Mathematics
Mathematics
Problem
A factory produces three types of widgets P, Q and R. In a particular week, the cost of producing a batch of each widget type is recorded as follows: • Producing 3 batches of P, 2 batches of Q and 1 batch of R costs ₹52,000. • Producing 2 batches of P, 3 batches of Q and 2 batches of R costs ₹60,000. • Producing 1 batch of P, 1 batch of Q and 2 batches of R costs ₹38,000. Assuming the cost per batch of each widget type remains constant throughout the week, (i) Formulate a system of three linear equations in the unknown costs per batch of widgets P, Q and R. (ii) Express the system in the compact form . (iii) Find the cost per batch of each widget type. (iv) Using the costs obtained in part (iii), calculate the total cost of producing 5 batches of P, 4 batches of Q and 3 batches of R.Intuitive explanation
This problem asks you to translate a real-world production scenario into algebra—the core skill of systems of linear equations. Each sentence about total costs gives you one equation; the key insight is to define variables for the unknown unit costs and let the batch quantities become coefficients. Once you have three equations in three unknowns, you can organize them into matrix form and solve using elimination, substitution, or matrix methods to find what each widget actually costs per batch.
Intuitive explanation
This problem asks you to translate a real-world production scenario into algebra—the core skill of systems of linear equations. Each sentence about total costs gives you one equation; the key insight is to define variables for the unknown unit costs and let the batch quantities become coefficients. Once you have three equations in three unknowns, you can organize them into matrix form and solve using elimination, substitution, or matrix methods to find what each widget actually costs per batch.