Suppose that 21 girls and 21 boys enter a mathemat- ics competition. furthermore, suppose that each entrant solves at most six questions, and for every boy-girl pair, there is at least one question that they both solved. show that there is a question that was solved by at least three girls and at least three boys.
Answers
Individuals shall draw a table consisting of 21 boys in each column and 21 girls in each row as shown on the image below.
The table will have 21x21 = 441 boxes. Mark each box with a letter showing the problem solved by both that boy and girl. Since at least one problem was solved by a girl and a boy, therefore each box will have a letter. Each entrant solved at most six questions, so there can be at most six letters in any row or column. This means six different letters can be there in a row only if at least 11 of the boxes contain letters appearing three or more times in the row. Individuals go through each row and color all the boxes, say blue. Therefore, 11 boxes in each row must be colored blue.
The number of the boxes that must be colored blue = 21 x 11 = 231. Individuals can apply the same process to the columns and color at least 231 boxes, say green. But the total boxes are 441 only. Therefore, by Pigeonhole principle, there will be some boxes which will be both blue and green. The problem of doubly colored boxes represents a problem solved by at least three boys and three girls.
The Pigeonhole Principle is used to show that there must be a question solved by at least three girls and three boys in a math competition with 21 girls and 21 boys.
To show that there is a question solved by at least three girls and three boys, we will use the Pigeonhole Principle.
We have 21 girls and 21 boys, each solving at most six questions.Let's assume no question is solved by at least three girls and three boys.The total number of questions solved would be at most 21 x 6 = 126, which is fewer than 21 x 2 + 21 x 2 = 84, a contradiction.Therefore, there must be a question solved by at least three girls and three boys.Related Questions
a ramp has a height of 3 foot and a base of 12 feet what is the angle of elevation
Answers
Let x = angle of elevation.
arctan(tan x) = arctan(3/12)
x = arctan(3/12)
x = 14.05°
In Seattle, the tax on a property assessed at $570,000 is $12,540. If tax rates are proportional in this city, how much would the tax be on a property assessed at $460,000?
Answers
cross multiply
570,000x = 5768400000
x = 5768400000 / 570,000
x = 10,120 <=== tax on 460,000
Final answer:
To find the property tax on a $460,000 property, first, find the tax rate by dividing the known tax of $12,540 by the assessed value of $570,000 to get approximately 0.022. Then multiply $460,000 by this tax rate to get an estimated tax of $10,120.
Explanation:
To calculate the property tax on a property assessed at $460,000 in Seattle, given the tax rate is proportional, we can use the tax information from another property as a reference. In this case, we know a property assessed at $570,000 has a tax of $12,540. This allows us to find the tax rate by dividing the tax amount by the assessed value of the property.
Step 1: Determine the tax rate.
Tax Rate = Tax / Assessed Value
Tax Rate = $12,540 / $570,000
Tax Rate ≈ 0.022
Step 2: Calculate the tax for the property assessed at $460,000 using the tax rate.
Tax = Assessed Value × Tax Rate
Tax = $460,000 × 0.022
Tax ≈ $10,120
Thus, the property tax on a property assessed at $460,000 would be approximately $10,120, assuming the tax rates are indeed proportional.
A Chinese restaurant offers buffet takeout for $4.99 per pound. How much does your takeout meal cost?
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Answer:
1.99
Step-by-step explanation:
If you need to cut 5 pieces of glass from a 14 feet length, how long should each piece be?
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14 / 5 = 2.8 feet each
2.8 feet = 2 feet 9.6 inches
2 feet 9 3/5 inches
One integer is 2 times another. If the product of the two integers is 32, then find the integers
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Final answer:
The two integers that satisfy the conditions of being twice of each other and having a product of 32 are either (4, 8) or (-4, -8).
Explanation:
The two integers as x and y, with y being 2 times x (y = 2x). According to the problem, the product of these two integers is 32 (x * y = 32). If we substitute y in the equation with 2x, we obtain the following: x * 2x = 32. Simplifying this equation gives us a quadratic equation, x² = 16.
To find x, we take the square root of both sides. The square root of 16 is 4, so x can be either 4 or -4. Consequently, y can be either 8 or -8, depending on whether x is positive or negative.
Therefore, the two sets of integers that satisfy the equation x*y = 32 are (4, 8) and (-4, -8), as both pairs have a product of 32 and one integer is exactly twice the other.
A summer baseball league is creating teams of 25 players each from a group of 265 players. How many players will be left over after the teams of 25 players are created?
A. 10
B. 11
C. 15
D. 250
Answers
265 divided by 25= 10 with 15 remaining
So the answer is C
Dave rented a limousine for his wife's birthday. The hourly rate is $60. They used the limousine for 4 hours, plus Dave gave the driver a 10% tip. How much did he spend in total for the hourly charges plus tip?
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Tip is added to that total.
240X.10 = $24
240+24 = $264
Consider the expression 3 * 2 ^ 2 < 16 + 5 andalso 100 / 10 * 2 > 15 - 3. which operation is performed second
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should be performed in the order PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction)/
Therefore, obtain
3 * 2 ^ 2 < 16 + 5
3 * 4 < 16 + 5 Exponentiation first
12 < 16 + 5 Multiplication second
Similarly, obtain
100 / 10 * 2 > 15 - 3
100 / 20 > 15 - 3 Multip[lication first
5 > 15 - 3 Division second
Andrew has 10 more goldfish than Todd. Together, they have 50. goldfish. How many goldfish does each boy have?
Answers
Andrew has x + 10 amount of gold fish
They have 50 in all
x +(x+10) = 50
2x + 10 = 50
2x +10 (-10) = 50 (-10)
2x/2 = 40/2
x = 20
Todd has x amount of goldfish, so he has 20 goldfish
Andrew has (x + 10) amount of goldfish, so he has 30, because 20 + 10 = 30
hope this helps
The line integral of (2x+9z) ds where the curve is given by the parametric equations x=t, y=t^2, z=t^3 for t between 0 and 1. Please don't respond if you can't explain how to get out of an impossible square root. Thanks
Answers
Then r' = (1, 2t, 3t^2)
General Line integral is:
[tex]\int_a^b f(r) |r'| dt[/tex]
The limits are 0 to 1
f(r) = 2x + 9z = 2t +9t^3
|r'| is magnitude of derivative vector [tex]\sqrt{(x')^2 + (y')^2 + (z')^2}[/tex]
[tex]\int_0^1 (2t+9t^3) \sqrt{1+4t^2 +9t^4} dt [/tex]
Fortunately, this simplifies nicely with a 'u' substitution.
Let u = 1+4t^2 +9t^4
du = 8t + 36t^3 dt
[tex]\int_0^1 \frac{2t+9t^3}{8t+36t^3} \sqrt{u} du \\ \\ \int_0^1 \frac{2t+9t^3}{4(2t+9t^3)} \sqrt{u} du \\ \\ \frac{1}{4} \int_0^1 \sqrt{u} du[/tex]
After integrating using power rule, replace 'u' with function for 't' and evaluate limits:
[tex]=\frac{1}{4} |_0^1 (\frac{2}{3}) (1+4t^2 +9t^4)^{3/2} \\ \\ =\frac{1}{6} (14^{3/2} - 1)[/tex]
To evaluate the line integral, calculate the arclength element and substitute it into the integral. Expand the expression inside the square root, multiply it by (2x+9z), and integrate term by term.
Explanation:To evaluate the line integral of (2x+9z) ds where the curve is given by the parametric equations x=t, y=t^2, z=t^3 for t between 0 and 1, we need to find the arclength element ds and substitute it into the integral. The arclength element ds can be calculated using the formula ds = sqrt(dx^2 + dy^2 + dz^2). In this case, dx = dt, dy = 2t dt, and dz = 3t^2 dt. Substituting these values into the arclength element formula, we get ds = sqrt(dt^2 + 4t^2 dt^2 + 9t^4 dt^2) = sqrt(1 + 4t^2 + 9t^4) dt.
Substituting ds into the line integral, we get the integral of (2x+9z) * sqrt(1 + 4t^2 + 9t^4) dt. To evaluate this integral, you can expand the expression inside the square root, multiply it by (2x+9z), and integrate term by term.
However, it seems that there might be a typo in the original question regarding the curve parametrization. The curve given by x=t, y=t^2, z=t^3 is actually a parabolic curve, not a line. If you need further clarification or assistance, please let me know.
If the maximum of 63 years is accidentally recorded as 630 years, how will this affect the mean?
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Travis wants to collect 20% more than Jessica. Robin wants to collect 35% more than Travis. If Travis collects $43, how much was collected in all?
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Add these together: 43.00 + 34.40 + 58.05 = 135.45.
$135.45 were collected in total assuming everyone collected exactly the percentage they 'wanted' to.
identify the number in the hundreth(1/100)position 15.386
Answers
When you are dealing with decimal place value, you simply need to know that all of the numbers to the left are classified as, ones, tens, hundred, etc. The numbers to the right of the decimal start at tenths, hundredths, thousandths, etc.
I am attaching an image to show what I mean, I hope this helps.
Take care,
Diana
(3x6÷2)v+10=3²v+9 (Need work)
Answers
The student's equation simplifies to 9v + 10 = 9v + 9. Upon further simplification, it becomes apparent that the equation is a contradiction and has no solution.
Explanation:The student is asking about the solution to the equation (3x6÷2)v+10=3²v+9. The first step in solving this equation is to simplify the terms. We start by simplifying the multiplication and division in the first part of the equation (3x6÷2), which equals 9 since 3 times 6 is 18 and 18 divided by 2 is 9. Therefore, the equation simplifies to 9v + 10 = 3²v + 9. Next, we simplify 3², which is 3 raised to the power of 2, meaning 3 times itself, which results in 9. The equation now reads 9v + 10 = 9v + 9.
With the equation simplified, we notice that the terms containing v are equal on both sides of the equation. This indicates that we may be dealing with an identity or a contradiction depending on the constants. Subtracting 9v from both sides yields 10 = 9, which is clearly not true. Therefore, we conclude that the equation is a contradiction, and there is no solution for v that would make the equation true.
Final answer:
To solve the equation (3x6÷2)v+10=3²v+9, we need to follow the order of operations and solve for the variable v. However, the equation is inconsistent and has no solution.
Explanation:
To solve the equation (3x6÷2)v+10=3²v+9, we need to follow the order of operations and solve for the variable v. Let's break it down step by step:
Start by performing the division 3x6÷2 = 18÷2 = 9.
Now rewrite the equation as 9v+10=9v+9.
Subtract 9v from both sides of the equation to isolate the constants: 10 = 9.
Since 10 is not equal to 9, the equation is inconsistent and has no solution.
Therefore, there is no value of v that satisfies the equation.
a painter charges $35 per hour for labor plus 40$ for a ladder rental when he paints a house.the costomer provides the paint.total charge to paint a costomer hoise was 950. how many hours did the painter spend painting this house
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First, subtract the $40 for the ladder. $950 - $40 = $910.
$910 is the charge just due to hours of painting.
Now divide $910 by $35/hour to find the hours.
910/35 =
Just do the division.
what is the quotient of 6/7 and 3/14
Answers
= ⁶/₇ × ¹⁴/₃
= ⁽⁶⁾⁽¹⁴⁾/₍₇₎₍₃₎
= ⁸⁴/₂₁
= 4
Which symbol makes the following statement true? 413,115 431,511
Answers
The first number is less than the second number
Suppose one bakery oven at a cookie manufacturer is being used to bake chocolate cookies and vanilla cookies. A batch of chocolate cookies bakes in 8 minutes, and a batch of vanilla cookies bakes in 10 minutes. Let x represent the number of batches of chocolate cookies and y represent the number of batches of vanilla cookies. Write a linear inequality for the number of batches of each type of cookie that could be baked in one oven in an 8-hour shift.
Answers
The amount of oven time required for the choc. cookies is 8x; that for the van. cookies is 10y.
The total oven time is 8 hours, or 8(60) minutes, or 480 minutes.
Thus, 8x + 6y (less than or equal to ) 480 (minutes).
Answer:
The answer is .......D
Step-by-step explanation:
d. 480
You have to remember that The answer is D
If 6 chocolates cost $0.93, how much do 22 chocolates cost?
Answers
I would be $3.41 first you figure out how much one cost by dividing 0.93 by 6 then you multiply your answer by 22
Help Yaya ASAP Please
Jessica and Marcus each earn the same amount of money every week mowing lawns. Jessica saves 50% of her money and Marcus saves 1/4 of his money.
Which statement is correct?
A. Marcus saves 4/5 as much money as Jessica.
B. Jessica saves 4/5 as much money as Marcus.
C. Marcus saves twice as much money as Jessica.
D. Jessica saves twice as much money as Marcus.
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marcus saves 1/4.....1/4 = 25%
Therefore, jessica saves twice as much as marcus
rewrite 3/8 and 5/14 to have a common denominator
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A meteor crater is 4100 feet in diameter. Approximate the distance around the crater. Use 3.14 for piπ
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Circle area= 2* 3.14 * radius= 2 *3.14 * 1/2 * 4100ft= 12874ft
HOPE THIS HELPS!!!
Which expression results in a product that is rational?
(a)square root of 7 times square root of 10
(b)square root of 8 times square root of 10
(c) square root of 9 times square root of 10
(d)square root of 10 times square root of 10
Answers
Final answer:
The expression that results in a rational product is √10 times √10, as this simplifies to the rational number 10.
Explanation:
To find which expression results in a product that is a rational number, we need to identify which pair of square roots can be multiplied to produce a non-radical number.
Multiplication of square roots is analogous to the multiplication of exponents. When the same base is multiplied, the exponents are added. Utilizing this property, we know:
√7 × √10 = √(7 × 10) = √70 (which is irrational since 70 is not a perfect square).
√8 × √10 = √(8 × 10) = √80 (which is irrational since 80 is not a perfect square).
√9 × √10 = √(9 × 10) = √90 (which is irrational since 90 is not a perfect square).
√10 × √10 = √(10 × 10) = √100 = 10 (which is rational since 100 is a perfect square).
Therefore, the expression that results in a rational product is √10 × √10, which simplifies to 10.
Write an equation relating the length of the legs of an isosceles triangle, x, to the length of the hypotenuse of the triangle, h. g
Answers
5-12-13
7-24-25
8-15-17Converse of the Pythagorean ConjectureIf the lengths of a triangle satisfy the Pythagorean Theorem, then the triangle is a right triangle.Isosceles Right Triangle ConjectureIn an isosceles right triangle, if the legs have length x, then the hypotenuse has length x-root-2.30-60-90 Triangle ConjectureIn a 30-60-90 triangle, if the shorter leg has length x, then the longer leg has length x-root-3 and the hypotenuse has length 2x.Rationalizing the DenominatorEquation for a Circlea squared + b squared is less than c squaredObtuse trianglea squared + b squared is greater than c squaredAcute triangle
An equation regarding the length of the sides of an isosceles right triangle is required.
The required equation is [tex]2x^2=h^2[/tex]
The length of the legs of triangle which are equal in length is [tex]x[/tex]
The length of the hypotenuse is [tex]h[/tex]
From the Pythagoras theorem we have
[tex]x^2+x^2=h^2\\\Rightarrow 2x^2=h^2[/tex]
Hence, in the above equation the length of the legs and the hypotenuse lengths are related.
Learn more:
https://brainly.com/question/15138986
https://brainly.com/question/343682
Jose is going to rent a truck for one day. There are two companies he can choose from, and they have the following prices. Company A charges $108 and allows unlimited mileage. Company B has an initial fee of $75 and charges an additional $0.60 for every mile driven. For what mileages will Company A charge less than Company B? Use m for the number of miles driven, and solve your inequality for m .
Answers
108 - 75 < 0.60m
33 < 0.60m
33 / 0.60 < m
55 < m...or m > 55...company A will charge less if it is driven 56 or more miles
Final answer:
Jose should rent a truck from Company A if he plans to drive more than 55 miles, as it will charge less than Company B beyond this mileage.
Explanation:
To determine for what mileages Company A will charge less than Company B, we set up an inequality comparing the total cost for each company and solve for m (the number of miles driven).
The cost for Company A is a flat rate of $108, so it does not depend on mileage. The cost for Company B includes an initial fee of $75 plus $0.60 per mile. We set up the inequality as follows:
Company A \< Company B
$108 < $75 + $0.60m
$108 - $75 < $0.60m
$33 < $0.60m
$33 / $0.60 < m
55 < m
Therefore, Company A will charge less than Company B for any number of miles greater than 55. So, if Jose plans to drive more than 55 miles, he should choose Company A for a cheaper rate.
Elsa is on the swim team. Each week she swims a total of 2000 meters. How many kilometers does she swim each week?
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2,000 meters = 2 kilometers
hope thid helps :)
Anna wants to call Holly. Holly is on vacation in Asia. It is a time difference of ten hours. Holly's time is always later than Anna's time. If it is 7:35 P.M. where Anna lives, then what time is it where Holly is?
This was everything that was on paper I hope i gave enough
information!
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3.24 repeating as a mixed number in simplest form
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So if you make a fraction out of the numbers to the right of the decimal..how would it look
A repeating mixed number is a mixed fraction, which after converted to decimal; the numbers after the decimal point repeat without end.
3.23 as a repeating mixed number is [tex]3\frac{23}{99}[/tex]
Given that
[tex]Number = 3.24[/tex]
First, we represent as a fraction
[tex]Number = \frac{324}{100}[/tex]
To represent the number as a repeating mixed number, we simply subtract 1 from the numerator
[tex]Number = \frac{324}{100 - 1}[/tex]
[tex]Number = \frac{324}{99}[/tex]
Represent as a mixed number
[tex]Number = 3\frac{23}{99}[/tex]
Hence, 1.12 as a repeating mixed number is [tex]3\frac{23}{99}[/tex]
Read more about repeating mixed numbers at:
https://brainly.com/question/4693336
How can ΔWXY be mapped to ΔMNQ?
Answers
Triangle WXY could be mapped to triangle MNQ by these two steps:First, you should translate vertex W to vertex M.And then, you reflect ΔWXY across the line containing line segment WX.
Two buses are traveling to a state park. One bus leaves the terminal at 4:00 p.m. and travels at 40 miles per hour. The second bus leaves the terminal at 5:00 p.m. and travels at 60 miles per hour.
How much time passes until the second bus catches up with the first bus?
Answers
40t = 60(t-1)
distribute the right side
40t = 60t-60
subtract 60t from each side
-20t =-60
divide both sides by -20
t = -60 / -20 = 3
3 hours