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37 pens and 53 pencils cost \[Rs.320\], while 53 pens and 37 pencils cost \[Rs.400\]. Find the cost of a pen and a pencil.

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Hint: Assume the cost of one pencil and one pen to be x and y respectively. To find the cost of ‘a’ pens or pencils, multiply ‘a’ with the cost of one pen or pencil. Write linear equations in two variables relating the cost of pens and pencils. Solve the equations by elimination method to find the cost of one pen and pencil.

Complete step-by-step answer:
We have 37 pens and 53 which cost \[Rs.320\], while 53 pens and 37 pencils cost \[Rs.400\]. We have to calculate the cost of each pen and pencil.
We will solve this by forming linear equations in two variables.
Let’s assume that the cost of one pen and pencil is \[Rs.x\] and \[Rs.y\] respectively. To find the cost of ‘a’ pens or pencils, we will multiply ‘a’ with the cost of one pen or pencil.
So, the cost of 37 pens \[=Rs.37x\] and cost of 53 pencils \[=Rs.53y\]. We know that the total cost of 37 pens and 53 pencils is \[Rs.320\].
Thus, we have \[37x+53y=320.....\left( 1 \right)\].
Similarly, the cost of 53 pens \[=Rs.53x\] and cost of 37 pencils \[=Rs.37y\]. We know that the total cost of 53 pens and 37 pencils is \[Rs.400\].
Thus, we have \[53x+37y=400.....\left( 2 \right)\].
We will now solve both the linear equations. Multiplying equation (1) by 53 and equation (2) by 37, we get \[1961x+2809y=16960\] and \[1961x+1369y=14800\].
Subtracting both the equations, we get \[1440y=2160\].
Thus, we have \[y=\dfrac{2160}{1440}=1.5\].

Substituting the value \[y=1.5\] in equation (1), we get \[37x+53\left( 1.5 \right)=320\].
Further simplifying the equation, we have \[37x=320-79.5\Rightarrow 37x=240.5\].
Thus, we have \[x=\dfrac{240.5}{37}=6.5\].

Hence, the cost of one pen \[=Rs.x=Rs.6.5\] and cost of one pencil \[=Rs.y=Rs.1.5\].

Note: We can also solve this question by writing linear equations in one variable, by assuming the cost of pen to be x and then writing the cost of pencil in terms of x and then solving equations to get the value of pen and pencil. We can also check if the calculated values are correct or not by substituting the calculated values in the equations and checking if they satisfy the given equations or not.