Answer to Question #189293 in Algebra for Aaqib

Question

There are 100 pappadam at a kalyaana beef.. for a man .. u should give 10 pappadams.. for a woman .. u should give 5 pappadams .. and for a kid .. you should give 1/2 ( half ) pappadam.. and 100 people who went to the kalyanam got pappadam too .. how many men, women and children were there in this 100 people ?

Answer 1

Let $x$ denotes the number of men, $y$ denotes the number of women and $z$ denotes the number of kids. Then we have the following equations: $x+y+z=100$ and $10x+5y+\frac{1}{2}z=100.$ It follows that $z=100-x-y$ and hence $10x+5y+\frac{1}{2}(100-x-y)=100.$ Consequently, we have the equation $19x+9y=100.$ It is easy to see that $x=1$ and $y=9$ is a particular solution of this equation. Therefore, the general solution is $x=1+9t, y=9-19t, t\in\mathbb Z.$ In our case $0\le x,y\le 100$ and hence $t=0.$ We conclude that $x=1,\ y= 9, z=100-1-9=90$ is a unique solution in our case. Therefore, there were 1 man, 9 women and 90 kids.