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Question 1190068: In school , the ratio of the number of pupils to the number of teachers is exactly 19.6:1 .
a) find the smallest possible number of pupils in the school.
b) the total number of teachers and pupils in the school is in fact 927 . find the number of teachers
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the ratio of the number of pupils to the number of teachers is exactly 19.6:1.
multiply both sides of this ratio by 10 to get 196:10.
this ratio can be simplified to 98:5, which are still whole numbers.
the total number of teachers and pupils in the school can't be less than 103 because you would then get a fractional part of a student or teacher which is not possible.
given that 103 is the minimum number of students and pupils, then, if the total number of students and pupil is 927, then divided 927 by 103 to get 9 sets of 103.
the number of pupils would be 9 * 98 = 882 and the number of teachers would be 9 * 5 = 45.
882 + 45 = 927 total.
if you let x equal a factor of the total, then you get:
19.6 * x + 1 * x = 20.6 * x.
you are looking for the smallest amount of x to make the total an integer.
in fact, you are looking for the smallest amount of x to make .6 an integer.
the smallest amount of x is 5 because .6 * 5 = 3.
therefore, 19.6 * x + 1 * x = 20.6 * x becomes 19.6 * 5 + 1 * 5 = 20.6 * 5.
this results in 98 + 5 = 103.
any amount of pupils and students less than that will not result in a whole number.
927 / 103 = 9 sets of 98 students and 5 teachers.
that becomes a total of 9 * 98 = 882 students and 9 * 5 = 45 teachers for a total of 882 + 45 = 927 students and teachers.
let me know if you have any questions.
theo
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