Question

Hashem puts boxes into small vans and into large vans.

He puts 5 boxes into each small van.
He puts 25 boxes into each large van.
Hashem puts a total of 400 boxes into the vans so that
number of boxes in small vans: number of boxes in large vans = 3:5
Hashem says that less than 30% of the vans filled with boxes are large vans.
Is Hashem correct?
You must show all your working.

Answer 1

Let's start by using algebra to solve for the number of boxes in small and large vans.

Let x be the number of small vans and y be the number of large vans. Based on the ratio given, we know that:

- Number of boxes in small vans = 3/8 * Total number of boxes

- Number of boxes in large vans = 5/8 * Total number of boxes

Using the information provided, we can write an equation:

5x + 25y = 400

Simplifying, we get:

x + 5y = 80

Now let's use the inequality provided to determine if less than 30% of the vans filled with boxes are large vans. We know that:

- Total number of vans filled with boxes = x + y

- Less than 30% of these vans are large vans, so y/(x+y) < 0.3

Substituting x + 5y = 80, we can simplify the inequality:

y/(x+y) < 0.3

y/80 < 0.3

y < 24

So if y (the number of large vans) is less than 24, then less than 30% of the vans filled with boxes are large vans.

Now we can solve for x and y using substitution. From x + 5y = 80, we get x = 80 - 5y. Substituting into 5x + 25y = 400, we get:

5(80 - 5y) + 25y = 400

400 - 20y = 400

20y = 0

y = 0

This means that y cannot be less than 24, as y = 0 would result in no large vans at all. Therefore, Hashem's statement is incorrect, and the number of large vans must be equal to or greater than 24.