If α and β are the Roots of a Quadratic Equation ax² + bx + c then :
✿ Sum of the Roots : α + β [tex]\mathsf{= \frac{-b}{a}}[/tex]
✿ Product of the Roots : αβ [tex]\mathsf{= \frac{c}{a}}[/tex]
Let the Quadratic Equation we need to find be : ax² + bx + c = 0
Given : The Roots of a Quadratic Equation are 6 and 3
⇒ α = 6 and β = 3
Given : The Leading Coefficient of the Quadratic Equation is 4
Leading Coefficient is the Coefficient written beside the Variable with Highest Degree. In a Quadratic Equation, Highest Degree is 2
Leading Coefficient of our Quadratic Equation is (a)
⇒ a = 4
⇒ Sum of the Roots [tex]\mathsf{: (6 + 3) = \frac{-b}{4}}[/tex]
⇒ -b = 9(4)
⇒ b = -36
⇒ Product of the Roots [tex]\mathsf{: (6 \times 3) = \frac{c}{4}}[/tex]
⇒ c = 18 × 4
⇒ c = 72
⇒ The Quadratic Equation is 4x² - 36x + 72 = 0
Arnolds workout consisted of 10 minutes of warm up exercises 25 minutes of lifting weights and 15 minutes on the treadmill. What was the ratio of the number of minutres he lifted weights to the total number of minitues of his entire workout
Can anybody help me with #14?
Answer:
The student made an error in step 3.
Step-by-step explanation:
Step 2 is correct but 42/-6 is 12.
given RST = NPQ R=-7x+9, N=-10x find R and N
Answer:
[tex]m\angle N=m\angle R=30^{\circ}[/tex]
Step-by-step explanation:
We are given that
[tex]\triangle RST=\triangle NPQ[/tex]
[tex]m\angle R=-7x+9,m\angle N=-10x[/tex]
We have to find the measure of angle R and measure of angle N.
When two triangles are equal then their angles are equal
Therefore, [tex]m\angle R=m\angle N[/tex]
[tex]-7x+9=-10x[/tex]
[tex]-7x+10x=-9[/tex]
By subtraction property of equality
[tex]3x=-9[/tex]
By combine like terms
[tex]x=\frac{-9}{3}=-3[/tex]
By division property of equality
Substitute x=-3 then we get
[tex]m\angle R=-7(-3)+9=21+9=30^{\circ}[/tex]
[tex]m\angle N=m\angle R=30^{\circ}[/tex]
Answer:
30
Step-by-step explanation:
Write the equation of the parabola in vertex form
Answer:
[tex]y^2=-12(x+2)[/tex]
Step-by-step explanation:
The general vertex form of parabola is,
[tex](y-k)^2=a(x-h)[/tex]
where,
[tex](h.k)[/tex] is the vertex,
[tex]y=h[/tex] is the axis of symmetry.
Given the coordinates of the vertex as [tex](-2,0)[/tex] and focus as [tex](-5,0)[/tex]
a is the 4 times the distance between the vertex and the focus.
The distance between the vertex and focus is -3. Negative is because we are calculating the distance to the left (-ve x direction) of the vertex.
Hence, [tex]a=4\times(-3)=-12[/tex]
Putting the values in the general equation,
[tex](y-0)^2=-12(x-(-2))[/tex]
i.e [tex]y^2=-12(x+2)[/tex]
The equation 4x2 – 24x + 4y2 + 72y = 76 is equivalent to
(1) 4(x – 3)2 + 4(y + 9)2 = 76
(2) 4(x – 3)2 + 4(y + 9)2 = 121
(3) 4(x – 3)2 + 4(y + 9)2 = 166
(4) 4(x – 3)2 + 4(y + 9)2 = 436
I know the answer is choice 1 but how do you get to that answer?
Answer:
Option 4 is correct.
The equation [tex]4x^2 -24x + 4y^2 + 72y = 76[/tex] is equivalent to [tex]4(x-3)^2 + 4(y+9)^2 =436[/tex]
Step-by-step explanation:'
Given equation: [tex]4x^2 -24x + 4y^2 + 72y = 76[/tex]
First group the terms with x and those with y;
[tex](4x^2-24x)+(4y^2+72y) = 76[/tex]
Next, we complete the squares.
We can do this by adding a third term such that the x terms and the y terms are perfect squares.
For this we must either add the same value on the other side of the equation or subtract the same value on the same side so that the equality is maintained.
⇒[tex]4(x^2-6x) +4(y+18y) = 76[/tex]
or
[tex]4(x^2 -6x +3^2 -3^2) + 4(y^2 +18y +9^2 -9^2) = 76[/tex]
[tex]4(x^2-6x + 3^2) - 36 + 4(y^2+18y +9^2) - 324 = 76[/tex]
[tex]4(x-3)^2 + 4(y+9)^2 - 360 =76[/tex]
Add 360 on both sides we get;
[tex]4(x-3)^2 + 4(y+9)^2 =360 +76[/tex]
Simplify:
[tex]4(x-3)^2 + 4(y+9)^2 =436[/tex]
Therefore, the given equation is equivalent to [tex]4(x-3)^2 + 4(y+9)^2 =436[/tex]
Describe how to find all the points on a baseball field that are equidistant from second base and third base. In the figure, m∠DBE=50.
The value of y and x is 11 and 6.
The m<BED = 40 degrees
The m<BEA = 40 degrees.
From the given diagram, the following statements are true:
AB = BC
3y = 5y - 22
3y - 5y = -22
-2y = -22
y = 11
Similarly, m<DBE = 50 degrees
m<BED = 90 - m<DBE
m<BED = 90 - 50
m<BED = 40 degrees
Since BE = BC = 3y
BE = BC = 3(11) = 33
Also, 7x - 2 = 40
7x = 40 + 2
7x = 42
x = 6
Get the measure of m<BEA
m<BEA = 7x - 2
m<BEA = 7(6) - 2
m<BEA = 40 degrees
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Question
Describe how to find all the points on a baseball field that are equidistant from second base and third base.
In the figure, m∠DBE=50. Find each of the following.
m∠BEDm angle b e d
m∠BEAm angle b e eh
x
y
BE
BC
If you shift the linear parent function, f(x) = x, down 7 units, what is the equation of the new function? A. G(x) = 7x B. G(x) = x + 7 C. G(x) = x – 7 D. G(x) = x
Answer:
The correct answer is C. f(x) = x - 7
Step-by-step explanation:
In order to find this, you simply need to know that vertical shifts can be added on to the end of parent equations as constants. Since this is a downward shift, we use a negative number.
Dakota earned ?$4.50 in interest in Account A and ?$18.00 in interest in Account B after 18 months. If the simple interest rate is 2?% for Account A and 4?% for Account? B, which account has the greater? principal
The account with the greater principal is Account B with a simple interest of $ 18
What is Simple Interest?
Simple interest is a method of calculating interest that ignores the impact of compounding. While interest frequently compounds throughout the course of a loan's set periods, simple interest does not. Simple interest is calculated by multiplying the principal amount by the interest rate, times the number of periods.
Simple Interest = ( Principal Amount x Rate x Time Period ) / 100
Given data ,
Let the first account be = A
Let the second account be = B
Now ,
The simple interest of A = $ 4.50
The simple interest of B = $ 18
The time period t = 18 months = 1.5 years
The rate of interest r for A = 2 %
The rate of interest r for B = 4 %
Now , the equation will be
Simple Interest of A = ( Principal of A x Rate of interest of A x Time ) / 100
Substituting the values in the equation , we get
4.50 = ( P x 2 x 1.5 ) / 100
Multiply by 100 on both sides of the equation , we get
450 = 3P
Divide by 3 on both sides of the equation , we get
Principal of A = $ 150
And ,
Simple Interest of B = ( Principal of B x Rate of interest of B x Time ) / 100
Substituting the values in the equation , we get
18.00 = ( P x 4 x 1.5 ) / 100
Multiply by 100 on both sides of the equation , we get
1800 = 6P
Divide by 6 on both sides of the equation , we get
Principal of B = $ 300
Therefore , the principal of B > principal of A
Hence , the simple interest of B with $ 18 has larger principal
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Shawn drew a rectangle that had a width of 4.2 inches and a length of 8.1 inches. Find the perimeter of Shawn's rectangle
Answer: 24.6 inches
Step-by-step explanation: Since it's a rectangle, and not a rectangular prism, that means it's 2D. Length is the long sides on the top and bottom, and width is the shorter sides on it's left and right. So since there's two of each, just do 8.1 + 8.1 + 4.2 + 4.2 and you have your answer!
Two poles of equal heights are standing opposite to each other on either side of a road, which is 100 meters wide .From a point between them on the road, the angles of elevation of their tops are 30? and 60? .Find the position of the point and also, the height of the poles. ( use = 1.73)
Answer:
Distance to the pole from the vertex of the 60o angle = 25 mDistance to the second pole from the vertex of the 30o angle = 75 mHeight of the pole = 43.3Step-by-step explanation:
The point is somewhere on the road. From that point, two poles can be seen at different angles of elevation. One is 30 degrees, and another is 60 degrees.
Call the height of the poles y and the point on the road as x meters away from the base of 60 degree angle to the base pole
Tan(60) = y/xTan(30) = y/(100 - x) Isolate y for both equations(100 - x) * tan(30) = x*tan(60)
Using exact values (from the unit circle) tan(30) = sqrt(3)/3tan(60) = sqrt(3)(100 - x) * sqrt(3)/3 = x*sqrt(3) Divide both sides by sqrt(3)(100 - x)/3 = x Multiply both sides by 3 100 - x = 3x Add x to both sides100 = 4x Divide by 4x = 25The distance to the other pole = 100 - 25 = 75
y = x * sqrt(3)y = 25 * 1.73y = 43.3How much less than 3x^2−7x+9 is 2x^2+4x−8? What is the value of the result when x=2?
^=power
Answer:
1
Step-by-step explanation:
well 3x2^2-7x2+9 = 7
well 2x2^2+4x2−8 = 8
8-7 =1
The difference is x² - 11x + 17 and the value is -1.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Difference = ( 3x²−7x+9 ) - ( 2x²+4x−8 )
Difference = x² - 11x + 17
The value of expression at x= 2 will be calculated as,
E = x² - 11x + 17
E = (2)² - ( 11 x 2 ) + 17
E = 4 - 22 + 17
E = -1
Therefore, the difference is x² - 11x + 17 and the value is -1.
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The lowest temperature ever recorded at Oymyakon in Russia was 96.2°F below 0°F. The lowest temperature ever recorded at Prospect Creek in Alaska was 80°F below 0°F. The thermometer reading of the lowest recorded temperature at Oymyakon was °F. The thermometer reading of the lowest recorded temperature at Prospect Creek was °F.
Answer:
Thermometer reading of the lowest recorded temperature at Oymyakon was -96.2° F
Thermometer reading of the lowest recorded temperature at Prospect Creek was -80° F
Step-by-step explanation:
If the temperature is x° F below 0° F then the thermometer reading is -x° F
It is given that the Lowest temperature recorded at Oymyakon in Russai was 96.2°F below 0°F
So the thermometer reading of the lowest recorded temperature at Oymyakon was -96.2° F
Also it is given that the Lowest temperature recorded at Prospect Creek in Alaska was 80°F below 0° F
So the thermometer reading of the lowest recorded temperature at Prospect Creek was -80° F
Answer:
-96.2 first, -80
Step-by-step explanation:
I took le test it was le correct
Guillermo bought some reams of paper for $5 each and a $200 printer.He spent a total of $450.Write and solve an equation to find the number reams of paper Guillermo purchased
PLEASE HELP PLEASE PLEASE
Answer: 25.13 (choice B)
================================
The formula we'll use is
s = r*theta
with
s = unknown arc length
r = radius = diameter/2 = 12/2 = 6 feet
theta = 4pi/3 radians <--- this angle must be in radian mode for the formula to work
Plug those values into the formula and simplify
s = r*theta
s = 6*(4pi/3)
s = 24pi/3
s = 8*pi
s = 8*3.1416
s = 25.1328
s = 25.13
which is approximate and rounded to the nearest hundredth.
Please help me out!!!!!!! :D
Answer:
decrease
Step-by-step explanation:
75 is a higher than 25% come man its ez
Given the system of equations.
4x+y=8
6x-9y=12
Which of the options represents the resulting equation after an equivalent expression for y is substituted into the second equation?
6x - 9(4x - 8) = 12
4x + 4x - 8 = 8
6(4x - 8) - 9y = 12
6x - 9(-4x + 8) = 12
Answer:
Choice D which is 6x - 9(-4x + 8) = 12
Step-by-step explanation:
The first equation solves to y = -4x+8 when you subtract 4x from both sides. This to undo the addition of 4x done to y.
Then you replace every copy of "y" in the second equation with (-4x+8). The parenthesis is important so that you multiply properly
We go from 6x - 9y = 12 to 6x - 9( -4x+8 ) = 12
The resulting equation after an equivalent expression for y is substituted into the second equation is 6x - 9(-4x + 8) = 12, option D.
The student asks about substituting an expression for y from the first equation into the second equation in a system of linear equations. The first equation gives us y in terms of x, which can be expressed as y = 8 - 4x. Substituting this expression into the second equation, 6x - 9y = 12, will result in the compound equation 6x - 9(8 - 4x) = 12. which is same as 6x - 9(-4x + 8) = 12 , option D.
Please help me with this!!!!
Answer: the no. is doubled every time we get an answer. therefore the next two no.s n the sequence are 48, 96
Answer:
3, 6, 12, 24 , 48, 96
Sarah and Gavyn win some money and share it in the ratio 5:3.Sarah gets ?26 more than Gavyn.How much did Gavyn get?
Answer:
$39 did Gavyn get
Step-by-step explanation:
Given the statement: Sarah and Gavyn win some money and share it in the ratio 5:3.
Let the number be x
Then, Sarah win money be 5x and Gavyn win money be 3x.
Also, it is given that Sarah gets $26 more than Gavyn.
⇒ 5x = 26 + 3x
Subtract 3x from both sides we get;
5x -3x = 26 + 3x -3x
Simplify:
2x = 26
Divide both sides by 2 we get;
[tex]\frac{2x}{2} = \frac{26}{2}[/tex]
Simplify:
x =13
∵Gavyn win money 3x = 3(13) = $39
Therefore, Gavyn get, $39.
Soft drinks cost $1.89 and refills cost $0.25 each. With $3.80 to spend on the soft drink and refills, what is the maximum number of refills that you can get? Refills (Hint: Do not answer with a fraction because you can not get a partial refill. Your answer should be an integer). 2 Game Bonus Solve equations with inequalities word problems
The number of times of the soft drink that refills will be 7.
What is inequality?Inequality is defined as an equation that does not contain an equal sign.
Soft drinks cost $1.89 and refills cost $0.25 each.
With $3.80 to spend on the soft drink and refills.
Then the maximum number of refills will be given as
Let x be the number of refills.
1.89 + 0.25x ≤ 3.80
0.25x ≤ 1.91
x ≤ 7.64
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Two angles are complementary . The first angle measures 35 Percent . What's the measurement of the second angle ?
Answer:
55 degrees
Step-by-step explanation:
Write the equation of the line that passes through the point (2, 6) and is perpendicular to the line x = 4
Answer:
The perpendicular line would be y = 6
Step-by-step explanation:
In order to get this, we must first recognize that the equation given is a vertical line. If our new line is to be perpendicular to it, it must be a horizontal line. All horizontal lines are written as y = (a number). Since we have a point it goes through, we can get that value in the ordered pair. The ordered pair has a y value of 6, which means the equation would be y = 6
Emily is making a meal for her family. She uses coconut milk for the dessert she is making. The coconut milk is in a can but she doesn't know how much the can holds. The can is 12cm high and 8cm in diameter. Using the formula V=pie r squared h. Work out how much coconut milk is in the can
What is the solution of the equation?
Answer: 7
Step-by-step explanation:
[tex]\sqrt{2x-5}+4 = x[/tex]
[tex]\sqrt{2x-5} = x-4[/tex] subtracted 4 from both sides
[tex](\sqrt{2x-5})^2 = (x-4)^2[/tex] squared both sides to eliminate square root
2x - 5 = x² - 8x + 16 expanded right side
0 = x² - 10x + 21 subtracted 2x and added 5 on both sides
0 = (x - 3) (x - 7) factored right side
0 = x - 3 0 = x - 7 applied zero product property
x = 3 x = 7 solved for x
Check:
x = 3
[tex]\sqrt{2(3)-5}+4 = (3)[/tex]
[tex]\sqrt{1}+4 = 3[/tex]
1 + 4 = 3
FALSE! x = 3 is NOT a valid solution
x = 7
[tex]\sqrt{2(7)-5}+4 = (7)[/tex]
[tex]\sqrt{9}+4 = 7[/tex]
3 + 4 = 7
TRUE! x = 7 IS a valid solution
Determine the first four terms of the sequence in which the nth term is
Answer:
The correct answer option is: [tex]\frac{1}{3} ,\frac{1}{4} ,\frac{1}{5} ,\frac{1}{6}[/tex].
Step-by-step explanation:
We know that the [tex]nth[/tex] term [tex]a_n[/tex] for an arithmetic sequence is given by:
[tex]a_n=\frac{(n+1)!}{(n+2)!}[/tex]
where [tex]n[/tex] is the number of the position of the term.
We are supposed to find the first four terms of the sequence so we will substitute the values of [tex]n[/tex] from 1 to 4 in the given formula to get:
1st term:
[tex]a_1=\frac{(1+1)!}{(1+2)!}=\frac{1}{3}[/tex]
2nd term:
[tex]a_2=\frac{(2+1)!}{(2+2)!}=\frac{1}{4}[/tex]
3rd term:
[tex]a_3=\frac{(3+1)!}{(3+2)!}=\frac{1}{5}[/tex]
4th term:
[tex]a_4=\frac{(4+1)!}{(4+2)!}=\frac{1}{6}[/tex]
Which function's graph passes through the points (1,4), (2,9), and (3,16)?
y=5x+4
y=(x+1)^2
y=(x+3)^2
y=7x-5
Put the coordinates of the points to the equations of the functions and check:
for (1, 4)
y = 5x + 4 → 4 = 5(1) + 4 → 4 = 5 + 4 → 4 = 9 FALSE
y = (x + 1)² → 4 = (1 + 1)² → 4 = 2² → 4 = 4 CORRECT
y = (x + 3)² → 4 = (1 + 3)² → 4 = 4² → 4 = 16 FALSE
y = 7x - 5 → 4 = 7(1) - 5 → 4 = 7 - 5 → 4 = 2 FALSE
Only y = (x + 1)².
Check other points:
for (2, 9)
9 = (2 + 1)² → 9 = 3² → 9 = 9 CORRECT
for (3, 16)
16 = (3 + 1)² → 16 = 4² → 16 = 16 CORRECT
Answer: Only y = (x + 1)²Answer:
second option is correct
Step-by-step explanation:
Let equation be y = [tex]ax^{2} +bx+c \\[/tex]
here plugging x =1 ,x=2 and x= 15
we have
[tex]a(1)^{2} +b(1)+c \\[/tex] = 4
a +b +c = 4 .................... equation (1)
similarly
[tex]a(2)^{2} +b(2)+c \\[/tex]= 9
4a+2b+c = 9 ....................... equation (2)
plugging x =3 ,we get
[tex]a(3)^{2} +b(3)+c \\[/tex] =16
9a+ 3b +c = 16 ......................... equation (3)
solving these equations simultaneously ,we have
a =1, b= 2 and c =1 ,
y= [tex](1)x^{2} +2x+1\\[/tex]
y = [tex](x+1)^{2} \\[/tex]
A cell phone company sold 4,000 cell phones in the month of November. In December, the company held its annual sale and sold 22% more cell phones than it did in November. In January, the company experienced its slowest month, managing to sell only 50% of the number of cell phones sold during the month of December. What is the total number of cell phones sold by the company during the three months combined? Select one: A. 10,450 B. 10,900 C. 11,320 D. 11,710
help please, 30 pts
Anyone please help!!!
15 points and will give brainliest to excellent answer
Thanks ^ ^
please see attachments
Problem 1
Answers:
Percentage of patients that were dogs = 46%
Standard Error = 0.07048404074682
Margin of error for 90% confidence interval = 0.11594624702851
Margin of error for 95% confidence interval = 0.13814871986377
Round the decimal values however you need them
------------
To get the first answer, you add up the numbers given (7,4,5,5,2) and divide that over 50. So 7+4+5+5+2 = 23 which leads to 23/50 = 0.46 = 46%; therefore phat = 0.46 is the sample proportion of dogs.
Use the SE (standard error) formula given to you with phat = 0.46 and n = 50 to get SE = 0.07048404074682
The critical z value at 90% confidence is 1.645; this value is found in your Z table (back of your stats textbook). Multiply the SE value by 1.645 to get 0.07048404074682*1.645 = 0.11594624702851
Also found in your textbook is 1.960 which is the z critical value at 95% confidence. Multiply this with the SE value to get 0.07048404074682*1.960 = 0.13814871986377
===============================================
Problem 2
Answer: Choice B) picking balls from a bin; the 60 randomly selected get chosen for the first bus, while the remaining 60 go to the second bus
-----------
Choice A is fairly vague on what the lower and upper boundaries are. What is the smallest number allowed? What about the largest? This isn't clear so it's possible that we could end up with more positive numbers than negative (eg: if we had an interval -10 < x < 110). So choice A is false. A similar issue shows up with choice D.
Choice B is true. Assuming the selection process is random and not biased, then each ball is equally likely for each selection. The fact that the balls are colored seems to be extra info which I'm not sure why your teacher threw that in there.
Choice C is false because choice B is true
Choice D is false for similar reasons as choice A. It's not clear where we start and where we end. If we had the interval 2 < x < 6 then x could take on the values {3, 4, 5} and we see that picking an odd number is twice as likely than picking an even. In this example, there is bias.
===============================================
Problem 3
Answer: Choice B) Roll a die; each number corresponds to a different class
------------
Choice A is false because choice B being the answer contradicts it
Choice B is true: there are 6 sides on the die, and each side is equally likely to be landed on, so each class is equally likely
Choice C is false because 2*2*2 = 8 represents the number of combos you can have when you flip three coins (one combo being HTH for heads tail heads) but there are 6 classes, not 8
Choice D is false because while we want 6 regions on the spinner. Each region must have the same area; otherwise, one class is weighted heavier than the others making it more likely you select that particular class.
The guy above me is correct but for the margin of error, I got 34%, 58% for the 90% interval and 32%, 60% for 95% interval. He just didnt take his margin of errors numbers and times them by 100 to get a percentage, and then round it so you get 12% for 90% and 14% for 95%. Then add and minus 12 percent to the 46% and thats how you get it. Then do the same with 95%
The width of a painting is 4 inches less than the length , and the surface area is 320 square inches. Find the length.
Simplify -3d^8(-4d^-14) . Assume d=/0 .
[tex]\( -3 d^8(-4 d^{-14}) \)[/tex] simplifies to [tex]\(12 d^{-6}\)[/tex].
To simplify the expression [tex]\( -3 d^8(-4 d^{-14}) \)[/tex], you can follow these steps:
1. Distribute the factor [tex]\(-3\)[/tex] to both terms inside the parentheses.
2. Combine the exponents with the same base [tex](\(d\))[/tex].
Let's work through it step by step:
[tex]\[ -3 d^8(-4 d^{-14}) \][/tex]
1. Distribute [tex]\(-3\)[/tex] to both terms inside the parentheses:
[tex]\[ (-3) \cdot d^8 \cdot (-4) \cdot d^{-14} \][/tex]
[tex]\[ = 12 d^8 d^{-14} \][/tex]
2. Combine the exponents with the same base [tex](\(d\))[/tex] by applying the rule [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[ = 12 d^{8 + (-14)} \][/tex]
[tex]\[ = 12 d^{-6} \][/tex]
Therefore, [tex]\( -3 d^8(-4 d^{-14}) \)[/tex] simplifies to [tex]\(12 d^{-6}\)[/tex].
The complete question is given below:
Simplify [tex]\( -3 d^8(-4 d^{-14}) \)[/tex]. Assume that [tex]d \neq 0[/tex].
the answer is 12/d^6 , D
A recipe for 9 banana dash nut muffins calls for 1 cup of flour. The number of muffins that can be made varies directly with the amount of flour used. There are 1 1/3 cups of flour available. How many muffins can be? made?
Answer:
12 muffins can be made.
Step-by-step explanation:
12 muffins because if one cup makes 9 muffins, and 1/3 of 9 is 3. Than 9+3=12 so you can make 12.