Answer to Question #117294 in Linear Algebra for mike

Question

Tin Tan Lollies makes three mixes of popular sweets:
The fruit mix contains 50% fruit jellies, 25% chocolates and 25% chews
The chocolate mix contains 30% fruit jellies, 45% chocolates and 25% chews.
The Lasting mix contains 40% fruit jellies, 25% chocolates and 35% chews.
Tin Tan Lollies buys the fruit jellies for $6 a kilogram, the chocolates for $10 a kilogram and the
chews for $5 a kilogram.

a) Write a 3 × 3 matrix, M, that represents the percentage of the three types of sweets in the three
mixes. Convert the percentage in decimal form in matrix M.

b) Write a 3 × 1 matrix C that represents the cost per kilogram of the three types of sweets.

F, C and L represent the cost price in dollars per kilogram of the Fruit, Chocolate and Lasting mixes
of sweets, respectively.
Use the matrix equation MC = P where P = F
C
L

to calculate the values of F, C, L. Give your answers to the nearest cent. Show entire working?

Answer 1

a) Let the columns of the matrix M represent the percentage of sweets- fruit jellies, chocolate and chews in the mixes and the rows represent the mixes- Fruit mix, chocolate mix and lasting mix.

$M=\begin{bmatrix} 50\% & 25\% & 25\% \\ 30\% & 45\% & 25\% \\ 40\% & 25\% & 35\% \end{bmatrix}$

Converting percentages to decimal values:

$M= \begin{bmatrix} 0.5 & 0.25 & 0.25 \\ 0.3 & 0.45 & 0.25 \\ 0.4 & 0.25 & 0.35 \end{bmatrix}$

b)

to find matrix C :

Since C represents the cost per kilogram of the three types of sweets- fruit jellies, chocolate and chews in the mixes

$C = \begin{bmatrix} 6 \\ 10 \\ 5 \end{bmatrix}$

$MC = \begin{bmatrix} 0.5 & 0.25 & 0.25 \\ 0.3 & 0.45 & 0.25 \\ 0.4 & 0.25 & 0.35 \end{bmatrix} \begin{bmatrix} 6\\ 10 \\ 5 \end{bmatrix}$

$= \begin{bmatrix} 6×0.5+.25×10 + .25×5 \\ 6×0.3+.45×10 + .25×5 \\6×0.4+.25×10 + .35×5 \end{bmatrix}$

$\begin{bmatrix} 6.75 \\ 7.55 \\ 6.65 \end{bmatrix}$

$=\begin{bmatrix} F \\ C \\ L \end{bmatrix}$

F = 6.75 , C= 7.55 , L= 6.65