Question

3 bags and 4 pens together cost Rs 257, whereas 4 bags and 3 pens together cost Rs 324. Find the total cost of 1 bag and 10 pens.

Answer 1

Hint- Here, we will proceed by assuming the cost of 1 bag and 1 pen as two different variables (say x and y respectively) and then we will form two linear equations in two variables and solve it by using elimination method.

Complete Step-by-Step solution:
Let us suppose that the cost of 1 bag and 1 pen be Rs x and Rs y respectively
Given, Cost of 3 bags and 4 pens = Rs 257
$ \Rightarrow $Cost of 3 bags + Cost of 4 pens = Rs 257 $ \to (1)$
As we know that the cost of n items is given by
Cost of n items = n(Cost of 1 item) $ \to (2)$
Using the formula given by equation (2) in equation (1), we get
$ \Rightarrow $3(Cost of 1 bag) + 4(Cost of 1 pen) = 257
$ \Rightarrow $3(x) + 4(y) = 257
Multiplying the above equation by 3 on both sides, we get
$
   \Rightarrow 3 \times 3x + 3 \times 4y = 3 \times 257 \\
   \Rightarrow 9x + 12y = 771{\text{ }} \to {\text{(3)}} \\
 $
Also, given that Cost of 4 bags and 3 pens = Rs 324 $ \to (4)$
Using the formula given by equation (2) in equation (4), we get
$ \Rightarrow $4(Cost of 1 bag) + 3(Cost of 1 pen) = 324
$ \Rightarrow $4(x) + 3(y) = 324
Multiplying the above equation by 4 on both sides, we get
$
   \Rightarrow 4 \times 4x + 4 \times 3y = 4 \times 324 \\
   \Rightarrow 16x + 12y = 1296{\text{ }} \to {\text{(5)}} \\
 $
By subtracting equation (3) from equation (5), we get
$
   \Rightarrow 16x + 12y - \left( {9x + 12y} \right) = 1296 - 771 \\
   \Rightarrow 16x + 12y - 9x - 12y = 525 \\
   \Rightarrow 7x = 525 \\
   \Rightarrow x = \dfrac{{525}}{7} = 75 \\
 $
Put x = 75 in equation (3), we get
\[
   \Rightarrow 9\left( {75} \right) + 12y = 771{\text{ }} \\
   \Rightarrow 3\left[ {3\left( {75} \right) + 4y} \right] = 3\left( {257} \right) \\
   \Rightarrow 225 + 4y = 257 \\
   \Rightarrow 4y = 257 - 225 = 32 \\
   \Rightarrow y = 8 \\
 \]
So, the cost of 1 bag and 1 pen are Rs 75 and Rs 8 respectively
 Cost of 1 bag and 10 pens = Cost of 1 bag + 10(Cost of 1 pen)
\[ \Rightarrow \]Cost of 1 bag and 10 pens = 75 + 10(8) = 75 + 80 = 155
Therefore, the cost of 1 bag and 10 pens is Rs 155.

Note- In this particular problem, we have used elimination method for solving the two linear equations in which the coefficient of variable y is made same in both these equations by multiplying the first equation and second equation by 3 and 4 respectively so that when these equations are subtracted we will be just left with variable x in the final equation.