There are six barrels, containing 15 gallons, 8 gallons, 17 gallons, 13
gallons, 19 gallons, and 31 gallons. Each barrel contains either oil or
vinegar. The oil sells for twice as much per gallon as the vinegar. A customer
buys $14 worth of each, leaving one barrel. Which barrel was left?
It doesn't matter what the price of oil or vinegar.
Since the customer bought the same dollar amount of vinegar and oil, and the vinegar costs twice as much as the oil, the number of gallons of vinegar and the number of gallons of oil must be in the ratio 2 / 1.
Barrels of vinegar twice as much as oil is:
17 + 31 + 8 = 56
This is twice as much as a barrel of oil:
13 + 15 = 28
The sold barrels are:
8 ,13 , 15 , 17 , 31
The unsold barrel is 19 gallon.
Well, it sounds like there might be some "sour" feelings about this situation, but let's "mix" some humor into it!
If we let O represent the price of oil per gallon and V represent the price of vinegar per gallon, we can set up a system of equations.
From the given information, we know that:
15O + 8V + 17O + 13V + 19O + 31V = 14
And since the oil sells for twice as much per gallon as vinegar, we know that:
O = 2V
Now, we can substitute O = 2V into the first equation and simplify:
15(2V) + 8V + 17(2V) + 13V + 19(2V) + 31V = 14
30V + 8V + 34V + 13V + 38V + 31V = 14
154V = 14
V = 14/154
V = 1/11
Now, we can substitute V = 1/11 back into O = 2V to find the value of O:
O = 2(1/11)
O = 2/11
So, the price of oil is 2/11 per gallon and the price of vinegar is 1/11 per gallon.
Now that we know the prices, we can figure out which barrel was left based on the $14 spent on each:
The customer spent $14 on vinegar, which means they bought 14/(1/11) = 154 gallons of vinegar.
Since the barrels contain a total of 15 + 8 + 17 + 13 + 19 + 31 = 103 gallons, there must be 103 - 154 = -51 gallons left.
Hmm, it seems like we have a bit of a "liquid disappearing" act here! With a negative amount of liquid left, it appears that there's been some sort of "magical" mishap. Maybe the customer accidentally spilled some vinegar on a clown's nose and caused a comedy catastrophe? Let's hope everyone can "keep their spirits up" and find a solution to this vinegar mystery!
To determine which barrel was left, we need to calculate the total value of oil and vinegar in each barrel.
Let's start by calculating the value per gallon for vinegar. Since the oil sells for twice as much as vinegar, we can assign a value of x per gallon for vinegar. This means the oil sells for 2x per gallon.
Now, let's calculate the total value of oil and vinegar in each barrel using the given capacities:
Barrel 1: 15 gallons * 2x (value per gallon of oil) + 15 gallons * x (value per gallon of vinegar)
Barrel 2: 8 gallons * 2x + 8 gallons * x
Barrel 3: 17 gallons * 2x + 17 gallons * x
Barrel 4: 13 gallons * 2x + 13 gallons * x
Barrel 5: 19 gallons * 2x + 19 gallons * x
Barrel 6: 31 gallons * 2x + 31 gallons * x
Now, let's simplify each calculation to determine the total value for each barrel:
Barrel 1: 30x + 15x = 45x
Barrel 2: 16x + 8x = 24x
Barrel 3: 34x + 17x = 51x
Barrel 4: 26x + 13x = 39x
Barrel 5: 38x + 19x = 57x
Barrel 6: 62x + 31x = 93x
Next, let's calculate the value of $14 worth of oil and vinegar in each barrel:
Barrel 1: 45x + 14
Barrel 2: 24x + 14
Barrel 3: 51x + 14
Barrel 4: 39x + 14
Barrel 5: 57x + 14
Barrel 6: 93x + 14
Since one of the barrels was left after the customer bought $14 worth of oil and vinegar from each barrel, we can now set up the equation:
45x + 14 + 24x + 14 + 51x + 14 + 39x + 14 + 57x + 14 + 93x + 14 = total value of all barrels
Simplifying the equation:
309x + 84 = total value of all barrels
Since we know the total value of all barrels, we can narrow down the possibilities for the barrel that was left.
However, the total value of all barrels is not provided. Could you please provide the total value so we can determine which barrel was left?
To determine which barrel was left, we can start by calculating the total value of oil and vinegar in each barrel.
Let's assign variables to the barrels:
Barrel 1: 15 gallons (oil or vinegar)
Barrel 2: 8 gallons (oil or vinegar)
Barrel 3: 17 gallons (oil or vinegar)
Barrel 4: 13 gallons (oil or vinegar)
Barrel 5: 19 gallons (oil or vinegar)
Barrel 6: 31 gallons (oil or vinegar)
Now, let's assume the vinegar sells for $X per gallon. As stated in the problem, the oil sells for twice as much per gallon, so it sells for $2X per gallon.
To find the total value of oil in each barrel:
Barrel 1: 15 gallons * $2X/gallon = $30X
Barrel 2: 8 gallons * $2X/gallon = $16X
Barrel 3: 17 gallons * $2X/gallon = $34X
Barrel 4: 13 gallons * $2X/gallon = $26X
Barrel 5: 19 gallons * $2X/gallon = $38X
Barrel 6: 31 gallons * $2X/gallon = $62X
To find the total value of vinegar in each barrel:
Barrel 1: 15 gallons * $X/gallon = $15X
Barrel 2: 8 gallons * $X/gallon = $8X
Barrel 3: 17 gallons * $X/gallon = $17X
Barrel 4: 13 gallons * $X/gallon = $13X
Barrel 5: 19 gallons * $X/gallon = $19X
Barrel 6: 31 gallons * $X/gallon = $31X
Now, let's calculate the total value of oil and vinegar in all barrels:
Total value of oil = $30X + $16X + $34X + $26X + $38X + $62X = $206X
Total value of vinegar = $15X + $8X + $17X + $13X + $19X + $31X = $103X
According to the problem, the customer buys $14 worth of each (oil and vinegar), which means the total value of oil and vinegar sold is $14 + $14 = $28.
Therefore, we have the equation: $206X + $103X = $28
Simplifying, we get: $309X = $28
Solving for X, we find: X ≈ $0.091
Now we can determine the value of each barrel using the value of X:
Barrel 1: $15X ≈ $1.365 (value of vinegar)
Barrel 2: $8X ≈ $0.728 (value of vinegar)
Barrel 3: $17X ≈ $1.547 (value of vinegar)
Barrel 4: $13X ≈ $1.183 (value of vinegar)
Barrel 5: $19X ≈ $1.729 (value of vinegar)
Barrel 6: $31X ≈ $2.821 (value of vinegar)
Since the customer left one barrel, we need to find which barrel has a value closest to $14. Let's compare the values of the barrels to $14:
Barrel 1: $1.365 (value of vinegar) - not equal to $14
Barrel 2: $0.728 (value of vinegar) - not equal to $14
Barrel 3: $1.547 (value of vinegar) - not equal to $14
Barrel 4: $1.183 (value of vinegar) - not equal to $14
Barrel 5: $1.729 (value of vinegar) - not equal to $14
Barrel 6: $2.821 (value of vinegar) - not equal to $14
None of the barrels have a value equal to $14. Therefore, the remaining barrel is not among the given options, or there may be some additional information missing to determine the exact answer.