indeterminate equations, JMO

The total age of some 7 years old children and some 5 years old children is 60 years. If a team is to be selected from these children such that their total age is 48 years, then in how many ways can it be done?




Answer:

let the number of children of age 7 be "x"

let the number of children of age 5 be "y"

So, 7x + 5y = 60

or, y = (60 - 7x)/5

since y is a whole number, so, (60 - 7x) must be a multiple of 5, (i.e. units place should be 0,5)

(x,y) = (5, 5)

Now, a team is to be selected. let the number of children of age 7 be "a" and that of age 5 be "b"

So, 7a + 5b = 48 and also "a" and "b" are less than or equal to 5.

or, b = (48 - 7a)/5, 

since b is a whole number, so, (48 - 7a) must be a multiple of 5, (i.e. units place should be 0,5)

(a,b) = (4, 4)

So, there is only one combination.