3 type of beans, whose prices per kg are respectively $15m,$25 and $30 are to be taken 2 at a time and mixed in the same proportion so that the resulting mixture are of equal value. How many they be mixed and the price /kg for resulting mixture. (ans; 2:3, $21/kg
To find the answer, we need to determine the ratio in which the beans should be mixed and the resulting price per kilogram (kg) of the mixture.
Let's assign variables to the quantities of each type of bean:
Let x represent the quantity in kg for the $15/kg beans
Let y represent the quantity in kg for the $25/kg beans
Let z represent the quantity in kg for the $30/kg beans
To ensure that the resulting mixtures have equal value, we need to find a proportion where the total value of each pair is the same.
Since we are looking for a proportional mix, we can write the equation:
15x + 25y = 25x + 30z (Equation 1)
Let's simplify Equation 1:
15x + 25y = 25x + 30z
10y = 10x + 30z
10(y - x) = 30z
y - x = 3z (Equation 2)
Now, we have two equations:
Equation 2: y - x = 3z (Equation 2)
Equation 1: 15x + 25y = 25x + 30z
We need to find values for x, y, and z that satisfy both equations.
To solve this system of equations, we will substitute Equation 2 into Equation 1:
15x + 25(y - x) = 25x + 30z
15x + 25y - 25x = 25x + 30z
25y - 25x = 25x + 30z - 15x
25(y - x) = 15x + 30z
5(y - x) = 3x + 6z
5(y - x) - 3x = 6z
Rearranging the equation, we get:
5(y - x) - 3x = 6z
Divide both sides by 6:
z = (5(y - x) - 3x) / 6 (Equation 3)
Now, let's substitute the values for the bean prices given in the question:
$15/kg beans: x
$25/kg beans: y
$30/kg beans: z
Substituting the values into Equation 3, we get:
z = (5(y - x) - 3x) / 6
Now, let's plug in the values and solve the equation:
z = (5(25 - 15) - 3(15)) / 6
z = (5(10) - 3(15)) / 6
z = (50 - 45) / 6
z = 5 / 6 = 0.83 (approximately)
Since we cannot have a fraction of a kg, we can round z to the nearest whole number:
z = 1
Now that we have the value of z, we can substitute it back into Equation 2 to find y:
y - x = 3z
y - x = 3(1)
y - x = 3
To find the ratio in which they should be mixed, we divide the quantities by their respective factors:
x : y : z ≈ 1 : 3 : 1 ≈ 1/5 : 3/5 : 1/5 ≈ 2/10 : 6/10 : 2/10 ≈ 2 : 6 : 2
Therefore, the beans should be mixed in a ratio of 2 : 6 : 2.
To find the price/kg for the resulting mixture, we calculate the weighted average:
Price/kg of the mixture = (price/kg of $15 beans * quantity of $15 beans + price/kg of $25 beans * quantity of $25 beans + price/kg of $30 beans * quantity of $30 beans) / (total quantity of beans)
Price/kg of the mixture = (15 * 2 + 25 * 6 + 30 * 2) / (2 + 6 + 2)
Price/kg of the mixture = (30 + 150 + 60) / 10
Price/kg of the mixture = 240 / 10
Price/kg of the mixture = $24
Therefore, the price/kg for the resulting mixture is $24/kg.
In conclusion, the beans should be mixed in a ratio of 2 : 6 : 2, and the price/kg for the resulting mixture is $24/kg.