During two years in a school,
7/8 of the students had malaria,
had catarrh and 5/6 had neither.
that fraction of the school had
both malaria and catarrh?
To solve this problem, we can use the concept of overlapping sets and the principle of inclusion-exclusion.
Let the total number of students in the school be represented as 1.
Given:
Fraction of students with malaria = 7/8 = (7/8) * 1 = 7/8
Fraction of students with catarrh = 5/6 = (5/6) * 1 = 5/6
Fraction of students with neither malaria nor catarrh = 5/6
To find the fraction of students with both malaria and catarrh, we can use the principle of inclusion-exclusion.
Total fraction = Fraction with malaria + Fraction with catarrh - Fraction with both malaria and catarrh
Total fraction = 7/8 + 5/6 - 5/6 (since the fraction with both malaria and catarrh is subtracted twice in the previous two fractions)
Total fraction = (7/8) + (5/6) - (5/6)
Total fraction = (7/8) + (5/6) - (5/6)
Total fraction = (7/8) + (5/6) - (5/6)
Total fraction = 42/48 + 40/48 - 40/48
Total fraction = 82/48 - 40/48
Total fraction = 42/48
Therefore, the fraction of the school that had both malaria and catarrh is 42/48.
To find the fraction of the school that had both malaria and catarrh, we need to subtract the fractions that had malaria and catarrh separately from the fraction that had neither.
Let's denote:
M = Fraction of students who had malaria
C = Fraction of students who had catarrh
N = Fraction of students who had neither malaria nor catarrh
Given:
M = 7/8
N = 5/6
We need to find the fraction of students who had both malaria and catarrh, which is the intersection of M and C, denoted as MC.
Since the total fraction of the school is 1, we can use the formula:
1 = M + C + N
Substituting the given values, we get:
1 = 7/8 + C + 5/6
Let's solve for C:
1 - 7/8 - 5/6 = C
To simplify, we need to find a common denominator for 8 and 6, which is 24:
24/24 - 21/24 - 20/24 = C
(24 - 21 - 20)/24 = C
3/24 = C
Now we have the fraction C, which represents the students who had only catarrh.
Since we want the fraction of students who had both malaria and catarrh (MC), we can subtract C from M:
MC = M - C
MC = 7/8 - 3/24
To simplify, we need a common denominator for 8 and 24, which is also 24:
(21/24) - (3/24) = MC
18/24 = MC
3/4 = MC
Therefore, the fraction of the school that had both malaria and catarrh is 3/4.