Question

Some students planned for a picnic. The budget for the food was 480dollars. As 8 of them failed to join the party,the cost of food for each member increased by 10 dollars .Find how many students went for the picnic

Answer 1

Answer: 16 students went for the party

Step-by-step explanation: First of all, let us represent the number of students that went for the party by x. The budget for the food was 480, that means each person was supposed to pay 480/x.Next we shall represent the cost per student by letter y, that means

480/x = y ----------(1)

However, 8 of them failed to show up and so the cost per person increased by 10. This can be expressed as follows;

480/(x - 8) = y + 10 ----------(2)

From equation (2) note that the number of students x, has reduced by 8. and then the cost per student y, has increased by 10.

We shall now solve for the pair of simultaneous equations

480/x = y ----------(1)

480/(x - 8) = y + 10 ----------(2)

Substitute for y = 480/x into equation (2)

480/(x - 8) = 480/x + 10

480/(x - 8) = (480 +10x)/x

By cross multiplication we now have,

480x = (480 + 10x) (x - 8)

480x = 480x - 3840 + 10x² - 80x

Collecting like terms we now have;

10x² - 80x -3840 = 0

Divide all through by 10

x² - 8x - 384 = 0

We now solve for the quadratic equation by factorization

(x + 16) (x - 24) = 0

(x + 16) = 0 OR

(x - 24) = 0

Therefore, x = -16 OR

x = 24

Knowing that we cant have a negative value, we shall take x = 24.

So, having determined that 24 students planned for the picnic, and 8 of them failed to join the party, then it turns out that only 16 students went for the picnic (that is, 24 - 8).

Answer 2

The number of people that eventually went on the picnic = 16

Explanation:

Let the number of students that planned the picnic initially be x

So, each person pays (480/x).

If 8 people don't show up, the number of students that will go on the picnic will be (x - 8)

The amount each person pays is now

(480/x) + 10

But note that, the total budget for the food for the picnic = 480

So,

(the amount paid by each person) × (the number of people) = 480

If everybody that planned showed up,

(480/x) × x = 480

if 8 people don't show up

[(480/x) + 10] × [x - 8] = 480

[480(x - 8)/x] + 10(x-8) = 480

480 - (3840/x) + 10x - 80 = 480

- (3840/x) + 10x - 80 = 0

multiply through by x

10x² - 80x - 3840 = 0

x² - 8x - 384 = 0

Solving the quadratic equation

x = 24 or -16

Obviously, x = 24 is the realistic answer as the number of humans cannot be negative.

The number of people that eventually go on the picnic = x - 8 = 24 - 8 = 16.

Hope this Helps!!!