Question

A group of mathematics teachers went to a restaurant. They agreed
to split the bill equally. When the bill arrived, two of them
discovered that they had forgotten their money. The others in the
group agreed to make up the difference by paying an extra $1.30
each. If the total bill was $78, how many teachers were in the
group?

Answer 1

let the number of teacher be n

so the bill per person should have been 78/n

Two did not have money, so they would be short 2*78/n = 156/n

The shortfall was made up by the remaining n-2 teachers each paying 1.30

so 1.3(n-2) = 156/n
1.3n^2 - 2.6n - 156 = 0
n^2 - 2n - 120 = 0
(n-12)(n+10)= 0

n=12 or n=-10

there were 12 teachers in the group.

Answer 2

The average, that is, the arithmetic mean, age of a group of
doctors and lawyers is 40. If the average age of the doctors is 35
and the average age of the lawyers is 50, then what is the ratio of
the number of doctors to the number of lawyers?