Question

PLEASE HELP
Two grain-collecting machines can collect all the grain from a field 9 days faster than if the first one was doing it alone and 4 days faster than if the second one was working alone. How long does it take each grain-collecting machine to collect all the grain by itself?

Answer 1

Given:

2 machines can collect the grains 9 days faster than first machine.

2 machines can collect the grains 4 days faster than second machine.

To Find:

Number of days each machine need if they collect alone

Solution

Let the first machine be Machine 1

Let the second machine be Machine 2

Define x :

Both machines working together = x days

Machine 1 alone = (x + 9) days

Machine 2 alone = (x + 4) days

Form the equation:

1 day of Machine 1 + 1 day of Machine 2 = 1 day of both together

$\dfrac{1}{x + 9} + \dfrac{1}{x + 4} = \dfrac{1}{x}$

Solve x:

$\dfrac{1}{x + 9} + \dfrac{1}{x + 4} = \dfrac{1}{x}$

$\dfrac{(x + 4) + (x + 9)}{(x + 9)(x + 4)} = \dfrac{1}{x}$

$\dfrac{x + 4 + x + 9}{(x + 9)(x + 4)} = \dfrac{1}{x}$

$\dfrac{2x + 13}{x^2 + 13x + 36} = \dfrac{1}{x}$

$x(2x + 13) = x^2 + 13x + 36$

$2x^2 + 13x = x^2 + 13x + 36$

$x^2 = 36$

$x = 6$

Find the number of days each machine need:

$\text{Machine 1} = x + 9$

$\text{Machine 1} = 6 + 9$

$\text{Machine 1} = 15$

$\text{Machine 2} = x + 4$

$\text{Machine 2} = 6 + 4$

$\text{Machine 2} = 10$

Answer: One machine needs 15 days and the other machine needs 10 days