The greatest common factor and least common multiple are the best used to simplify a fraction.
Suppose we wanted to add the fractions:
10/12 and 20/15.
Now, consider the fraction:
10/12
We can eliminate that component from both the numerator and the denominator and simplify the fraction if we can identify the greatest common factor between 10 and 12.
So,
The factors of 10 are 1, 2, 5, 10.
The factors of 12 are 1, 2, 3, 4, 6, 12.
Therefore, GCF of 10 and 12 is 2.
Hence,
10/12 = ( 10 ÷ 2 ) / ( 12 ÷ 2 ) = 5/6
Similarly,
For 20/15,
The factors of 20 are 1, 2, 4, 5, 10, 20.
The factors of 15 are 1, 3, 5, 15.
The greatest common factor is 5.
So,
20/15 = ( 20 ÷ 5 ) / ( 15 ÷ 5 ) = 4/3
Now, when we add the fractions:
10/12 + 20/15 = 5/6 + 4/3
Now, as the fractions are already simplified.
By using the least common multiple.
10/12 + 20/15 = 5/6 + ( 4/3 ) × ( 2/2 )
10/12 + 20/15 = 5/6 + 8/6
10/12 + 20/15 = 13/6
Hence, using the greatest common factor we can simplify the fractions, and the least common multiple we can perform operations on the fractions.
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(4, -13) and (8,-8) slope formula
Point A (4, -13)
Point B (8, -8)
Slope formula
[tex]\begin{gathered} m=\frac{-8-(-13)}{8-4} \\ m=\frac{-8+13}{4} \\ m=\frac{5}{4} \\ \end{gathered}[/tex]The slope would be 5/4
Solve the system of equation by substitution. x=7y+45x-4y=-11
Mila, this is the solution:
x = 7y + 4
5x - 4y = -11
_________________
Step 1: Let's substitute x in the second equation and solve for y, as follows:
For which value(s) of x will the rational expression below be undefined? Check all that apply
Given:
[tex]\frac{(x+3)(x+6)}{x+7}[/tex]Required:
We need to find the value of x that makes the given expression is undefined.
Explanation:
Recall that a rational expression is undefined when the denominator is equal to zero.
The denominator of the given expression is x+7.
Equate it to zero.
[tex]x+7=0[/tex]Subtract 7 from both sides of the equation.
[tex]x+7-7=0-7[/tex][tex]x=-7[/tex]Final answer:
[tex]x=-7[/tex]You have 10 gallons of lemonade to sell. [tex](1gal = 378 {cm}^{2})[/tex] Each customer uses 1 paper cup. The cups are sold in packages of 50. How many packages should you buy?
The volume of a cilinder (like the cup) is given by:
[tex]V=\pi r^2h[/tex]Then, each cups can hold:
[tex]V=\pi(8)^2(11)=2211.68[/tex]Now, since each gallon is equivalent to 3785.41 cubic cm then we have a total of 10 times 3785.41=37854.1 cubic cm of lemonade.
Therefore we need:
[tex]\frac{37854.1}{2211.68}=17.11[/tex]Hence, we need 18 cups of lemonade and we only have to buy one package.
Write this as a decimal number.
1 hundred + 7 tens + 3 tenths + 8 thousandths
Answer:
Step-by-step explanation:
17.308
this is the decimaal
write the letter of the table that corresponds with the graph.Explain your answer.
for the table X :
for the table R :
For tabel V :
For tabel Q :
An animal shelter currently only has 19 dogs and 13 cats. What is the ratio of cats to animals?
Answer:
13:32
Step-by-step explanation:
Given: x-8> -3.
Choose the solution set.
OXIXER, Xx>-5]
OXIXER,X > 14]
OXIXER,X > 5)
OXIXER,X> -9)
Solution set for x - 8 > -3 will be [ x|x E R, x > 5 ] --(option 3)
We have given rule x - 8 > -3.
Adding 8 on both side of given rule.
∴ x - 8 + 8 > -3 + 8
By Simplification of given rule.
∴ x > 5
Solution set : -The solution set of an inequality is the set of all solutions. An inequality usually has an infinite number of solutions, and the set of solutions can be easily written using interval notation.
A solution set is the set of all variables that makes the equation true.
Here as given solution set will be all the numbers greater that 5 excluding 5.
∴ Solution set will be [ x|x E R, x > 5 ] --(option 3)
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Explain the difference between the two different function transformations represented by
f(x+2)
f(x-2)
Include an example of each using the parent cubic equation,
f(x) = x³
For full credit, include the following:
1.
A description of each transformation including direction and number of units
2.
3 cubic equations and one graph showing the 3 curves: f(x), f(x+ 2), and f(x - 2)
The difference between the functions is explained below
What are functions?
A function in mathematics from a set X to a set Y allocates precisely 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. The set of all pairs, also known as the function's graph, is the only way to express a function in a unique way. In science, engineering, and the majority of the branches of mathematics, functions are often utilized. Functions are allegedly "the principal objects of inquiry" in the majority of mathematical disciplines. Maps and mappings are other names for functions.
Two transformations of the function f(x) are given f(x+2) and f(x-2)
The first one increments the function by 2 and the next one decrements the function by 2.
Let us understand with the help of an example,
The function f(x) = x³
Then f(x+2) = (x+2)³ = x³ + 4x + 8
and f(x-2) = (x-2)³ = x³ - 4x + 8
We can clearly see that both functions expanded differently
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Jane wants to make a a cake for her friend's birthday she had $9 and spent it on decoration with tax included a pack of 10 candles is $1.50 and which tube of of colored frosting is 2.75 if she has to bye two packs and candles how many tubes of frosting can she buy ?
Given:
Total money she had = $9
Cost of each pack of candles = $1.50
Cost of each tube of colored frosting = $2.75
Let's find the number of colored frosting she can buy.
Since she has to buy 2 packs of candles, the total cost she would spend on candles is:
Cost of candles = $1.50 x 2 = $3.00
Thus, we have the inequality:
2.75x + 3.00 ≤ 9
Where x represents the number of tubes of colored frosting she can buy.
Let's solve for x.
2.75x + 3 ≤ 9
Subtract 3 from bith sides:
2.75x + 3 - 3 ≤ 9 - 3
2.75x ≤ 6
Divide both sides by 2.75:
[tex]\begin{gathered} \frac{2.75x}{2.75}\le\frac{6}{2.75} \\ \\ x\le2.18 \end{gathered}[/tex]Since x is less than or equal to 2.18, this means Jane can buy no more than 2 tubes of colored frosting.
ANSWER:
No more than 2 tubes
The perimeter of an isosceles triangle is 30 cm. The length of each congruent side is 3cm more than the length of its base. Find the lengths of all the sides.
Given:
the perimeter of an isosceles triangle 30cm
The length of each congruent side is 3cm more than the length of its base.
Let the length of the base side = x
So, the length of each congruent side = x + 3
the perimeter is the sum of the lengths of the sides of the triangle
So,
[tex](x+3)+(x+3)+x=30[/tex]Solve the equation to find the value of x
[tex]\begin{gathered} x+3+x+3+x=30 \\ 3x+6=30 \\ 3x=30-6 \\ 3x=24 \\ \\ x=\frac{24}{3}=8 \end{gathered}[/tex]So, the length of the base side = 8 cm
And the length of each congruent side = 8 + 3 = 11 cm
So,
The lengths of all sides are: 11 cm, 11 cm and 8 cm
4. Jacob bought some tickets to see his favorite group, and it cost $76. The relationship between the adult tickets, a, and the student's tickets, s, can be expressed by the equation 10a + 8C = 76. If he bought 4 adult ticket, then how student's tickets did he buy? If he bought 2 student ticket, then how adult's tickets did he buy? Which equation shows the number of student tickets as a function of the number of adult tickets? A. C= 68 – 10a B.C=76 – 10a C. C= -4/5a +38/5 D. C=-5/4a+19/2
Given the equation:
[tex]10a+8c=76[/tex]If Jacob bought 4 adult tickets, then a = 4, so we can solve for c:
[tex]\begin{gathered} a=4 \\ \Rightarrow10(4)+8c=76 \\ \Rightarrow8c=76-40=36 \\ \Rightarrow c=\frac{36}{8}=\frac{9}{2}=4.5 \\ c=4.5 \end{gathered}[/tex]therefore, Jacob bought 4 or 5 students tickets.
Now, if Jacob bought 2 student tickets, then c=2 and for 'a' we have the following:
[tex]\begin{gathered} c=2 \\ \Rightarrow10a+8(2)=76 \\ \Rightarrow10a=76-16=60 \\ \Rightarrow a=\frac{60}{10}=6 \\ a=6 \end{gathered}[/tex]therefore, Jacob bought 6 adult tickets.
Finally, to find the equation that shows the number of student tickets as a function of adult tickets, we have to solve for 'c' to get the following:
[tex]\begin{gathered} 10a+8c=76 \\ \Rightarrow8c=76-10a \\ \Rightarrow c=-\frac{10}{8}a+\frac{76}{8}=-\frac{5}{4}a+\frac{19}{2} \\ c=-\frac{5}{4}a+\frac{19}{2} \end{gathered}[/tex]therefore, the function would be c = -5/4a +19/2
A linear function that models a relationship where the x-values increases by 4 units for every 2 units the y-value increases. The value of the function at x=0 is -2.
The most appropriate choice for equation of line in slope intercept form will be given by
x -2y = 4 is the required linear function.
What is equation of line in slope intercept form?
Equation of line in slope intercept form is given by y = mx + c
Where, m is the slope of the line and c is the y intercept of the line
The distance from the origin to the point where the line cuts the x axis is called x intercept
The distance from the origin to the point where the line cuts the y axis is called y intercept
Slope of a line is the tangent of the angle which the line makes with the positive direction of x axis
If [tex]\theta[/tex] is the angle which the line makes with the positive direction of x axis, then slope of the line is given by [tex]tan \theta[/tex]
If the line passes through ([tex]x_1, y_1[/tex]) and ([tex]x_2, y_2[/tex])
slope = [tex]\frac{y_2 - y_1}{x_2 - x_1}[/tex]
Here,
Increase in x value = 4 units
Increase in y value = 2 units
Slope = [tex]\frac{2}{4}[/tex]
Slope = [tex]\frac{1}{2}[/tex]
Let the equation of line be [tex]y = \frac{1}{2}x + c[/tex]
When x = 0, y = -2
[tex]-2 = \frac{1}{2}\times 0+c[/tex]
c = -2
Equation of line
[tex]y = \frac{1}{2} x - 2\\2y = x - 4\\x - 2y = 4[/tex]
This is the required linear function.
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According to Greg, perfect cherry pies have a ratio of 240 cherries to 3 pies. How many cherries does Greg need to make 15 perfect cherry pies?
Answer: 1200 cherries.
Step-by-step explanation:
If 240 cherries are need for 3 pies, multiply 240 by 5 to get the answer.
Your answer is 1200 cherries.
Greg needs 1200 cherries to make 15 perfect cherry pies.
Question 15 of 15Taniah is trying to find the area of the following square whereWY=10. What is the area?
The area of a square given its diagonal (d) is computed as follows:
[tex]A=\frac{1}{2}d^2[/tex]In this case, the diagonal is 10 units long, then the area is:
[tex]\begin{gathered} A=\frac{1}{2}\cdot10^2 \\ A=\frac{1}{2}\cdot100 \\ A=50\text{ square units} \end{gathered}[/tex]Jerry tries to exercise each day of the week. He burns 850 calories when he runs and 200 calories when he lifts weights. Last week he exercised for a total of 6.5 hours and burn 2,925 total calories. How long does he spend on each exercise? Which augmented matrix can be used to represent the system of equations used to solve the problem above?
SOLUTION:
Case: Augmented Matrix
Given:
In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.
Given:
He burns 850 calories when he runs.
he burns 200 calories when he lifts weight
he exercised for a total of 6.5 hours
he burns 2,925 total calories
Method:
First, we make a linear equation:
Assumption: Let the number of hours he runs be r. The number of hours he lifts weights be w.
The equation therefore is:
Hours:
r + w = 6.5
Calories:
850x + 200y = 2925
Final answer:
Augmented Matrix
[tex]\begin{bmatrix}{1} & {1} & {|6.5} \\ {850} & {200} & {|2925} \\ {} & {} & {}\end{bmatrix}[/tex]factoring is writing an expression as the product of two factors; what are those factors ? 6(5+3)
Given the expression 6(5+3)
The expression shows that 6 was factored out from the bracket
evaluate the expression in parenthesis
= 6(5+3)
= 6(8)
This shows that the factors of the expression are 6 and 8
Determine if the function is an example of exponential growth or exponential decay:A) f(x) = (1/3)^xB)f(x) =8^xC) f(x) =0.3^x
a) decay
b) growth
c) decay
Explanation:Exponential function is given as:
[tex]y=ab^x[/tex]To determine if the function is an exponential growth or decay:
when a is positive and b is greater than 1, it is a growth
when a is positive and b is less than 1, it is a decay
[tex]\begin{gathered} a)\text{ f(x) = (}\frac{1}{3})^x \\ a\text{ in this case 1 (positive)} \\ b\text{ = 1/3 (it is less than 1)} \\ \text{Hence, it is a decay} \end{gathered}[/tex][tex]\begin{gathered} b)f(x)=8^x \\ a\text{ in this case 1 (positive)} \\ b\text{ = 8 (it is greater than 1)} \\ \text{Hence, it is a growth} \end{gathered}[/tex][tex]\begin{gathered} c)\text{ }f\mleft(x\mright)=0.3^x \\ a\text{ in this case 1(positive)} \\ b\text{ = 0.3 (it is less than 1)} \\ \text{Hence, it is a decay} \end{gathered}[/tex]Thor goes to the store and buys a game station for $75. The store is running a discount of 10% What is the price of the game station after the discount?
To find the price after the discount, find at first, the amount of discount then subtract it from the original price
Since the discount is 10% of $75, then
Change 10% to decimal by divide it by 100, then multiply it by 75
[tex]\begin{gathered} \frac{10}{100}\times75=0.1\times75 \\ =7.5 \end{gathered}[/tex]The discount is $7.5
Subtract it from $75 to find the price after the discount
The price after discount = 75 - 7.5 = 67.5
The price after the discount is $67.5
Show how to use place value to add 354 + 271.
Answer:
Step-by-step explanation:
300+50+4+200+70+1
=500+120+5
=625
The length of a rectangle i 5 centimeters more than the width. The area of the rectangle is 36 square centimeters. What is the length?
Answer: 9 cm
Step-by-step explanation:
l=length. A=area. w=width.
l*w=A
l*(l-5)=36 ==> w=l-5
l^2-5l=36
l^2-5l-36=0
l^2-9l+4l-36=0
l(l-9)+4(l-9)=0
(l+4)(l-9)=0
9*4=36
4=9-5 ==> w=l-5
l=9 cm
Pls help fast it is due in a few minutes
The day in which Chloe would have more money than Kenzi is after 15 days.
When would Chloe have more money?The equation that represents the amount left after the purchase of Frutti Tutti and lunch is:
Amount left = amount she began with - (amount spent x number of days)
Amount that Chloe has left : $75 - $3x
Amount that Kenzi has left : $90 - $4x
The inequality that can be used to represent when Kenzi would have less money than Chloe is:
$75 - $3x > $90 - $4x
$4x - 3x > $90 - $75
x > 15
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The complex number w is given by w= p-4i/2-3i where p is a real constant. Express w in the form a+bi where a and b are real constants. give your answer in simplest terms of p.b) given that arg w= pi/4 find the value of p
Step 1
Given;
[tex]w=\frac{p-4i}{2-3i}[/tex]Required; To express w in the form of a+bi.
Step 2
Express w in the form of a+bi
Multiply the numerator and the denominator by the binomial conjugate of (2-3i)
The binomial conjugate of (2-3i) = (2+3i)
[tex]w=(\frac{p-4i}{2-3i})\times(\frac{2+3i}{2+3i})=\frac{2p+3pi-8i-12i^2}{4+6i-6i-9i^2}[/tex][tex]\begin{gathered} w=\frac{2p+3pi-8i-12(-1)}{4-9(-1)} \\ \text{Note; i}^2=(\sqrt[]{-1})^2=-1 \end{gathered}[/tex][tex]\begin{gathered} w=\frac{2p+3pi-8i+12}{4+9}=\frac{2p+3pi-8i+12}{13} \\ w=\frac{(2p+12)+(3p-8)i}{13} \\ w=\frac{(2p+12)}{13}+\frac{(3p-8)i}{13} \\ w=\frac{2(p+6)}{13}+\frac{(3p-8)i}{13} \end{gathered}[/tex]where;
[tex]\begin{gathered} a=\frac{2(p+6)}{13}_{} \\ bi=\frac{(3p-8)i}{13} \end{gathered}[/tex]Kite ABDC is shown. What is the value of x?
In a kite, one diagnal bisects the angle its passes through. This means that
4x - 5 = 2x + 35
4x - 2x = 35 + 5
2x = 40
x = 40/2
x = 20
For the given kite ABDE, the value of x will be 20. Option D is correct.
What is quadrilateral?It is defined as a four-sided polygon in geometry having four edges and four corners.
It is given in the figure, the two angles are (4x-5) and (2x+35).
Since the kite has two sets of congruent sides, we know it is a quadrilateral. It has a single pair of incongruent, opposing angles. A kite's diagonals are perpendicular. One diagonal is split in half. Although the top and bottom angles are divided in half, the diagonals are not congruent.
One diagonal in a kite divides the angle it traverses. It follows that
4x - 5 = 2x + 35
4x - 2x = 35 + 5
2x = 40
x = 40/2
x = 20
Thus, for the given kite ABDE, the value of x will be 20. Option D is correct.
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Krissy owns a burger restaurant in Chicago, Illinois. The number of burgers sold at Krissy's restaurant was 605 in June. The number sold in July was 20% more than the number sold in June. How many burgers did Krissy's restaurant sell in July? burgers.
Answer:
100% ~ 605
20% ~ 121
120% ~ 726
The restaurant sells 726 burgers in July.
Which of the following statements is NOT true regarding an expression written in scientific
notation in the form of ax10*?
O The value of a must be greater than or equal to 1 and smaller than 10.
O The value of n must be an integer.
O Doubling n results in a doubling of the value of the expression.
O Doubling a results in a doubling of the value of the expression.
Step-by-step explanation:
not true is " doubling n results in a doubling of the value of the expression".
e.g.
2.3 × 10² = 230
now I double n :
2.3 × 10⁴ = 23,000
the value of the expression has increased much more than just doubling.
Naomi is driving on a long road trip. She currently has 10 gallons of gas in her car. Each hour that she drives, her car uses up 0.75 gallons of gas. How much gas would be in the tank after driving for 4 hours? How much gas would be left after t hours?
7 Gallons of gas would be in the tank after driving for 4 hours. And,
10 - 0.75t gas would be left after t hours.
We have given that the total amount of gas in her car is 10 gallons and the car consumes 0.75 gallon per hour.
First we will determine the amount of gas left in the car after driving for four hours.
Since 0.75 gallon is consumed in an hour, to find the amount of gas consumed in 4 hours we will multiply the amount consumed in an hour to 4 hours.
That would be 0.75 × 4 = 3 Gallons of Gas
Now, the amount of gas left is simply the difference between the initial amount of gallons in the car and the amount of gallons spend in 4 hours.
That would be 10 - 3 = 7 Gallons of Gas
Now with the help of above concept, we will determine the amount of gas left after t hours.
Since 0.75 is used up per hour, the amount used up in t hours would be 0.75 × t = 0.75t
Now, the amount of fuel that would be left in the car would be the difference between what we had initially and what was used up.
That would be 10 - 0.75t .
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Calculate the area of each figure. Which figure has the greatest area? 9 cm 6.8 cm 7 cm l-_- 10 cm 15 cm 6 cm 13 cm
For the frst one is: 63 cm^2
[tex]A_{F1}=\text{ 9cm}\cdot7\operatorname{cm}=63cm^2[/tex]For the second figure we have that the area is: 136cm^2
[tex]\begin{gathered} A_{F2}=2\cdot(6.8cm)(10cm) \\ =2(68cm^2)\text{ = 136}cm^2 \end{gathered}[/tex]And for the third one, the area is: 84cm^2
[tex]\frac{(15cm\text{ + 13cm)}\cdot6\operatorname{cm}}{2}=\frac{28\operatorname{cm}\cdot6\operatorname{cm}}{2}=\frac{168\operatorname{cm}^2}{2}=84\operatorname{cm}[/tex]Does the table show direct variation? If so, state the constant of variation.
Okay, here we have this:
Considering the provided table, we are going to analize if the table shows direct variation, so we obtain the following:
To identify if there is direct variation then we will calculate the ratios between the points, and if they are all the same then the table does show direct variation, then we have:
y=kx
k=y/x
5/2=2.5
45/18=2.5
80/30=2.66
Since the ratios between the points are not the same, then it does not represent a direct variation.
write equation in standard form
y= - 9/11x- 4
The standard form of the linear equation y = -9/11x - 4 is 9x + 11y = -44.
What is the given equation in standard form?The standard form of a linear equation is expressed as;
Ax + By = C
Where A and B are the coefficient of x and y respectively, C is the constant term.
Given the equation in the question;
y = -9/11x - 4
First, get rid of the fraction by multiplying both sides by //
11( y ) = 11( -9/11x - 4 )
Apply distributive property
11y = -9x - 44
Now, move all terms containing variables to the left and side of the equation.
11y + 9x = -44
Reorder the equation in form of Ax + By = C
9x + 11y = -44
Therefore, the standard form of the equation is 9x + 11y = -44
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