Question

In a development plan of a city, a contractor has taken a contract to construct certain houses for which he needs building materials like stones, sand, etc. There are three firms $A, B, C$ that can supply him these materials. At one time these firms A, B, C supplied hum \[40,{\text{ }}35,\] and $35$ truckloads stones and $10,5$ and $8$ truckloads of sand respectively. If the cost of one truckload of stone and sand are Rs.$1200$ and Rs.$500$ respectively, then find the total amount paid by the contractor to each of these firms A, B, C separately be a, b, c respectively, find $a + b - 2c$ in Rs

Answer 1

Hint: First find the amount paid by the contractor to the firm A, B, and C separately, which gives the values of a, b and c, then use these values to find the value of $a + b - 2c$.

Complete answer:
It is given in the problem that at one time the firms A, B, C supplies \[40,{\text{ }}35,\] and $35$ truckloads stones and $10,5$ and $8$ truckloads of sand respectively to the contractor. It is also given that the cost of one truckload of stone is Rs.$1200$ and the cost of one truckload of sand is Rs.$500$.
If the total amount paid by the contractor to each of these firms $A,B$ and $C$ separately be $a,b$ and $c$, then the goal of the problem is to find the value of $\left( {a + b - 2c} \right)$ in rupees.
According to the given data, at one time the firm $A$ supplies, $40$ truckloads of stone, and the cost of one truckload of stone is Rs. $1200$, also the firm $A$ supplies $10$ truckloads of sand and the cost of one truckload of sand is RS.$500$, then the total cost paid by the contractor to the firm $A$ is given as:
$ \Rightarrow $ $A = 40 \times 1200 + 10 \times 500$
$ \Rightarrow $ $A = 4800 + 500$
$ \Rightarrow $ $A = {\text{Rs}}{\text{. }}53000$
According to the given data, at one time the firm $A$ supplies $40$ truckloads of stone and the cost of one truckload of stone is Rs. $1200$, also the firm $A$ supplies $10$ truckloads of sand and the cost of one truckload of sand is Rs.$500$, then the total cost paid by the contractor to the firm $A$ is given as:
$ \Rightarrow $ $a = 40 \times 1200 + 10 \times 500$
$ \Rightarrow $ $a = 48000 + 5000$
$ \Rightarrow $ $a = {\text{Rs}}{\text{. }}53000$
According to the given data, at one time the firm $B$ supplies $35$ truckloads of stone and the cost of one truckload of stone is Rs. $1200$, also the firm $B$ supplies $5$ truckloads of sand and the cost of one truckload of sand is Rs.$500$, then the total cost paid by the contractor to the firm $B$ is given as:
$ \Rightarrow $ $b = 35 \times 1200 + 5 \times 500$
$ \Rightarrow $ $b = 42000 + 2500$
$ \Rightarrow $ $b = {\text{Rs}}{\text{. 44500}}$
According to the given data, at one time the firm $C$ supplies $35$ truckloads of stone and the cost of one truckload of stone is Rs. $1200$, also the firm $C$ supplies $8$ truckloads of sand and the cost of one truckload of sand is Rs.$500$, then the total cost paid by the contractor to the firm $C$ is given as:
$ \Rightarrow $ $c = 35 \times 1200 + 8 \times 500$
$ \Rightarrow $ $c = 42000 + 4000$
$ \Rightarrow $ $c = {\text{Rs}}{\text{. 46000}}$
Now, we have the amount paid by the contractor to the firms $A,B$ and $C$. These amounts are given below:
$ \Rightarrow $ $a = {\text{Rs}}{\text{. }}53000$,$b = {\text{Rs}}{\text{. 44500}}$ and $c = {\text{Rs}}{\text{. 46000}}$
We have to find the value of the expression:
$ \Rightarrow $ $\left( {a + b - 2c} \right)$
Substitute the values of $a,b$ and $c$ in the above expression:
$ \Rightarrow $ $\left( {a + b - 2c} \right) = Rs.53000 + Rs.44500 - 2\left( {Rs.46000} \right)$
$ \Rightarrow $ $\left( {a + b - 2c} \right) = Rs.97500 - 92000$
$ \Rightarrow $ $\left( {a + b - 2c} \right) = Rs.5500$

The value of the required expression is $Rs.{\text{ 5500}}$.

Note: This problem can also be solved using the method of a matrix. First express one matrix as the number of trucks of sand and stone supplied by the firms and one matrix can be used to express the amount of one truckload of sand and stone. Assume that $X$ defines the matrix that expresses the number of truckloads supplied by the firms to the contractor and $Y$ defines the matrix which represents the cost of one truckload of sand and stone.
 Then $X$ and $Y$ are given as:
\[X = \left[ {\begin{array}{*{20}{c}}
  {40}&{35}&{35} \\
  {10}&5&8
\end{array}} \right]\]
In this matrix, the first row shows the number of truckloads of stone supplied by the firms A, B, and C respectively, and the second row shows the number of truckloads of sand supplied by the firms A, B, and C respectively.
$Y = \left[ {\begin{array}{*{20}{c}}
  {1200}&{500}
\end{array}} \right]$
In the above matrix, the first element shows the cost of one truckload of stone, and the second element shows the cost of one truckload of sand.
The required total amount paid to each of the firms A, B and C are given by the product matrix YX.
$BA = \left[ {\begin{array}{*{20}{c}}
  {1200}&{500}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  {40}&{35}&{35} \\
  {10}&5&8
\end{array}} \right]$
$BA = \left[ {\begin{array}{*{20}{c}}
  {1200 \times 40 + 500 \times 10}&{1200 \times 35 + 500 \times 5}&{1200 \times 35 + 500 \times 8}
\end{array}} \right]$
$BA = \left[ {\begin{array}{*{20}{c}}
  {53000}&{44500}&{46000}
\end{array}} \right]$
Now, we have the amount paid by the contractor to the firms $A,B$ and $C$. These amounts are given below:
$a = {\text{Rs}}{\text{. }}53000$,$b = {\text{Rs}}{\text{. 44500}}$ and $c = {\text{Rs}}{\text{. 46000}}$
We have to find the value of the expression:
$ \Rightarrow $ $\left( {a + b - 2c} \right)$
Substitute the values of $a,b$ and $c$ in the above expression:
$ \Rightarrow $ $\left( {a + b - 2c} \right) = Rs.53000 + Rs.44500 - 2\left( {Rs.46000} \right)$
$ \Rightarrow $ $\left( {a + b - 2c} \right) = Rs.97500 - 92000$
$ \Rightarrow $ $\left( {a + b - 2c} \right) = Rs.5500$