-step explanation:
To get the indicated angle we need to use the knowledge of trigonometry.
Since the required angle has both opposite and adjacent sides we use Tangent
Tan ? = opposite/adjacent
= 37/24
= 1.5412
To get the angle we now need to to find the tan inverse of the value we found
Tan inverse of 1.5412
= 57.03⁰
to the nearest degree it will be 57⁰
Select the equation in standard form for the line that passes through (-9, 8) and (1, -12).
A) x + 2y = -20
B) 2x + y = -13
C) 2x+y = 7
D) 2x+y = -10
E) x + 2y = 20
Answer: D
Step-by-step explanation:
Since we are given 2 points, we can use that to find slope, convert to slope-intercept form, and then manipulate it to be standard form. Standard form is ax+by=c.
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{-12-8}{1-(-9)} =\frac{-20}{10} =-2[/tex]
With m=-2, we can plug it into slope-intercept form to find the y-intercept.
[tex]y=-2x+b[/tex] [plug in point]
[tex]8=-2(-9)+b[/tex] [multiply]
[tex]8=18+b[/tex] [subtract both sides by 18]
[tex]b=-10[/tex]
With m and b, we can plug in the equation.
[tex]y=-2x-10[/tex] [add both sides by 2x]
[tex]2x+y=-10[/tex]
This gives D as the answer.
Given: △ACE,BD¯¯¯¯¯∥AE¯¯¯¯¯ Prove: BACB=DECD A triangle with vertices labeled as A, C, and E, with base A E. Sides C A and C E contain midpoints B and D, respectively. A line segment parallel to base A E is drawn from point B to D. Angle C A E is labeled as 4 and angle C E A is labeled as 3. Angle C B D is labeled as 1 and angle C D B is labeled as 2. Drag an expression or phrase to each box to complete the proof. Statement Reason △ACE,BD¯¯¯¯¯∥AE¯¯¯¯¯ Given Response area Corresponding Angles Postulate △ACE∼△BCD Response area CACB=CECD Definition of similar triangles CA=CB+BACE=CD+DE Segment Addition Postulate CB+BACB=CD+DECD Substitution Property of Equality CBCB+BACB=CDCD+DECD Addition of fractions 1+BACB=1+DECD Simplification of fractions Response area Subtraction Property of Equality
For triangle ACE, if BD ∥AE , then BA/CB = DE/CD is proved.
For given triangle ACE,
BD || AE ...............(Given)
∠4 ≅ ∠1 and ∠3 ≅ ∠2 ..............(Corresponding Angles Postulates)
ΔACE ~ ΔBCD .............(Angle-Angle Similarity Postulate)
CA/CB = CE/CD ............(Definition of similar triangles)
CA = CB + BA, CE = CD + DE ..........(Segment Addition Postulate)
(CB + BA)/CB = (CD + DE)/CD ..........(Substitution Property of Equality)
CB/CB + BA/CB = CD/CD + DE/CD .........(Addition of Fractions)
1 + BA/CB = 1 + DE/CD .............(Siplification of fractions)
BA/CB = DE/CD ...............(Subtraction Property of equality)
Hence proved.
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There are 96 girls in a junior class of 176 students. Find the ratio of girls to boys.a.11:6c.6:11b.5:6d.6:5
The ratio tells us how often one number is contained in another. In the class, there are six girls for every five boys. D is the right answer.
What is a Ratio?A ratio is an ordered pair of numbers a and b, expressed as a / b, where b is not equal to 0. A percentage is an equation in which two ratios are made equal to one another. For instance, you could express the ratio as 1: 3 if there is 1 guy and 3 girls (for every one boy there are 3 girls) One-fourth are boys and three-fourths are girls.
The ratio tells us how often one number is contained in another.
Considering that 96 of the 176 junior students are female. Consequently, the class's gender distribution is,
96 girls are present.
80 boys out of 176, or 96 boys.
Currently, the junior class's gender distribution is as follows:
Girls to Boys in Ratio
96 girls and 80 boys.
Divide the numerator and denominator by 16 to simplify.
Boys/Girls: 6 to 5.
Therefore, the answer is option d) 6:5.
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1. Calculate the unit rate if you are
shopping and you buy 10 pounds of
bananas for $8.50. What is the
cost per pound?
The cost per pound of bananas is $0.85.
To calculate the unit rate or cost per pound, we divide the total cost by the number of pounds.
You purchased 10 pounds of bananas for $8.50.
To find the cost per pound, we divide the total cost by the number of pounds:
Cost per pound = Total cost / Number of pounds
The total cost is $8.50 and the number of pounds is 10. So we can calculate the cost per pound as follows:
Cost per pound = $8.50 / 10 pounds
To perform the division, we divide the numerator ($8.50) by the denominator (10 pounds):
Cost per pound = $0.85
This means that for every pound of bananas you purchase, it will cost you $0.85.
The unit rate allows you to compare the cost of different quantities.
It tells you that if you want to know the cost of a certain number of pounds of bananas, you can multiply that number by $0.85 to find the total cost.
It's important to note that unit rates are helpful for comparing prices and making informed purchasing decisions, especially when dealing with different quantities or sizes of items.
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Suppose a, b, c, and d are constants such that a is not zero and the system below is consistent for all possible values of f and g . What can you say about the numbers a, b, c, and d?
Justify your answer. axı + bx2 = f cxı + dx2 = g
a and d are non-zero and ad = bc for the system to be consistent for all possible values of f and g.
By definition, a system is consistent if there exists a set of solutions that satisfy the equations. In this system, the constants a, b, c, and d are the coefficients of the two equations. Since the system is consistent for all possible values of f and g, then the coefficients must satisfy certain conditions. For example, a and d must both be non-zero in order for the equations to be solvable. Additionally, the coefficients of both equations must be in proportion to each other, meaning that ad must equal bc. This is necessary for the equations to have the same slope, which is necessary for them to be solvable. Thus, we can conclude that a and d are non-zero and ad = bc for the system to be consistent for all possible values of f and g.
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