Question
A manufacturer produces two types of furniture — chairs and tables. Each chair requires 1 hour of skilled labour and 2 hours of unskilled labour. Each table requires 2 hours of skilled labour and 1 hour of unskilled labour. The factory has at most 10 hours of skilled and 14 hours of unskilled labour available per day. The profit is ₹30 per chair and ₹20 per table.
How many chairs and tables should the factory produce daily to maximise profit? What is the maximum profit?
(CBSE 2025 Sample Paper — 5 marks)
Solution — Step by Step
Let = number of chairs, = number of tables produced per day.
We always start by naming what we’re optimising — this becomes our objective function.
We want to maximise . Every chair adds ₹30, every table adds ₹20.
From the labour limits:
Read constraints directly from the problem table. One resource = one inequality.
Plot the boundary lines and find where they intersect axes and each other.
Boundary 1: → cuts axes at and
Boundary 2: → cuts axes at and
Intersection of both lines: Solve simultaneously.
Multiply by 2: . Subtract : , so .
Substituting back: .
Intersection point: .
Corner points of feasible region: , , , .
| Corner Point | |
|---|---|
| ₹0 | |
| ₹210 | |
| ₹220 | |
| ₹100 |
Maximum value is Z = ₹220 at .
The factory should produce 6 chairs and 2 tables daily for a maximum profit of ₹220.
Why This Works
Linear programming works on one key theorem: the optimal value of a linear objective function over a convex polygonal region always occurs at a corner point (vertex). So we never need to check interior points — just the vertices.
The feasible region here is the intersection of both half-planes (all four constraints together). Every point inside satisfies all constraints, but profit is maximised only at the boundary — specifically at a corner.
This is why finding all corner points and testing at each one is a complete method. We are guaranteed to find the global maximum this way.
Alternative Method — Iso-Profit Line
Instead of testing corners, draw lines of the form for increasing .
Start with a small , say (the line ). Keep sliding this line outward (away from the origin, parallel to itself) while it still touches the feasible region.
The last point where this “iso-profit line” touches the feasible region is the optimal solution. Geometrically, you can see it grazes the corner before leaving the feasible region entirely.
The iso-profit method is faster for verifying answers in MCQ format — you just need to check which corner is “furthest out” in the direction of increasing . For 5-mark board questions, always use the corner-point table since you need to show working.
Common Mistake
Mixing up which constraint is which. Students often write (swapping the coefficients for skilled labour). Always re-read: “each chair needs 1 hr skilled, each table needs 2 hrs skilled” → coefficient of is 1, coefficient of is 2 → .
A wrong constraint gives a completely different feasible region and a wrong answer that still looks valid. If your corner point doesn’t give integer values here, go back and re-check your constraint setup — this problem is designed to have clean integer answers.