Answer
x = (2 + 2√5)
OR
x = (2 - 2√5)
Explanation
In order to use the quadratic formula for the general quadratic equation,
ax² + bx + c = 0 is given as
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]For x² - 4x - 16 = 0,
a = 1
b = -4
c = -16
[tex]\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x=\frac{-(-4)\pm\sqrt[]{(-4)^2-4(1\times-16)}}{2(1)} \\ x=\frac{4\pm\sqrt[]{16+64}}{2} \\ x=\frac{4\pm\sqrt[]{80}}{2} \\ \sqrt[]{80}=\sqrt[]{16\times5}=\sqrt[]{16}\times\sqrt[]{5}=4\sqrt[]{5} \\ x=\frac{4\pm\sqrt[]{80}}{2} \\ x=\frac{4\pm4\sqrt[]{5}}{2} \\ x=2\pm2\sqrt[]{5} \\ x=2+2\sqrt[]{5} \\ OR \\ x=2-2\sqrt[]{5} \end{gathered}[/tex]Hope this Helps!!!
Simplify [tex](\sqrt{5})(\sqrt[3]{5})[/tex]
The value of the indices given will be
[tex](\sqrt{5}) (\sqrt[3]{5})[/tex] simplifies to give [tex]5^{\frac{5}{6} }[/tex] or [tex]\sqrt[6]{3125}[/tex][tex]\sqrt[6]{3125}[/tex]
How can to simply [tex](\sqrt{5}) (\sqrt[3]{5})[/tex]The indices concept can be used for this simplification. An index is the small number which tells us how many times a term had been multiplied by itself.
[tex](\sqrt{5}) (\sqrt[3]{5}) = 5^{\frac{1}{2} } . 5^{\frac{1}{3} }\\\\(\sqrt{5}) (\sqrt[3]{5}) =5^{\frac{1}{2} +\frac{1}{3} }\\\\(\sqrt{5}) (\sqrt[3]{5}) =5^{\frac{2+3}{6}[/tex]
[tex](\sqrt{5}) (\sqrt[3]{5}) = 5^{\frac{5}{6}} = \sqrt[6]{5^{5} } = \sqrt[6]{3125}[/tex]
Therefore, the result of the simplification is [tex]5^{\frac{5}{6} }[/tex] or [tex]\sqrt[6]{3125}[/tex]
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A professor tells a student that he has a 90% chance of getting an A for the course that he score an 85 or better on the midterm exam. The student thinks he has only a 50% chance of getting 85 better. what is the probability that the student scores an 85 or better and he receive an A in the score?
The probability that the student scores an 85 or better and he receive an A in the score is 0.45.
In the question it is given that the professor tells that the student has 90% chance of getting A, if he gets 85 or better in mid term exam.
The student is 50% sure that he will get 85 or better in the examination.
To get the probability of the student scores an 85 or better and he receives an A grade, we use multiplication rule.
P( student getting A grade and 85 or better)= P(getting A) x P(getting 85 or better)
P(getting A)=0.9
P(getting 85 or better)=0.5
= 0.9x0.5
P( student getting A grade and 85 or better)=0.45
Therefore, probability that the student scores an 85 or better and getting A is 0.45.
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In a cricket match, you have a squad of 15 players and you need to select 11 for a game. The two opening batsmans are fixed and the rest of the players are flexible. How many batting orders are possible for the game?
1365 batting orders are possible for the game.
Given,
In the question:
In a cricket match, you have a squad of 15 players and you need to select 11 for a game.
To find the how many batting orders are possible for the game?
Let's Know:
What are combination?
Combinations are also referred to as selections. Combinations imply the selection of things from a given set of things. In this case, we intend to select the objects.
Combination formula
ⁿCr = n! / ((n – r)! r!
n = the number of items.
r = how many items are taken at a time.
Substitute the values in above formula :
15! / 11! (15 - 11)!
= 15! / 11! 4!
= 15 × 14 × 13 × 12 / 4 × 3 × 2
= 1365 orders
Hence, 1365 batting orders are possible for the game.
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Find the distance between the points (6,-3) and (4,3)
Give an exact answer in simplest radical form. Do not round.
The distance between point (6,-3) and (4,3) is 3.162 units using the point formula for distance that is √((x2-x1)²+(y2-y1)²) units.
What is point?Since it only represents a dot, a point is a dimensionless shape, whereas a line is one dimension. Both points are lines that can be used to depict various sizes and shapes in a plane.
What is Coordinates?Coordinates are numerical distances or angles that uniquely identify points on two-dimensional (2D) surfaces or in three-dimensional (3D) space ( 3D ). Mathematicians, scientists, and engineers frequently use a variety of coordinate systems.
Here,
distance between point (6,-3) and (4,3)
=√((x2-x1)²+(y2-y1)²)
=√((4-6)²+(3+3)²)
=√((-2)²+(6)²)
=√(4+36)
=√(40)
=2√(10)
=3.162
According to the point formula for distance, which is √((x2-x1)²+(y2-y1)²) units, the distance between point (6,-3) and point (4,3) is 3.162 units.
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Calculate the area of each figure. Which figure has the greatest area?
a) Area of the parallelogram = b x h
= 9 cm x 7 cm
= 63 cm^2
Find the percent of change
Original $85 new $68
Answer: -20%
Step-by-step explanation:
new-old divided by old
68-85=-17
-17/85=-0.2
-0.2x100=-20
-20%
Find the y intercept of the line y
4x+12
Answer:
12
Step-by-step explanation:
y = mx + b is the slope intercept form of a line. The b is the y-intercept.
In the equation y = 4x + 12, 12 is in the b spot.
Answer:
y- intercept = 12
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = 4x + 12 ← is in slope- intercept form
with y- intercept c = 12
Suppose you have $10,000 in savings when the price level index is 100.
Instructions: Round your responses to the nearest whole number.
a. What is the real value of your savings if the price level increases by 10 percent for the year?
b. What is the real value of your savings if, instead, the price level declines by 5 percent for the year?
Answer:
a. $9,090.91
b. $10,526.32
Step-by-step explanation:
You want to know the value of your $10,000 savings when the price level (a) increases by 10%, and (b) decreases by 5%.
Price levelAs the price level goes up, the number of items that can be purchased with a given dollar amount goes down in inverse proportion.
a. Price level increaseWhen the price level increases by 10%, it is multiplied by 1 +10% = 1.10. The effective purchasing power of the savings will be inversely proportional to that:
$10,000/1.10 = $9,090.91 . . . "real value" of savings
b. Price level decreaseWhen the price level decreases by 5%, it is multiplied by 1 -5% = 0.95. The effective purchasing power of the savings will be inversely proportional, so will be ...
$10,000/0.95 = $10,526.32 . . . "real value" of savings
__
Additional comment
The price level index is usually normalized to a value of 100 at some specified point in time. An increase of 10% in price level would be reflected in an index value of 100×1.10 = 110. Effectively, this means that what did cost $100 at the reference time now costs $110.
With $10,000, you could have purchased 100 of those items, but now you can purchase only 90.91 of those items. Your purchasing power has changed in inverse proportion to the price level.
For the entire year, $9091 will be spent if prices rise by 10%.
In the event that prices drop by 5% for the entire year, $10,526.32
What is a Whole number?Integers include zero, a positive natural number, and a negative integer represented by a minus sign. The negative numbers are the inverse additions of the comparable positive numbers.
Given,
If the price index is 100, savings equal $10,000.
a) If prices increase by 10% annually, the following formula will be used to calculate the true value of the savings:
Real Value = Nominal Value x Prior Year CPI / Current Year CPI
According to the CPI:
= 100 + (100 x 10/100)
For the current year, = 110%.
The resultant value will be
= $10000 x 100/110
= $9090.90
b) If prices decline by 5% annually, the following formula will be used to calculate the true value of the savings:
Real Value = Nominal Value x Prior Year CPI / Current Year CPI
According to the CPI:
= 100 - (100 x 5/100)
for the current year, = 95%
Consequently, the true value will be:
= $10000 x 100/95
=$10,526.32
As a result, your savings would actually be worth $9090.90 if prices rose by 10% annually.
Alternatively, if prices fall by 5% over the course of the year, the actual worth of your savings is $10,526.32.
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Arithmetic sequence is given by
and .
What is the sum of the first terms of that arithmetic sequence?
The sum of the first four terms of the arithmetic sequence is 24.
How to calculate the sequence?From the information illustrated, the arithmetic sequence is given by 2n + 1.
The first term will be:
= 2n + 1
= 2(1) + 1 = 3
The second term will be:
= 2n + 1
= 2(2) + 1 = 5
The third term will be:
= 2n + 1
= 2(3) + 1
= 7
The fourth term will be:
= 2n + 1
= 2(4) + 1
= 9
The sum of the first four terms will be:
= 3 + 5 + 7 + 9
= 24
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An arithmetic sequence is given by 2n + 1. What is the sum of the first 4 terms of that arithmetic sequence?
A species of animal is discovered on an island. Suppose that the population size P (t) of the species can be modeled by the following function, where time t is measured in years.P (t) = 800/(1+4e^-0.31t)
To find the initial population, we evaluate
[tex]P(t)=\frac{800}{1+4e^{-0.31t}}[/tex]at t=0:
[tex]\begin{gathered} P(0)=\frac{8_{}00}{1+4e^{-0.31\cdot0}}=\frac{8_{}00}{1+4e^0} \\ =\frac{8_{}00}{1+4}=\frac{800}{5}=160. \end{gathered}[/tex]Therefore, the initial population was 160 individuals.
To find the population after 10 years, we evaluate the given function at t=10:
[tex]\begin{gathered} P(10)=\frac{800}{1+4e^{-0.31\times10}}=\frac{800}{1+4e^{-3.1}} \\ \approx678. \end{gathered}[/tex]Therefore, the population after 10 years is 678 individuals.
Answer:
The initial population was 160 individuals.
The population after 10 years is 678 individuals.
Which angle relationships are correct? Check all that apply.
∠1 and ∠8 are alternate exterior angles.
∠4 and ∠6 are same side interior angles.
∠5 and ∠7 are vertical angles.
∠2 and ∠8 are corresponding angles.
∠3 and ∠6 are alternate interior angles.
The angle relationships which are correct are:
∠4 and ∠6 are same side interior angles.
∠3 and ∠6 are alternate interior angles.
Side interior angles
Two angles that are on the same side of the transversal and written the same side of the transversal when two lines are intersected by a third line are called side angles.
Alternate interior angles
Alternate interior angles are the angles formed on the opposite sides of the transversal.
Hence the correct relationship are ∠4 and ∠6 are same side interior angles and ∠3 and ∠6 are alternate interior angles.
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how do you write 4.501795324E^-6 in simpler form
I need help with geometry. Were learning similarity and i have a test soon but im really confused. I've been trying to figure it out for the past 2 hours but I really have no idea. I attached a photo of my assignment from today if anyone could help me with that.
Consider the upper triangle
if we want to find b side, consider the following trigonometric identity:
[tex]\cos \text{ (}\theta\text{) = }\frac{adjacent\text{ side}}{hypotenuse}[/tex]in this case, we have:
[tex]\cos \text{ (45) = }\frac{b}{20}[/tex]solve for b:
[tex]b\text{ = cos(45) x 20 = }\frac{\sqrt[]{2}}{2}\text{ x 20 = 10}\sqrt[]{2}[/tex]then, we can conclude that
[tex]b\text{ = 10}\sqrt[]{2}[/tex]Now, for a-side, consider the following trigonometric identity:
[tex]\sin \text{ (}\theta\text{) = }\frac{opposite\text{ side}}{hypotenuse}[/tex]in this case, we have:
[tex]\sin \text{ (}\theta\text{) = }\frac{a}{20}[/tex]solve for a:
[tex]a\text{ = }\sin \text{(45) x 20 = }\frac{\sqrt[]{2}}{2}\text{ x 20 = 10}\sqrt[]{2}[/tex]then, we can conclude that
[tex]a\text{ = 10}\sqrt[]{2}[/tex]Now, for the c-side, consider the greater triangle :
if we denote the hypotenuse by h, then by Pythagorean theorem we have:
[tex]h^2=20^2+15^2[/tex]but h = a + c, then, replacing this in the previous equation we have
[tex]h^2=(a+c)^2=20^2+15^2[/tex]but, we know that a is
[tex]a\text{ = 10}\sqrt[]{2}[/tex]then we have:
[tex](10\sqrt[]{2}+c)^2=20^2+15^2\text{ = 625}[/tex]now, taking the square root of both sides of the equation we obtain:
[tex]10\sqrt[]{2}+c^{}=\text{ 25}[/tex]solve for c:
[tex]c^{}=\text{ 25}-\text{ 10}\sqrt[]{2\text{ }}\text{ }\approx10.85[/tex]then we can conclude that :
[tex]a\text{ = 10}\sqrt[]{2}[/tex]
[tex]b\text{ = 10}\sqrt[]{2}[/tex]
and
[tex]c^{}=\text{ 25}-\text{ 10}\sqrt[]{2\text{ }}\text{ }\approx10.85[/tex]
Suppose that the cost of an item, excluding tax, is $2.75x describes the cost , in dollars, of purchasing x units of that item. Evaluate the algebraic expression when x=90
The cost of the item is $247.5 when x=90
Let x represent the number of units of the item
Cost of item excluding tax = $2.75x
Algebra: The area of mathematics known as algebra aids in the representation of circumstances or problems as mathematical expressions. Creating a meaningful mathematical expression, it requires variables like x, y, and z as well as mathematical operations like addition, subtraction, multiplication, and division. Algebra is used in every discipline of mathematics, including trigonometry, calculus, and coordinate geometry.
Finding the cost of the item excluding tax when x=90 we get:
= 2.75*90
= 247.5
So, the cost is 247.5
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Angelo's kayak travels 10km/h in still water. If the river's current flows at a rate of 4km/h, How long will it take to travel 35km downstream? it'll take [____] hours.
Given that Angelo's kayak travels 10km/h in still water, and the river's current flows at a rate of 4km/h.
Travelling downstream means Angelo is travelling with the current, that is the current of the water will add to Angelo's speed.
Their combined speed will be;
[tex]\begin{gathered} v=10\text{ km/h + 4 km/h} \\ v=\text{ 14 km/h} \end{gathered}[/tex]To travel 35 km downstream;
[tex]\text{distance = 35km}[/tex]Recall that;
[tex]\begin{gathered} \text{speed = }\frac{\text{ distance}}{\text{time}} \\ \text{time = }\frac{\text{ distance}}{\text{ speed}} \end{gathered}[/tex]substituting the given values;
[tex]\begin{gathered} \text{time = }\frac{35\operatorname{km}}{14\text{ km/h}} \\ \text{time =2.5 hours} \end{gathered}[/tex]Therefore, it'll take 2.5 hours
[tex]undefined[/tex]Which of the following is equal to the expression below?(4 53ОА.1415B.1OB.142O c. – 42D. -415
Recall the following properties of exponentials:
[tex]\begin{gathered} (a^n)^m=a^{n\times m} \\ \\ a^{-n}=\frac{1}{a^n} \end{gathered}[/tex]Use these two properties in that order to find another expression for the same number:
[tex](4^{-5})^3=4^{-5\times3}=4^{-15}=\frac{1}{4^{15}}[/tex]Therefore, the correct choice is option A)
[tex]\frac{1}{4^{15}}[/tex]To indirectly measure the distance across a river, Kiran stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Kiran draws the diagram below to show the lengths and angles that he measured. Find PR, the distance across the river. Round your answer to the nearest foot.
Okay, here we have this:
Considering the provided information and figure, we are going to calculate the requested measure, so we obtain the following:
We can see that the two triangles are similar because they have the same angles, so from this we have:
PR/RE=PO/OC
PR/195=(PR+160)/305
PR*305=(PR+160)195
305PR=195PR+31200
305PR-195PR=31200
110PR=31200
PR=31200/110
PR=3120/11
PR≈294 ft
Finally we obtain that PR, the distance across the river is approximately 294 ft.
[20] A food label shows a total of 448 calories for 4 servings. How many
calories would that be per serving?
Answer:
112
Step-by-step explanation:
448/4=112
Given: F(x) = 2x and G(x) = x^2+ 2Find f/g(-1)
Given data:
[tex]f(x)=2x,g(x)=x^2+2[/tex]First find f( -1 ) and g( -1 ),
[tex]\begin{gathered} f(-1)=2(-1) \\ =-2 \end{gathered}[/tex][tex]\begin{gathered} g(-1)=(-1)^2+2 \\ =1+2 \\ =3 \end{gathered}[/tex]Therefore, we have
[tex]\begin{gathered} \frac{f}{g}(-1)=\frac{f(-1)}{g(-1)} \\ =\frac{-2}{3} \end{gathered}[/tex]What is the functional value of the ceiling function: f(1.5)
f is the ceiling function
f(1.5) = 2
In the sale room at a store, every item is on sale for half the original price, plus 3 dollars.
Complete parts a through d.
(a) Write a function g(x) that finds half of x.
(b) Write a function f(x) that adds 3 to x.
(c) Write and simplify the function (f o g)(x).
(d) Use the function from part (c) to find the sale price of a shirt at the store that has the original
price $60
The solution to the questions are
g(x) = 1/2xf(x) = x + 3(f o g)(x) = 1/2x + 3The sales of shirt that originally costs $60 is $33How to find the equation of function g(x)?The given parameters are:
Every item is on sales for half the original price, plus 3 dollars.
In this case, the function g(x) is half the original price
Let the original price be x
So, we have the following equation
Price = 1/2x
Express as a function
g(x) = 1/2x
How to find the equation of function f(x)?The given parameters are:
Every item is on sales for half the original price, plus 3 dollars.
In this case, the function f(x) is the 3 dollars added
Let the original price be x
So, we have the following equation
Price = x + 3
Express as a function
f(x) = x + 3
The function (f o g)(x)In (A) and (B), we have
g(x) = 1/2x
f(x) = x + 3
The function (f o g)(x) is calculated as
(f o g)(x) = f(g(x))
So, we have
(f o g)(x) = g(x) + 3
Substitute g(x) = 1/2x
(f o g)(x) = 1/2x + 3
The sales of shirt that originally costs $60Here, we have
(f o g)(x) = 1/2x + 3
This means that x = 60
So, we have
(f o g)(60) = 1/2 x 60 + 3
Evaluate
(f o g)(60) = 33
Hence, the value of the shirt is $33
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Determine the equation of the circle
graphed below.
-10 -9
-8 -7 -6 -5 -4 -3
Ņ
-1
10
9
y
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2 3 4 5 6 7 8 9 10
X
Answer: [tex](x-5)^2 +(y+4)^2 =25[/tex]
Step-by-step explanation:
The equation of a circle with center [tex](h, k)[/tex] and radius [tex]r[/tex] is [tex](x-h)^2 +(y-k)^2 =r^2[/tex].
So, the equation is [tex](x-5)^2 +(y+4)^2 =25[/tex].
The equation of the circle in the figure is (x-5)²+(y+4)²=25
What is Circle?A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre.
From the figure the circle has a center at (5, -4) and radius 5
The standard form equation of the circle is (x-h)²+(y-k)²=r²
(h, k) is the center of the circle and r is the radius of circle
Plug in the values of center and r
(x-5)²+(y-(-4))²=5²
(x-5)²+(y+4)²=25
Hence, the equation of the circle in the figure is (x-5)²+(y+4)²=25
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Which situation could NOT represent a proportional relationshipAnswerA The number gallons of water in x barrels with 42 gallons of water in each barrelB The amount an employee who makes 58.50 per hour eams in h hoursC The weight in x weeks of a puppy that gains 2 pounds per week it is starting weight is 5 poundsD The cost of purchasing p pounds of bananas for 50 55 per pound
In a proportional relationship, the ratio between the two variables are equaivalent.
We can also say that one variable is a constant times the second variable in a proportional relationship.
Therefore, the situation that does not represent a proportional relationship from the given choices is:
C. The weight in x weeks of a puppy that gains 2 pounds per week it is starting weight is 5 pounds.
This is because there is an initial value of 5 pounds here
If f (14) = 2(14) ^2−7, which function gives f(x)?
f(x) = 2x^2
f(x) = 2x
f(x) = 2x^2 − 7
f(x) = 14x^2 −7
The function that gives f(x) = 2x² - 7
What is Function?The core concept of mathematics' calculus is functions. The unique varieties of relations are the functions. In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation. Typically, these functions are identified by letters like f, g, and h. The collection of all the values that the function may input while it is defined is known as the domain. The entire set of values that the function's output can produce is referred to as the range. The set of values that could be a function's outputs is known as the co-domain.
Given:
f (14) = 2(14)²−7
Now, here the changing value is 14 according to which the resultant values changes.
so, every time it depends on the function value which can be any number.
Then, let us consider that number is x.
Thus, the function that gives f(x) = 2x² - 7
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I'll give brainliest!
Answer: it is 40 times greater
Step-by-step explanation:
please give brainlyist
Answer:
40x larger
Step-by-step explanation:
9.6 is 4 times larger than 2.4
and 10^4 is 10 times larger than 10^3
Find the greatest common factor for the list of terms.x4, x7, x9
To find the greatest common factor (GCM) for monomials, as you have in this case, you can write the complete factorization of each monomial and find the common factors. Then:
[tex]\begin{gathered} x^4=x\cdot x\cdot x\cdot x \\ x^7=x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x \\ x^9=x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x \end{gathered}[/tex]As you can see, all of these terms have x*x*x*x in common, and this is equal to x^4.
Thus, the GCF for this list of terms is:
[tex]x^4[/tex]help with this question in attachment below
The value of x is 7 units
Length = 4x + 2 = 4 x 7 + 2
=> length = 30units
Breadth = 5x = 5(7 )= 35
Breadth = 35 units
Given,
Perimeter of the rectangle is 130 units
Length of the rectangle = 4x + 2
Breadth of the rectangle = 5x
To find the value of x
Now According to the question:
We know that;
Perimeter of a Rectangle = 2(l + b)
where, l = length of the rectangle
b = breadth of the rectangle
Now, plug the values in above formula :
130 = 2(4x + 2 + 5x)
130 = 2(9x + 2)
130 = 18x + 4
130 - 4 = 18x
126 = 18x
x = 126/ 18
x = 7
Hence, The value of x is 7 units .
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Which expression is NOT equivalent to 40 + 20?
O 5(8+20)
O2(20 + 10)
○ 5(8+4)
○ 10(4+2)
Submit Answer
Answer:
Option 1
Step-by-step explanation:
[tex]40+20=60 \\ \\
5(8+20)=5(28)=140 \\ \\ 2(20+10)=2(30)=60 \\ \\ 5(8+4)=5(12)=60 \\ \\ 10(4+2)=10(6)=60[/tex]
which of the following is the graph of y=-5+2x
we have the equation
y=5+2x
this is the equation of the line
the slope is positive m=2
the y-intercept is (0,5)
the x-intercept is (-2.5,0)
using a graphing tool
the graph in the attached image
please wait a minute
answer is option CRonald's lemon cookie recipe calls for 3 1/3 cups of sugar. How much sugar would Ronald use to make 4 batches of cookies?
To make 4 batches of lemon cookie, Ronald will need 13.3 cups of sugar.
Firstly converting mixed fraction to fraction, thus getting the amount of cups of sugar required for 1 batch.
Number of cups of sugar required for 1 batch = (3×3 + 1)/3
Performing multiplication and addition in numerator
Number of cups of sugar required for 1 batch = 10/3
Number of cups of sugar required for 4 batches = 10/3 × 4
Performing multiplication to find the number of cups of sugar required
Number of cups of sugar required for 4 batches = 40/3
Performing division to find the number of cups in decimal
Number of cups of sugar required for 4 batches = 13.3
Thus, 13.3 cups of sugar will be needed by Ronald.
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