# Diophantine eqution problem

• Dec 27th 2009, 05:59 PM
dhammikai
Diophantine eqution problem
Hi I need help to find the answer for the following question this is need to be solving by using the Diophantine equation

Find the number of men, woman and children in a company of 20 persons if together they pay 20 coins, each man paying 3, each woman 2, and each child 1/2.
• Dec 27th 2009, 06:11 PM
aman_cc
Quote:

Originally Posted by dhammikai
Hi I need help to find the answer for the following question this is need to be solving by using the Diophantine equation

Find the number of men, woman and children in a company of 20 persons if together they pay 20 coins, each man paying 3, each woman 2, and each child 1/2.

m - men
w- women
c - children

so we have,
m+w+c=20
3m+2w+(1/2)c=2

Can you proceed now?
• Dec 27th 2009, 06:48 PM
Soroban
Hello, dhammikai!

Quote:

Find the number of men, woman and children in a company of 20 persons
if together they pay 20 coins, each man paying 3, each woman 2, and each child $\displaystyle \tfrac{1}{2}$.

Let: $\displaystyle M$ = number of men, $\displaystyle W$ = number of women, $\displaystyle C$ = number of children.

There are 20 people: .$\displaystyle M + W + C \:=\:20$ .[1]

They paid a total of 20 coins: .$\displaystyle 3M + 2W + \tfrac{1}{2}C \:=\:20 \quad\Rightarrow\quad 6M + 4W + C \:=\:40$ .[2]

Subtract [1] from [2]: .$\displaystyle 5M + 3W \:=\:20 \quad\Rightarrow\quad M \:=\:\frac{20-3W}{5} \quad\Rightarrow\quad M \:=\:4 - \frac{3W}{5}$

Since $\displaystyle M$ is an integer, $\displaystyle W$ must be a multiple of 5.

There are two solutions: . $\displaystyle \begin{Bmatrix}W = 0,\:M = 4,\;C = 16 \\ \\ W = 5,\:M = 1,\:C = 14 \end{Bmatrix}$

• Dec 27th 2009, 08:25 PM
dhammikai
Thank you very much for your help thanks again