It takes a bear 9 more days to eat a barrel of honey alone than it takes the bear and a second bear to eat the barrel of honey together. It takes the first bear 7 fewer days to eat a barrel of honey alone than it takes the second bear to eat a barrel of honey alone. In how many days can the two bears eat a barrel of honey together?

Answer:

it takes the bears 12 days to eat honey together.

Step-by-step explanation:

1/x+9+1/x+16=1/x

(solve this using cross multiplication)

= x^2 = 144

x=12

The number of days that it takes for both of the considered bears to eat a barrel of honey together is 12 days.

More people to do a task means less time it will take.

Less people to do a task means more time it will take.

Thus, they are inversely related.

We can define a constant as "Manpower" needed for doing that specific work.

Each worker has his/her own strength. That strength is applied per unit of time.

Let we define:

Manpower needed for a work = time taken for that work  × strength used for that work per unit of time

(we used multiplication as it represents linear use of strength to finish the task, so using that strength for a finite units of time means you stacked up your strengths that many times to finish that work)

Here, we have:

Assume that:

Then, we get:

Manpower needed for a work = time taken for that work  × strength used for that work per unit of time

[tex]M = x \times S\\M = y \times T[/tex]

If bear A and bear B both do the work together, then their strenghts will add up.

That means, the combined strength would be S + T per day

The manpower the work needs is still M, therefore, the time needed is:

[tex]M = \text{Time needed} \times (S + T)\\\\\text{Time needed} = \dfrac{M}{S+T}[/tex]

This is the time needed for both bears to do the work together.

Also, it is given that:

It takes a bear 9 more days to eat a barrel of honey alone than it takes the bear and a second bear to eat the barrel of honey together.

or

Time needed by bear A to do work = 9 + time needed by both bears to do the work

or

[tex]x = 9 + \dfrac{M}{S + T}[/tex]

Also, it is given that:

It takes the first bear 7 fewer days to eat a barrel of honey alone than it takes the second bear to eat a barrel of honey alone.

or

Time taken by bear A to do the work = Time taken by bear B for that work - 7 days

or

[tex]x = y - 7[/tex]

Thus, we got these equations:

[tex]M = S \times x\\M = T \times y\\\\x = \dfrac{M}{S + T} +9\\\\x = y - 7[/tex]

call them first, second, third and fourth equation as per their ordering.

From the first and second equation, we get:

[tex]S = \dfrac{M}{x}\\\\T = \dfrac{M}{y}[/tex]

Putting these values in the third equation, we get:

[tex]x = \dfrac{M}{M/x + M/y} + 9\\\\\\x = \dfrac{xy}{x+y} + 9\\\\(x-9)(x+y) = xy\\x^2 -9x -9y + xy = xy\\x^2 -9x = 9y[/tex]

From the fourth equation, we know that:

[tex]x = y - 7[/tex]

or [tex]y = x +7[/tex]

Putting this value of y in the  equation [tex]x^2 -9x = 9y[/tex], we get:
[tex]x^2 -9x = 9(x+7)\\x^2 -9x = 9x + 63\\x^2 -18x -63 =0\\\\x^2 -21x + 3x -63 = 0\\x(x-21) + 3(x - 21) = 0\\(x+3)(x-21) = 0\\(x+3) = 0, (x-21) = 0\\x = -3, x = 21[/tex]

'x' represents time in number of days. So it cannot be negative.

Thus, the value of x obtained is 21.

Also, we need: Time needed for both the bears to do the work together.

This time is the value of [tex]\dfrac{M}{S+T}[/tex], which is equal to [tex]x - 9[/tex] days (from the third equation)

Thus, the time needed  for both the bears to do the work together = [tex]x-9 = 21-9 = 12 \; \rm days[/tex]

Thus, the number of days that it takes for both of the considered bears to eat a barrel of honey together is 12 days.

Learn more about work and time here:

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