There are 11649 solutions for the given binomial series x¹ + x² + x³ + x⁴ + x⁵ = 21.
x¹ + x² + x³ + x⁴ + x⁵ = 21, where xi, i = 1, 2, 3, 4, 5, is a non-negative integer such that: 0≤ x1 ≤10 and x², x³, x⁴, x⁵ can range in between 0 to 21.
Now we have to choose the coefficient of x²¹ in the following expression
(1 + x + x² + x³.....+ x¹⁰) (1 + x + x² + x³.......+x²¹)(1 + x + x² + x³.....+ x²¹)(1 + x + x² + x³.....+ x²¹)(1 + x + x² + x³.....+ x²¹)
also we can extend term of x², x³, x⁴, x⁵ from x²¹ upto infinty because it didn't change the value of x²¹ coefficient .
(1 + x + x² + x³.....+ x¹⁰) (1 + x + x² + x³.......∞)(1+ x + x² + x³.....+ ∞)(1 + x + x² + x³.....+ ∞)(1 + x + x² + x³.....+ ∞)
(1 - x¹¹)/(1 - x)[1/(1 - x)⁴]
(1 - x¹¹) [1/(1 - x)⁵]
now we have to find only the coefficient of x²¹ for which we consider the coefficient of x²¹ and x¹⁰ in [tex]1-x^{-5}[/tex]
general term of the binomial coefficient is [tex]\binom{n+r-1}{r}[/tex]
[tex]\binom{5+21-1}{21}-\binom{5+10-1}{10}[/tex] = 12650-1001 = 11649
--The given question is incomplete; the complete question is
"How many solutions are there to the equation x¹ + x² + x³ + x⁴ + x⁵ = 21,
where xi, i = 1, 2, 3, 4, 5, is a nonnegative integer such that: 0≤ x¹≤ 10?"--
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how many solutions does 3(2x+1)=2(3x+2) have
The given expression 3(2x+1) = 2(3x+2) does not have a solution.
What is Linear equation ?
A linear equation is an equation that can be written in the form ax + b = 0, where x is a variable and a and b are constants. This type of equation is used to describe relationships between two variables, and it can be used to find the solution to a variety of problems.
Given expression ,
3(2x+1) = 2(3x+2)
3*2x+3 = 2*3x + 4
shifting 6x to the LHS , we get
6x+3 = 6x+ 4
6x-6x = 4-3
As 6x-6x is 0 , so there is no solution
0 = -1
Here we do not have a solution as there is no x exponent.
Hence, The given expression 3(2x+1) = 2(3x+2) does not have a solution.
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Segment VU is parallel to segment LM. Find the missing length.
L
M24 U
44
27
24
39
?
11
8
N
The missing length LN for the similar triangles is found to be 33 units.
Define the term similarity of triangles?A pairing of triangles are said to be comparable if pairs of equivalent angles in those triangles are congruent.We can conclude that the next pair must indeed be equal if the first two angle pairs are. When all three pairs of angles are equal, the sides of the 3 triangles must therefore be proportionate.For the stated triangles.
Using the similarity of triangles, the ratios of the sides of the triangles are equal.
Let the missing length LN be 'x'.
Thus,
8/ 24 = 11 / x
x = 24*11 / 8
x = 33
Thus, the missing length LN for the similar triangles is found to be 33 units.
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The correct question is-
Segment VU is parallel to segment LM. Find the missing length LN.
The answer is 44. By setting up a proportion, it would look like 8/24 = 11/33. The question is asking for the length of LN. So you would add 11 + 33, which will give you the answer of 44.
In a baket of muffin 15% of the muffin are blueberry. How many are not blueberry?
The percentage that is not blueberry in a bake of muffin is 85%
What is a percentage?Percentage is defined as the number that represents a ratio of a total that is divided by 100 equal parts. It is represented by the symbol %.
To solve this problem we must perform the following algebraic operations with the given information:
Information about the problem:
%Total = 100%Blueberry = 15%Not blueberry = ?Calculating the percentage that is not blueberry in a bake of muffin:
% Not blueberry = %Total - %blueberry
% Not blueberry = 100 - 15
% Not blueberry = 85
85% of the bake is not blueberry
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The following triangles are similar. Which of the properties could be used to solve x:
if the triangles are similar , then ,
The small trapezoid is a scaled-down version of the large trapezoid, but is also a mirror image.
So the side x corresponds to the length 45 in the larger trapezoid.
[tex]45 = 2x + 4 \\ 2x = 45 - 4 \\ x = \frac{41}{2} \\ x = 20.5[/tex]
Madelyn earns $159.25 for 7 hours of work. If she makes a constant hourly wage, which table represents the relationship between the number of hours she works and her total earnings?
Answer: She makes around $22 dollars per hour.
Step-by-step explanation:
You can divide her wage by the amount of hours she works.
What are the 4 types of division?.
There are four types of division are:
DividendDivisorQuotient RemainderNow, According to the question:
Division is one of the four basic math operations with addition, subtraction, and multiplication. Division is an important operation because it is common in everyday life and helps people understand multiplication better. Division is splitting a group into equal parts. Division is used to split up a class into equal groups for a game or activity or even split a pizza into equal portions between a group of friends.
The dividend is the number that is being divided in the division process.The number by which the dividend is being divided by is called the divisor. The quotient is a result obtained in the division process.Sometimes, we cannot divide things exactly.Learn more about Division at:
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2x^5 different from (2x)^5
The difference between the two expressions:
2x^5 and (2x)^5
Is the coefficient that multiplies the variable.
How are the two expressions different?We want to see how the expressions:
2x^5 and (2x)^5
Are different.
Remember that the exponents can be distributed, then we can write the rule:
(a*b)^n = (a^n)*(b^n)
Then we can rewrite the second expression as:
(2x)^5 = (2^5)*x^5 = (2*2*2*2*2)*x^5 = 32x^5
So the difference between the two expressions is the coefficient that is multiplying the variable x.
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9. A water tank already contain 55 gallon of water when Baxter begin to fill it. Water flow into the tank at a rate of 8 gallon per minute. Write a linear equation to
model thi ituation. Find the volume of water in the tank 25 minute after Baxter
begin filling the tank
So, 25 minutes after Baxter began filling the tank, the volume of water in the tank is 255 gallons.
A linear equation can be used to model the situation where water is flowing into a tank at a constant rate. The equation would be in the form:
V = 55 + 8t
Where V is the volume of water in the tank (in gallons), t is the time (in minutes) that has passed since Baxter began filling the tank, and 55 and 8 are the y-intercept and slope respectively
To find the volume of water in the tank 25 minutes after Baxter began filling it, you would substitute 25 for t in the equation above:
V = 55 + 8(25) = 55 + 200 = 255 gallons
So, 25 minutes after Baxter began filling the tank, the volume of water in the tank is 255 gallons.
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How do I convert improper fractions into mixed numbers?
Answer:
Step-by-step explanation: Divide the numerator by the denominator.
Write down the whole number part of the quotient.
Take the remainder and write it over the original denominator.
What are the zeroes of f/x x2 − X − 2?.
The zeros of function f(x) = x² -x - 2 are: x = 2 and x = -1
Consider given polynomial function f(x)=x² -x - 2
As we know the zeros of a function are nothing but the values of the variable of the function that makes the function value equal to 0.
To find the zeros of function f(x), we need to equate f(x) to zero then solve it for x.
⇒ f(x) = 0
⇒ x² -x - 2 = 0
using the Quadratic Formula where
a = 1, b = -1, and c = -2
[tex]\Rightarrow x=\frac{-b\pm \sqrt{b^2 -4ac} }{2a}\\\\\\ \Rightarrow x = \frac{1\pm \sqrt{9} }{2}\\\\\\\Rightarrow x = \frac{1\pm 3 }{2}[/tex]
Thus x = 2 and x = -1 are zeros.
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The complete question is:
What are the zeros of f(x)=x² -x - 2 ?
How do you write greater than or equal to 6?.
The correct way to write greater than or equal to 6 is A ≥6 where it tells that A having greater than or equal to value than 6.
In mathematics,
→The greater than symbol (>) is used to indicate that one value is larger than another. For example, if we say "A > B," this means that the value of A is greater than the value of B.
→The less than symbol (<) is the opposite, it indicates that one value is smaller than another. For example, if we say "A < B," this means that the value of A is less than the value of B.
→The greater than or equal to symbol (≥) indicate that one value is larger than or equal to another. For example, if we say "A ≥ B," this means that the value of A is greater than or equal to the value of B.
→The less than or equal to symbol (≤) indicate that one value is smaller than or equal to another. For example, if we say "A ≤ B," this means that the value of A is less than or equal to the value of B.
These symbols are commonly used in mathematical expressions, equations, and inequalities.
To write "greater than or equal to 6" in mathematical notation, you would use the greater than or equal to symbol (≥) followed by the letter 6. For example:
A ≥6
This means that A having greater than or equal to 6.
Therefore, the correct way to write greater than or equal to 6 is A ≥6 where it tells that A having greater than or equal to value than 6.
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Kwame and Olivia each combine orange juice and mango juice. Kwame makes a
total of 12 cups, and Olivia makes a total of 20 cups. Kwame uses less orange
juice than Olivia, but his orange-mango juice tastes more like oranges than hers
does. Give an example of the number of cups of each ingredient each person
could have used. Explain your thinking.
The number of cups of each ingredient Kwame and Olivia could have used is 6 cups and 10 cups respectively.
What do you mean by possible combinations?In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter. For example, the combination of three fruits from a basket containing apples, oranges, and bananas would be the set {apple, orange, banana}, {apple, banana, orange}, {orange, apple, banana}, {orange, banana, apple}, {banana, apple, orange}, and {banana, orange, apple}. The number of possible combinations is often represented using the combination formula, which is often written as "nCr", where "n" is the total number of items in the collection.
It's not possible to give an exact example of the number of cups of each ingredient each person used, as there are infinitely many ways that Kwame and Olivia could have combined orange juice and mango juice to make 12 and 20 cups respectively.
However, one example of the number of cups of each ingredient each person could have used is:
Kwame: 6 cups orange juice, 6 cups mango juice Olivia: 10 cups orange juice, 10 cups mango juice
This satisfies the constraints that Kwame uses less orange juice than Olivia and his orange-mango juice tastes more like oranges than hers does.
This is an example of the possible combination, but it might not be the only one. Also, it's possible that there are other combinations that could satisfy the given information, but this is one example.
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Prepaid off-peak call cost = R41,67 (rounded off) + R1.30 x airtime used. O 1 Time (in min) Prepaid costs Contract costs 0 41,67 108,42 10 54,67 108,42 20 30 67.67 80.67 108.42 C 40 50 93,67 107 108,42 108.42 60 B D 10214 . Use your completed table to draw a graph of the airtime in minutes versus the prepaid contract call costs on different sets of axes.
Answer: I can provide you with the data in tabular format if you want to create the graph with the provided data.
Airtime (min) | Prepaid costs | Contract costs
0 | 41.67 | 108.42
10 | 54.67 | 108.42
20 | 67.67 | 108.42
30 | 80.67 | 108.42
40 | 93.67 | 108.42
50 | 102.14 | 108.42
60 | 114.64 | 108.42
This table shows the airtime (in minutes) used, and the corresponding costs for both prepaid and contract call plans. The prepaid costs are calculated using the formula: R41.67 + R1.30 x airtime used, whereas the contract costs are fixed at R108.42.
It's worth noting that the costs for the contract call plan is fixed and doesn't change with the airtime used, while for the prepaid plan the costs increase as the airtime increases.
Step-by-step explanation:
Can someone help me with this please?
Pleas show steps.
Solving the system of equations:
x + y = 45
y = 2x
We can see that he made a total of 30 hot dogs and 15 hamburgers.
How to write and solve the system of equations?We can define the variables:
x = number of hamburgersy = number of hot dogs.We know that he cooked a total of 45 dishes, then we can write the equation
x + y = 45
And he cooked twice as many hot dogs as hamburgers, then we can write:
y = 2x
Then we can write a system of equations at:
x + y = 45
y = 2x
We can replace the second equation into the first one, then we have:
x + (2x) = 45
3x = 45
x = 45/3
x = 15
And the solution of y is:
y = 2x
y = 2*15 = 30
The solution of the system is:
x = 15
y = 30
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Draci sets up a room for a banquet. She has 54 chairs. She places 9 chairs at each table. How many tables have 9 chairs?
Draci sets up a room for a banquet with 6 tables.
Linear FunctionAn equation can be represented by a linear function. The standard form for the linear equation is: y= mx+b , for example, y=15x+7. Where:
m= the slope.
b= the constant term that represents the y-intercept.
The problem is a literal math question, that you can convert into a linear equation. Where:
x= number of tables
y= number of chairs
From the information of the question, you can write:
y=mx+b, where:
y= total number of chairs= 54
m= number of chairs in each table = 9
b=0 , because you have a constant function since Draci places 9 chairs at each table. Thus:
54=9x
x=54/9
x=6
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What percentage profit did she make on costs
Answer:21
Step-by-step explanation:
A worker spent 2 hours and 40 minutes making 20 products. If he works at the same productivity, how long will it take the worker to make 30 units?
Answer:
Step-by-step explanation:
The Time Taken to make 20 products = 2hours and 40 minutes = 160 minutes
Time taken to make 10 units = 80 minutes
Time taken to make 30 units = time taken to make 20 units + time taken to make 10 units
= 160 minutes + 80 minutes
= 240 minutes or 4 hours.
The worker needs 4 hours to make 30 units.
The ratio can be calculated as follows:
The first is to convert hours to minutes
2 hours 40 minutes = (2×60)+40 = 120+40 = 160 minutes
The next step is to compare worth comparison
[tex]\frac{a1}{b1}[/tex] = [tex]\frac{a2}{b2}[/tex]
[tex]a_{1} = 160 minutes\\a_{2} = X\\b_{1} = 20 products\\b_{2} = 30 products[/tex]
So,
[tex]\frac{160}{20} = \frac{X}{30}[/tex]
X = [tex]\frac{(160.30)}{20}[/tex]
X = [tex]\frac{4800}{20}[/tex]
X = 240 minutes
The next is to convert minutes into hours
240 minutes ÷ 60 minutes
= 4 hours
The worker needs 4 hours to make 30 units.
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I need help I’ll give BRAINLIESt
Player A is more consistent with a range of 7.
What is IQR or interquartile range?The interquartile range serves as a gauge for where a data set's "middle fifty" lies. An interquartile range (IQR) measures where the majority of the values lay, whereas a range measures where a set's start and finish are.
Arranging the data in ascending order for player A,
1, 1, 2, 2, 3, 3, 4, 4, 8.
The range of this data set is the last term minus the first term which is,
= 8 - 1.
= 7.
Arranging the data in ascending order for player B,
1, 1, 2, 4, 4, 5, 5, 5, 10.
The range of player B is 10 - 1 = 9.
We know less range signifies more consistency therefore, Player A is more consistent.
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x is an obtuse angle and sin x = 0.43.
Find the value of x.
Answer: 154.53
Step-by-step explanation:
sin x° = 0.43
t = arcsin (0.43) = 25.47
x= 18 - t= 18 - 25.47 = 154.53
What is modulus of 4i?.
The modulus of the complex number 4i is 4.
Now, According to the question:
Consider the given number 4i .
The number is a complex number which is of the form z = x + iy .
We can write the above question as z = 4i where x = 0 and y = 4 . The modulus of the complex number is given by:
[tex]|z| =\sqrt{Re(z)^2+Im(z)^2}[/tex]
where real part is x and imaginary part is y. Hence the modulus of a complex number is :
[tex]|z| =\sqrt{0^2+4^2}[/tex]
[tex]|z| = \sqrt{16} \\[/tex]
|z| = 4
Hence the modulus of the complex number 4i is 4.
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What is a cube of 12?.
1728 is the cubic of the number 12.
Cube of given number = [tex]12^{3}[/tex] which is equal to 1728.
The cubes are made up of values that are obtained by multiplying integers by themselves three or four times. The cube's original number can be found by raising any natural number to the power of three.
Cubes are the numbers that are produced when an integer is multiplied by itself three times. We can find cubes for any number, whether it be an integer or a fraction. In order for the cube of a whole number (x) to produce a perfect cube, the cube root of x3 must equal x. The procedure of locating cubes is thus the opposite of locating the cube root.
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tangent lines help (15 pts)
The required perimeter of the polygon is 92 cm.
What is a polygon?Polygon is defined as a geometric shape that is composed of 3 or more sides these sides are equal in length, and an equal measure of angle at the vertex,
Examples of polygons, equilateral triangles, squares, pentagons etc
here,
∠B ≅ ∠C
tangent drawn from D = tangent drawn from B
tangent drawn from D = 11.5 cm
The perimeter of the polygon is given = as the sum of a tangent from all points
= 11.5 + 11.5 + 11.5 + 11.5 + 10.5 + 10.5 + 12.5 + 12.5
= 92 cm
Thus, the required perimeter of the polygon is 92 cm.
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Write the fraction that repreent the colored part of each hape. Then, anwer and urround the larger fraction of each pair
The fraction that represents the colored part of each shape is i) ½ ii) 8/9 iii) ½ iv) ¼ v) 3/7 vi) ¼ vii) 1 viii) 4/9 ix) ½ x) 1/2.
Fractions refer to the parts of a whole. A fraction is a ratio between specified parts of a figure to all parts of a figure. The numerator indicates the parts of the whole and the denominator indicates the total number of equal parts. In the given shapes, as everything is equally divided, the fraction of shaded region is equal to the number of shaded parts divided by the number of total parts.
i) The number of shaded regions = 2
The total number of regions = 4
Fraction = 2/4 = 1/2
ii) The number of shaded regions = 8
The total number of regions = 9
Fraction = 8/9
iii) The number of shaded regions = 4
The total number of regions = 8
Fraction = 4/8 = ½
iv) The number of shaded regions = 1
The total number of regions = 4
Fraction = 1/4
v) The number of shaded regions = 3
The total number of regions = 7
Fraction = 3/7
vi) The number of shaded regions = 3
The total number of regions = 12
Fraction = 3/12 = 1/4
vii) The number of shaded regions = 10
The total number of regions = 10
Fraction = 10/10 = 1
viii) The number of shaded regions = 4
The total number of regions = 9
Fraction = 4/9
ix) The number of shaded regions = 4
The total number of regions = 8
Fraction = 4/8 = 1/2
x) The number of shaded regions = 1
The total number of regions = 2
Fraction = 1/2
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Mihka i on a long road trip, and he average 65 mph for 3 hour while he' driving on the highway. While he' driving on ide road for 1 hour, he only average 45 mph. What i the total ditance that he cover on her road
trip?
Mishka travels 230 kilometres in total on her road trip.
Mishka travels at a 65 km/hr average speed on the highway.
Mishka's time spent driving at this speed on the highway is 3 hours.
Mishka travels at an average rate of 35 km/h on the side road.
Driving at this speed on a side road took Mishka one hour.
Mishka travelled a total distance.
Having said that,
Speed equals Distance/Time
Or, Distance = Speed X Time.
Mishka travelled a total of Mishka miles on the roadway =
= Highway Speed X Highway Travel Time
= 65 X 3
= 195 km
Mishka travelled the following distance via side roads:
= Speed on side street X Travel time
= Speed on side road X Time taken on side road
= 35 X 1
= 35 km
Total distance travelled by Mishka = Distance travelled on highway + Distance travelled on side road
= 195 + 35
= 230 km
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There are 14 dogs at the dog park on a busy Saturday. 2 of them are springer spaniels.
What is the probability that a randomly selected dog is a springer spaniel?
Write your answer as a fraction or whole number.
Answer:1/7
Step-by-step explanation: You take the amount that are springer spaniels and you put them with the whole of the dogs in the park 14 so you would have 2/14. You can reduce this fraction to 1/7 by dividing both the top and bottom of the fraction by 2.
hi can you help it will help me
The exact value of the letter D is three and three-tenths.
What is a number line?A number line is a one-dimensional horizontal line where we can represent any real number. The origin of the number line is represented by zero left to it are all negative real numbers and right to it are all positive real numbers.
By observing the number line there are 10 subsections between two integers.
The letter D is in between two integers 3 and 4 so it can be from 3.1 to 3.9.
In between those ten subsections, the letter D is in the third subsection.
Therefore it is Three and three-tenths.
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How much is a 90 angle?.
A 90-degree angle, also known as a right angle, is an angle that measures exactly 90 degrees.
A 90-degree angle is an angle that measures exactly 90 degrees. It is also known as a right angle, since it is an angle that forms a right angle.
In geometry, angles are measured in degrees, and a full rotation is equal to 360 degrees. The angle formed by two intersecting lines is measured by the amount of rotation needed to bring one line to overlap the other. A 90-degree angle is formed by rotating one line by exactly 90 degrees, so that the two lines form a right angle.
A right angle is an angle that measures 90 degrees and it is represented by a small square at the angle's vertex. This square is a symbol of the angle's measure. In addition, the right angle is the only angle that can be used in geometry to create perpendicular lines which are lines that meet at 90 degrees.
In real-world applications, right angles are found in many things such as the corners of a room, the corners of a square or rectangle, and the corners of tools like a square, a ruler, and a protractor.
Therefore, In short, a 90-degree angle, also known as a right angle, is an angle that measures exactly 90 degrees. It's represented by a small square at the angle's vertex, and it's the only angle that can be used in geometry to create perpendicular lines.
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you want to sell 1/lb jars of mixed peanuts and cashews for five dollars . you paid three dollars per pound for peanuts and six dollars per pound of cashews . you plan to combine four parts peanuts and one part cashews to make your mix . you have spent $70 on materials to get started . how many jars must you sell to break even ?
Answer:
To calculate the number of jars you need to sell to break even, we can use the following formula:
(Cost of materials + Cost of ingredients per jar) / Price per jar = Number of jars needed to break even
We know that the cost of materials is $70, the cost of ingredients per jar is $18 ( $12 for peanuts + $6 for cashews) and the price per jar is $5.
Plugging in these values into the formula we get:
($70 + $18) / $5 = (88/5) = 17.6 jars
So, you need to sell at least 17.6 jars to break even.
--------------------
Cost of materials / (Price per jar - Cost of ingredients per jar) = Number of jars needed to break even
We know that the cost of materials is $70, the cost of ingredients per jar is $18 ( $12 for peanuts + $6 for cashews) and the price per jar is $5.
Plugging in these values into the formula we get:
$70 / ($5 - $18) = 70 / -13 = -5.38 jars
It is not possible to sell -5.38 jars, which means you are making a loss. This indicates that your cost of materials and ingredients is higher than the price you are selling the jars for. To break even, you will have to lower your costs or increase the price of the jars.
y is directly porportional to x
when y=30 and x=6
a)work out an equation connecting y and x
b)work out the value of y when x =12
a. The equation connecting y and x is: y = 5x.
b. The value of y = 60.
What is the Equation of a Direct Proportional Relationship?Given that y is directly proportional to x, the equation that is connecting y and x can be expressed as y = kx, where k is the constant of proportionality.
a. We are given that when y = 30, x = 6, therefore, find the value of k by substituting the values of x and y into y = kx:
30 = k(6)
30/6 = k
5 = k
k = 5
Substitute the value of k into y = kx to write the equation:
y = 5x
b Substitute x = 12 into y = 5x:
y = 5(12)
y = 60
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6) Se tienen 60 lápices, 90 esferos y 120 borradores y se quieren distribuir paquetes en los que haya
estos tres tipos de artículos. ¿cuál es el máximo número de paquetes que se puede armar usando
todos los artículos? ¿Cuántos lápices, esferos y borradores deben ir en cada paquete?
5 pencils, 2 pens, and 3 erasers should go in each package.
The goal is to determine the maximum number of packages that can be built using 60 pencils, 90 pens, and 120 erasers, each package containing 5 pencils, 2 pens, and 3 erasers. To achieve this, we need to calculate the number of packages each item can be used in and take the minimum result.
Starting with pencils, we can make 60/5 = 12 packages. For pens, we can make 90/2 = 45 packages. Finally, for erasers, we can make 120/3 = 40 packages. The minimum result is 40 packages, meaning that the maximum number of packages we can build is 40.
Therefore, each package should contain 5 pencils, 2 pens, and 3 erasers. Any extra items will not be used in the packages. By taking the minimum result, we ensure that all items are used in an equal manner and no items are left unused. In conclusion, the maximum number of packages that can be built using 60 pencils, 90 pens, and 120 erasers is 40, each containing 5 pencils, 2 pens, and 3 erasers.
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