9. A grocer wants to mix a type of spice which costs £22 per kilogram with another
type which costs £12 per kilogram, to obtain 20 kilograms of mixture which will
cost £15 per kilogram. What quantity of each spice must the grocer take?
let a = 22 / kg, and b = 12 / kg
a + b = 20 kg
22a + 12b = 15 * 20
12a + 12b = 240
10a = 60
To solve this problem, we can set up a system of equations.
Let's assume x represents the quantity (in kilograms) of the spice that costs £22 per kilogram, and y represents the quantity (in kilograms) of the spice that costs £12 per kilogram.
Given:
1. The grocer wants to obtain 20 kilograms of mixture.
2. The mixture will cost £15 per kilogram.
We can now set up the equations.
Equation 1: x + y = 20
This equation represents the total quantity of the mixture, which must be 20 kilograms.
Equation 2: (22*x + 12*y) / 20 = 15
This equation represents the average cost per kilogram of the mixture, which must be £15.
Now, let's solve this system of equations.
Rearrange Equation 1 to solve for x:
x = 20 - y
Substitute this value of x into Equation 2:
(22*(20 - y) + 12*y) / 20 = 15
Simplify the equation:
(440 - 22*y + 12*y) / 20 = 15
(440 - 10*y) / 20 = 15
Multiply both sides by 20 to get rid of the fraction:
440 - 10*y = 300
Subtract 440 from both sides:
-10*y = -140
Divide both sides by -10:
y = 14
Now, substitute the value of y into Equation 1:
x + 14 = 20
x = 20 - 14
x = 6
Therefore, the grocer must take 6 kilograms of the spice that costs £22 per kilogram, and 14 kilograms of the spice that costs £12 per kilogram to obtain the desired mixture.