Question

In a pollution study of 1500 Indian rivers, the following data were reported. 520 were polluted by sulphur compounds, 335 were polluted by phosphates, 425 were polluted by crude oil, 100 were polluted both by crude oil and sulphur compounds, 180 were polluted by both sulphur compounds and phosphates, 150 were polluted by both phosphates and crude oil and 28 were polluted by sulphur compounds, phosphates and crude oil. Let “n” be the number of the rivers that were polluted by at least one of three impurities. Find the sum of digits of “n”.

Answer 1

Hint: We will use a Venn diagram to solve this question. Venn diagram is a way of representing data using circles and rectangles. For example, we have 12 mangoes and 10 apples and the total number of fruits as 22, then it can be represented by Venn diagram as follows:


Complete step by step answer:
It is given in the question that in a pollution study of 1500 Indian rivers, the following data were reported:
520 were polluted by sulphur compounds.
335 were polluted by phosphates.
425 were polluted by crude oil.
It is also mentioned that,
100 were polluted by crude oil and sulphur compounds.
180 were polluted by sulphur compounds and phosphates.
150 were polluted by phosphates and crude oil.
28 were polluted by sulphur compounds, phosphates, and crude oil.
We will assume that n is the number of the rivers that were polluted by at one of three impurities. Let us consider the Venn diagram as below,



In this Venn diagram we have,
A as the number of rivers that were polluted by only sulphur compounds.
B as the number of rivers that were polluted by sulphur compounds and phosphate.
C as the number of rivers that were polluted by only phosphate.
D as the number of rivers that were polluted by phosphate and crude oil.
E as the number of rivers that were polluted by only crude oil.
F as the number of rivers that were polluted by sulphur compounds and crude oil.
And G as the number of rivers that were polluted by sulphur compounds, phosphate, and crude oil.
We know that 520 rivers were polluted by sulphur compounds, which means,
A + B + F + G = 520 …………… (1)
335 rivers were polluted by phosphate, so,
 B + C + D + G = 335 …………. (2)
425 rivers were polluted by crude oil, so,
E + D + F + G = 425 ……………. (3)
180 rivers were polluted by sulphur compounds and phosphate, so,
B + G = 180 ………………. (4)
100 rivers were polluted by sulphur and crude oil, so,
F + G = 100 .…………. (5)
150 rivers were polluted by crude oil and phosphate, so,
D + G =150 ……………. (6)
28 rivers were polluted by the three impurities, sulphur compounds, phosphate, and crude oil, so,
G = 28 ……………… (7)
Putting the value of G in equation (6), we get,
D + 28 = 150
D = 150 – 28
D = 122
Putting the value of G in equation (5), we get,
F + 28 = 100
F = 100 – 28
F = 72
Putting the value of G in equation (4), we get,
B + 28 = 180
B = 180 – 28
B = 152
Putting the value of B, F, and G in equation (1), we get,
A + 152 + 72 + 28 = 520
A = 520 – 252
A = 268
Now putting the values of B, D, and G in equation (2), we get,
152 + C + 122 + 28 = 335
C = 335 - 302
C = 33
Putting the values of D, F, and G in equation (3), we get,
E + 122 + 72 + 28 = 425
E = 425 – 242
E = 203
Thus, we have all the values required to find the number of rivers which were polluted by at least one of three impurities. So,
n = A + B + C + D + E + F + G
n = 268 + 152 + 33 + 122 + 203 + 72 + 28
n = 878
The number of rivers polluted by at least one of the impurities is 878. Now we have been asked to find the sum of the digits of n. We have found out the value of n = 878. Therefore, the sum of the digits of n = 8 + 7 + 8 = 23.
Thus, the required answer is 23.

Note: We can solve this question using set properties, but it is recommended to use a Venn diagram as it is easier to solve. We can solve this question without using any formula by using Venn diagrams, but if we use set properties, then a lot of formulas of sets have to be used to expand and solve this question. Also there is a possibility that we might not calculate the sum of digits of “n” and loose marks as we have been asked to find the sum of digits of “n” in the question.