Question Details

In a school with 1500 students, each student chooses any one of the streams out of science, arts, and commerce, by paying a fee of Rs 1100, Rs 1000, and Rs 800, respectively. The total fee paid by all the students is Rs 15,50,000. If the number of science students is not more than the number of arts students, then the maximum possible number of science students in the school is

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Correct Answer :

700

Solution :

The correct option/answer is 700.

Let the number of students who choose Science, Arts, and Commerce be represented by $S$ , $A$ , and $C$ , respectively.

According to the problem, the total number of students in the school is 1500. This gives us our first equation:
$S + A + C = 1500$
(Equation 1)

The fees for Science, Arts, and Commerce streams are Rs 1100, Rs 1000, and Rs 800, respectively. The total fee paid by all the students is Rs 15,50,000. This gives us our second equation:
$1100 S + 1000 A + 800 C = 1550000$
Dividing the entire equation by 100 to simplify, we get:
$11 S + 10 A + 8 C = 15500$
(Equation 2)

We want to find the maximum possible number of Science students ( $S$ ), subject to the constraint:
$S \leq A$
Also, since $S$ , $A$ , and $C$ represent the number of students, they must be non-negative integers:
$S \geq 0, A \geq 0, C \geq 0$

To eliminate $C$ , we multiply Equation 1 by 8:
$8 S + 8 A + 8 C = 12000$
Subtracting this from Equation 2:
$(11 S + 10 A + 8 C) - (8 S + 8 A + 8 C) = 15500 - 12000$
Simplifying this yields:
$3 S + 2 A = 3500$
(Equation 3)

From Equation 3, we can express $2 A$ in terms of $S$ :
$2 A = 3500 - 3 S$
Since $A \geq S$ , multiplying both sides by 2 gives:
$2 A \geq 2 S$
Substituting $2 A = 3500 - 3 S$ into the inequality:
$3500 - 3 S \geq 2 S$
Adding $3 S$ to both sides:
$3500 \geq 5 S$
Dividing by 5:
$S \leq 700$

Therefore, the maximum possible value of $S$ is 700. When $S = 700$ , we have:
$2 A = 3500 - 3 (700) = 3500 - 2100 = 1400$
Which gives $A = 700$ (satisfying $S \leq A$ ).
Using Equation 1, the number of Commerce students is:
$C = 1500 - S - A = 1500 - 700 - 700 = 100$
Since all student values are non-negative integers, the maximum possible number of science students is indeed 700.