Question

b) During the first market day, Fatuma bought 30 oranges and 12 mangoes and paid Ksh. 936 for all the fruits. In the second market day, the price of an orange increased by 20% while that of a mango reduced in the ratio 3:4. Fatuma bought 15 oranges and 20 mangoes and paid Ksh. 780 for all the fruits. Given that the cost of an orange and that of a mango during the first market day was Ksh. x and Ksh. y respectively: (i) Write down simultaneous equations to represent the information above. (2 marks) (ii) Use matrix in (a) above to find the cost of an orange and that of a mango in the first market day. (4 marks) (iii) Fatuma sold all the fruits bought on the second market day at a profit of 10% per orange and 15% per mango. Calculate the total amount of money realized for the sales. (2 marks)​

Answer 1

Answer:Let the cost of an orange and that of a mango during the first market day be Ksh. x and Ksh. y respectively.

From the first market day:

30x + 12y = 936

From the second market day:

15(1.2x) + 20(3/4y) = 780

Simplifying the second equation:

18x + 15y = 780

(ii) Using matrix to find the cost of an orange and that of a mango in the first market day:

Rewriting the equations in matrix form:

|30 12| |x| |936|

|18 15| x |y| = |780|

Multiplying the matrices:

|30 12| |x| |936|

|18 15| x |y| = |780|

|30x + 12y| |936|

|18x + 15y| = |780|

Using matrix inversion:

| x | |15 -12| |936 12|

| y | = | -18 30| x |780 15|

|x| |270 12| |936 12|

| | = |-360 30| x |780 15|

|y|

Simplifying the matrix multiplication:

|x| |1194| |12|

| | = | 930| x |15|

|y|

Therefore, the cost of an orange in the first market day was Ksh. 39 and the cost of a mango in the first market day was Ksh. 63.

(iii) Calculation of the total amount of money realized for the sales:

On the second market day, Fatuma bought 15 oranges and 20 mangoes.

Cost of 15 oranges = 15(1.2x) = 18x

Cost of 20 mangoes = 20(3/4y) = 15y

Total cost of fruits bought on the second market day = 18x + 15y = 18(39) + 15(63) = Ksh. 1629

Profit earned on 15 oranges at 10% = 1.1(1.2x)(15) - (1.2x)(15) = 0.18x(15) = 2.7x

Profit earned on 20 mangoes at 15% = 1.15(3/4y)(20) - (3/4y)(20) = 0.15y(20) = 3y

Total profit earned = 2.7x + 3y

Total amount of money realized for the sales = Total cost + Total profit

= Ksh. 1629 + 2.7x + 3y.

Explanation: