Ones in 1 through one million and handshake problem..

[Jaskaran]

Hi there, it's Jaskaran again!

"If you write the integers from 1 through 1,000,000 inclusively, then how many ones will you write?"

The book says 600,001 but Jaskaran has no idea how they got this answer, anyone?

The other; "At the start of an economic conference between the eastern delegation and the western delegation, each delegate shook hands with every other member of his own delegation for a total of 466 handshakes.Next, each delegate shook hands with every person in the other delegation for 480 more handshakes. What was the total number of delegates at the conference?"

The book says 44(20 on one side and 24 on the other), but Jaskaran has no idea how they got this number!

Anyone? Thank you!

 

[Deleted member 4993]

Hi there, it's Jaskaran again!

"If you write the integers from 1 through 1,000,000 inclusively, then how many ones will you write?"

The book says 600,001 but Jaskaran has no idea how they got this answer, anyone?
Think it rhrough

How many ones are there from 0 - 9 ................................................................1

How many ones are there from 10 - 19 .............................................................11

How many ones are there from 20 - 99 .............................................................8

How many ones are there from 100 - 109 ..........................................................????

Make table:

Find pattern and then find equation

Test equation against the table

The other; "At the start of an economic conference between the eastern delegation and the western delegation, each delegate shook hands with every other member of his own delegation for a total of 466 handshakes.Next, each delegate shook hands with every person in the other delegation for 480 more handshakes. What was the total number of delegates at the conference?"

Start with small example:

Start with in a crowd of n people how many handshakes will be there?

The book says 44(20 on one side and 24 on the other), but Jaskaran has no idea how they got this number!

Anyone? Thank you!

 

[soroban]

Hello, Jaskaran!

The Handshake Problem is more complex than it seemed on first reading.

At the start of an economic conference between the eastern delegation and the western delegation, each delegate shook hands with every other member of his own delegation for a total of 466 handshakes. Next, each delegate shook hands with every person in the other delegation for 480 more handshakes. What was the total number of delegates at the conference?"

The book says 44 (20 on one side and 24 on the other).

\(\displaystyle \text{Formula: }\:\text{The number of handshakes among }n\text{ people is: }\:\frac{n(n-1)}{2}\)


\(\displaystyle \text{Let }a\text{ = number of delegates from the East.}\)
. . \(\displaystyle \text{There were: }\:\tfrac{a(a-1)}{2}\text{ handshakes among them.}\)

\(\displaystyle \text{Let }b\text{ = number of delegates from the West.}\)
. . \(\displaystyle \text{There were: }\:\tfrac{b(b-1)}{2}\text{ handshakes among them.}\)

\(\displaystyle \text{There was a total of 466 handshakes: }\:\boxed{\frac{a(a-1)}{2} + \frac{b(b-1)}{2} \:=\:466}\)


\(\displaystyle \text{The two delegations shook hands with each other: }\:\boxed{\frac{ab}{2}\:=\:480}\)


\(\displaystyle \text{And we must solve this system of equations . . . }Good\:luck!\)

 

[Deleted member 4993]

Hello, Jaskaran!

The Handshake Problem is more complex than it seemed on first reading.

At the start of an economic conference between the eastern delegation and the western delegation, each delegate shook hands with every other member of his own delegation for a total of 466 handshakes. Next, each delegate shook hands with every person in the other delegation for 480 more handshakes. What was the total number of delegates at the conference?"

The book says 44 (20 on one side and 24 on the other).

\(\displaystyle \text{Formula: }\:\text{The number of handshakes among }n\text{ people is: }\:\frac{n(n-1)}{2}\)


\(\displaystyle \text{Let }a\text{ = number of delegates from the East.}\)
. . \(\displaystyle \text{There were: }\:\tfrac{a(a-1)}{2}\text{ handshakes among them.}\)

\(\displaystyle \text{Let }b\text{ = number of delegates from the West.}\)
. . \(\displaystyle \text{There were: }\:\tfrac{b(b-1)}{2}\text{ handshakes among them.}\)

\(\displaystyle \text{There was a total of 466 handshakes: }\:\boxed{\frac{a(a-1)}{2} + \frac{b(b-1)}{2} \:=\:466}\)


\(\displaystyle \text{The two delegations shook hands with each other: }\:\boxed{\frac{ab}{2}\:=\:480}\) ... shouldn't this be just "a*b"

\(\displaystyle \text{And we must solve this system of equations . . . }Good\:luck!\)

 

[Deleted member 4993]

One approximation method can be used from the following assumption - the number of delegates in two parties would be approximately equal and the numbers are integers.

Then from the fact that a*b = 480 and ?(480) ? 22, we get two sets of nubers to try (20, 24) and (30,16) - others are further apart.

Then try these sets in the first equation derived by Soroban to get the answer (20,24) or (24,20).

To solve analytically - of course there will be a fourth order equation - and well ...

(Soroban - how come you are not waiting in the car anymore? Getting too cold???)