Math question

Question

Pirates found a treasure of golden coins and divided them equally.If there were 4 pirates less then each one of them would've got 10 coins more and if there were 50 less coins each pirate would've got 5 less coins. How many golden coins did pirates find?

Answer 1

A = average

c = # of coins
p = # of pirates

 

Pirates found a treasure of golden coins and divided them equally

A = c / p

 

If there were 4 pirates less then each one of them would've got 10 coins more

A + 10 = c / (p - 4)

 

if there were 50 less coins each pirate would've got 5 less coins

A - 5 = (c - 50) / p

 

We have three unknowns and three equations, so we can solve for each variable. To summarize, our system of equations is:

 

A = c / p

A + 10 = c / (p - 4)

A - 5 = (c - 50) / p

 

We can begin solving by method of elimination or substitution. I'll substitute the first equation into the third.

 

(c / p) - 5 = (c - 50) / p

⇒ c - 5p = c - 50

⇒ -5p = -50

⇒ p = 10

 

Next, I'll substitute the first equation into the second, then substitute the value of p we just found.

(c / p) + 10 = c / (p - 4)

⇒ (c / 10) + 10 = c / (10 - 4)

⇒ (c / 10) + 10 = c / 6

⇒ (6c / 10) + 60 = c

⇒ 6c + 600 = 10c

⇒ 4c = 600

⇒ c = 150

 

The pirates found 150 golden coins.

 

[Note: We can check our answer by substituting the values of p and c into our three equations. If they're the correct values, then A should be the same in all three equations. I'll go ahead and do it just to boost our confidence.

 

First, I'm going to rearrange all the equations for A.

 

A = c / p
A + 10 = c / (p - 4) ⇒ A = (c / (p - 4)) - 10
A - 5 = (c - 50) / p ⇒ A = ((c - 50) / p) + 5

 

Now, substitute:

 

A = c / p = 150 / 10 = 15

A = (c / (p - 4)) - 10 = (150 / 6) - 10 = 25 - 10 = 15

A = ((c - 50) / p) + 5 = (100 / 10) + 5 = 10 + 5 = 15]

 

 

Answer 2

Suppose x is the number of coins and y is the number of pirates.

 

So the number of coins each pirate gets is x/y.

 

Now, if there were 4 fewer pirates, there would be y-4 of them.  So then each would get x / (y-4) and that would be 10 more than the number they got originally,  x/y + 10.  So x / (y-4) = x/y + 10.

 

And if there were 50 fewer coins there would be x - 50 of them.  So each pirate would get (x - 50) / y and that would be 5 less than the number the got originally, x/y - 5.  So (x - 50) / y = x/y - 5

 

So we have two equations:

 

x / (y-4) = x/y + 10

(x - 50) / y = x/y - 5

 

Let's simplify the second equation:

(x - 50) / y = x/y - 5

x - 50 = x - 5y

-50 = -5y

10 = y

 

Now use y = 10 in the first equation:

x / (y-4) = x/y + 10

x/6 = x/10 + 10

 

Multiply both sides by 60

10x = 6x + 600

4x = 600

x = 150

 

So there were 150 coins and 10 pirates.