When Mr. Smith cashed a check at his bank, the teller mistook the number of cents for the number of dollars and vice versa. Unaware of this, Mr. Smith spent 68 cents and then noticed to his surprise that he had twice the amount of the original check. Determine the smallest value for which the check could have been written. [Hint: If x denotes the number of dollars and y the number of cents in the check, then 100y +x-68 = 2(100x + y).]
When Mr. Smith cashed a check at his bank, the teller mistook the number of cents for the number of dollars and vice versa. Unaware of this, Mr. Smith spent 68 cents and then noticed to his surprise that he had twice the amount of the original check. Determine the smallest value for which the check could have been written. [Hint: If x denotes the number of dollars and y the number of cents in the check, then 100y +x-68 = 2(100x + y).]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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Question
7
![[Hint: There exist integers xo and yo such that axo + byo = c. For any integer 1,
4. If a and b are relatively prime positive integers, prove that the Diophantine equar
ax - by =c has infinitely many solutions in the positive integers.
which is larger than both xol/b and | yo1/a, a positive solution of the given equation
x = X+ bt, y = -(yo- at).]
5. (a) A man has $4.55 in change composed entirely of dimes and quarters. What are n
maximum and minimum number of coins that he can have? Is it possible for
number of dimes to equal the number of quarters?
(b) The neighborhood theater charges $1.80 for adult admissions and $.75 for childres
On a particular evening the total receipts were $90. Assuming that more adults thes
children were present, how many people attended?
(c) A certain number of sixes and nines is added to give a sum of 126; if the number of
sixes and nines is interchanged, the new sum is 114. How many of each were ther
originally?
6. A farmer purchased 100 head of livestock for a total cost of $4000. Prices were as follow-
calves, $120 each; lambs, $50 each; piglets, $25 each. If the farmer obtained at least one
animal of each type, how many of each did he buy?
7. When Mr. Smith cashed a check at his bank, the teller mistook the number of cents for
the number of dollars and vice versa. Unaware of this, Mr. Smith spent 68 cents and then
noticed to his surprise that he had twice the amount of the original check. Determine the
smallest value for which the check could have been written.
[Hint: If x denotes the number of dollars and y the number of cents in the check, then
100y +x - 68 = 2(100x +y).]
8. Solve each of the puzzle-problems below:
(a) Alcuin of York, 775. One hundred bushels of grain are distributed among 100 persons
in such a way that each man receives 3 bushels, each woman 2 bushels, and each child
bushel. How many men, women, and children are there?
(b) Mahaviracarya, 850. There were 63 equal piles of plantain fruit put together and 7
single fruits. They were divided evenly among 23 travelers. What is the number of
fruits in each pile?
[Hint: Consider the Diophantine equation 63x + 7 = 23 y.]
(c) Yen Kung, 1372. We have an unknown number of coins. If you make 77 strings of
them, you are 50 coins short; but if you make 78 strings, it is exact. How
are there?
[Hint: If N is the number of coins, then N =77x +27 = 78y for integers
x and y.]
(d) Christoff Rudolff, 1526. Find the number of men, women, and children in a company
of 20 persons if together they pay 20 coins, each man paying 3, each woman 2. and
each child.
(e) Euler, 1770. Divide 100 into two summands such that one is divisible by 7 and t
other by 11.
many coins](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39c12f77-fd5e-41ae-b73a-621f9f2bab30%2Fe3ff3daf-5ec2-417b-b053-d2ec0a857181%2Fqwkzfz4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[Hint: There exist integers xo and yo such that axo + byo = c. For any integer 1,
4. If a and b are relatively prime positive integers, prove that the Diophantine equar
ax - by =c has infinitely many solutions in the positive integers.
which is larger than both xol/b and | yo1/a, a positive solution of the given equation
x = X+ bt, y = -(yo- at).]
5. (a) A man has $4.55 in change composed entirely of dimes and quarters. What are n
maximum and minimum number of coins that he can have? Is it possible for
number of dimes to equal the number of quarters?
(b) The neighborhood theater charges $1.80 for adult admissions and $.75 for childres
On a particular evening the total receipts were $90. Assuming that more adults thes
children were present, how many people attended?
(c) A certain number of sixes and nines is added to give a sum of 126; if the number of
sixes and nines is interchanged, the new sum is 114. How many of each were ther
originally?
6. A farmer purchased 100 head of livestock for a total cost of $4000. Prices were as follow-
calves, $120 each; lambs, $50 each; piglets, $25 each. If the farmer obtained at least one
animal of each type, how many of each did he buy?
7. When Mr. Smith cashed a check at his bank, the teller mistook the number of cents for
the number of dollars and vice versa. Unaware of this, Mr. Smith spent 68 cents and then
noticed to his surprise that he had twice the amount of the original check. Determine the
smallest value for which the check could have been written.
[Hint: If x denotes the number of dollars and y the number of cents in the check, then
100y +x - 68 = 2(100x +y).]
8. Solve each of the puzzle-problems below:
(a) Alcuin of York, 775. One hundred bushels of grain are distributed among 100 persons
in such a way that each man receives 3 bushels, each woman 2 bushels, and each child
bushel. How many men, women, and children are there?
(b) Mahaviracarya, 850. There were 63 equal piles of plantain fruit put together and 7
single fruits. They were divided evenly among 23 travelers. What is the number of
fruits in each pile?
[Hint: Consider the Diophantine equation 63x + 7 = 23 y.]
(c) Yen Kung, 1372. We have an unknown number of coins. If you make 77 strings of
them, you are 50 coins short; but if you make 78 strings, it is exact. How
are there?
[Hint: If N is the number of coins, then N =77x +27 = 78y for integers
x and y.]
(d) Christoff Rudolff, 1526. Find the number of men, women, and children in a company
of 20 persons if together they pay 20 coins, each man paying 3, each woman 2. and
each child.
(e) Euler, 1770. Divide 100 into two summands such that one is divisible by 7 and t
other by 11.
many coins
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