Applied Word Problems — Diet, Manufacturing, Allocation

The Three Archetypes

Note: the rationalized NCERT trimmed these applied problem types from the exercises, but boards continue to set them — the formulation skill is exactly Section 1's recipe, applied to a story. This section restores the three classic archetypes with full solutions.

1. Diet problems (minimise cost)

Two foods, each carrying known amounts of nutrients per unit; the diet must meet minimum nutrient requirements. Nutrients give $\geq$ constraints (floors); the objective is total cost, minimised. The feasible region is typically unbounded (you can always overeat), so the half-plane test is mandatory before declaring the minimum.

2. Manufacturing problems (maximise profit)

Two products, each consuming known amounts of limited resources (machine hours, labour hours, raw material). Resources give $\leq$ constraints (ceilings); the objective is total profit, maximised. The region is bounded by the resource ceilings, so Theorem 2 applies directly.

3. Allocation problems (capital, space, count)

A merchant or planner splits limited capital/space between two options. Constraints typically pair a money ceiling (unit costs as coefficients) with a count or space ceiling ($x + y \leq$ capacity). Maximise profit or the number of items.

The translation table

Key Point (board marking): applied questions award marks in two halves — the formulation (variables defined in words, objective, every constraint, non-negativity) and the graphical solution (graph, corner table, conclusion in the story's language: "the dealer should buy 8 fans and 12 sewing machines for a maximum profit of ₹392"). Ending with bare coordinates loses the final mark.

Diet Problems

Example 1: Two foods, two vitamins

Food X costs ₹16/kg and contains $1$ unit of vitamin A and $2$ units of vitamin C per kg. Food Y costs ₹20/kg and contains $2$ units of vitamin A and $1$ unit of vitamin C per kg. The diet needs at least $10$ units of vitamin A and $12$ units of vitamin C. Find the least-cost mixture.

Solution:

1. Formulate: minimise $Z = 16x + 20y$ subject to $x + 2y \geq 10$, $2x + y \geq 12$, $x, y \geq 0$.
2. Corners of the unbounded region: $(0, 12)$, $\left(\dfrac{14}{3}, \dfrac{8}{3}\right)$ (solving the lines), $(10, 0)$.
3. Table: $240$, $\dfrac{224 + 160}{3} = 128$, $160$; smallest $m = 128$.
4. Half-plane test: $16x + 20y < 128$ shares no point with the region. ✓

Answer: buy $\dfrac{14}{3}$ kg of X and $\dfrac{8}{3}$ kg of Y for the minimum cost of ₹128 — fractional kilograms are perfectly sensible for foodstuffs.

Example 2: A feed-mix problem

A farmer mixes two brands of cattle feed. Brand P (₹250 per bag) gives $3$ units of nutrient A, $2.5$ of B and $2$ of C per bag; brand Q (₹200 per bag) gives $1.5$ of A, $11.25$ of B and $3$ of C. Minimum requirements are $18$ of A, $45$ of B and $24$ of C. Find the least-cost mix.

Solution:

1. Formulate and simplify: minimise $Z = 250x + 200y$ subject to $3x + 1.5y \geq 18$ i.e. $2x + y \geq 12$; $2.5x + 11.25y \geq 45$ i.e. $2x + 9y \geq 36$; $2x + 3y \geq 24$; $x, y \geq 0$.
2. Corners: $(0, 12)$, $(3, 6)$, $(9, 2)$, $(18, 0)$.
3. Table: $2400, 1950, 2650, 4500$; smallest $m = 1950$.
4. Test: $250x + 200y < 1950$ misses the region. ✓

Answer: mix $3$ bags of P with $6$ bags of Q for the minimum cost of ₹1,950 — with three floors, the middle corners matter; tabulate all four.

Manufacturing Problems

Example 3: Rackets and bats

A factory makes tennis rackets and cricket bats. A racket needs $1.5$ hours of machine time and $3$ hours of craftsman's time; a bat needs $3$ hours of machine time and $1$ hour of craftsman's time. Daily availability: $42$ machine hours and $24$ craftsman hours. Profits: ₹20 per racket, ₹10 per bat. Maximise profit.

Solution:

1. Formulate: maximise $Z = 20x + 10y$ subject to $1.5x + 3y \leq 42$ i.e. $x + 2y \leq 28$; $3x + y \leq 24$; $x, y \geq 0$.
2. Corners: $(0,0)$, $(8, 0)$, $(4, 12)$ (solving the lines), $(0, 14)$.
3. Table: $0, 160, 200, 140$.

Answer: make $4$ rackets and $12$ bats daily for the maximum profit of ₹200 — at the optimum both resources are used exactly ($4 + 24 = 28$ and $12 + 12 = 24$): no idle hours.

Example 4: Nuts and bolts

A manufacturer produces nuts and bolts. A kg of nuts needs $1$ hour on machine A and $3$ hours on machine B; a kg of bolts needs $3$ hours on A and $1$ hour on B. Each machine runs at most $12$ hours a day. Profit is ₹17.50 per kg of nuts and ₹7 per kg of bolts. Maximise profit.

Solution:

1. Formulate: maximise $Z = 17.5x + 7y$ subject to $x + 3y \leq 12$, $3x + y \leq 12$, $x, y \geq 0$.
2. Corners: $(0,0)$, $(4, 0)$, $(3, 3)$, $(0, 4)$.
3. Table: $0, 70, 73.50, 28$.

Answer: produce $3$ kg of nuts and $3$ kg of bolts for the maximum profit of ₹73.50 per day.

Allocation Problems

Example 5: Cakes from limited ingredients

One kind of cake requires $200$ g of flour and $25$ g of fat; another requires $100$ g of flour and $50$ g of fat. Find the maximum number of cakes that can be made from $5$ kg of flour and $1$ kg of fat.

Solution:

1. Formulate: maximise $Z = x + y$ (total cakes) subject to $200x + 100y \leq 5000$ i.e. $2x + y \leq 50$; $25x + 50y \leq 1000$ i.e. $x + 2y \leq 40$; $x, y \geq 0$.
2. Corners: $(0,0)$, $(25, 0)$, $(20, 10)$, $(0, 20)$.
3. Table: $0, 25, 30, 20$.

Answer: $20$ cakes of the first kind and $10$ of the second — $30$ cakes in all. Here the objective counts items, not rupees; the recipe is unchanged.

Example 6: Fans and sewing machines

A dealer has ₹5,760 to invest and space for at most $20$ items. A fan costs ₹360 and yields a profit of ₹22; a sewing machine costs ₹240 and yields ₹18. How should the dealer invest?

Solution:

1. Formulate: maximise $Z = 22x + 18y$ subject to $360x + 240y \leq 5760$ i.e. $3x + 2y \leq 48$; $x + y \leq 20$; $x, y \geq 0$.
2. Corners: $(0,0)$, $(16, 0)$, $(8, 12)$ (solving the lines), $(0, 20)$.
3. Table: $0, 352, 392, 360$.

Answer: buy $8$ fans and $12$ sewing machines for the maximum profit of ₹392 — the conclusion must name the items, not just the point $(8, 12)$.