Canteen Prices and a Singular Matrix
A school runs two canteens, North and South. On Monday the North canteen sold 40 samosas and 25 glasses of juice for a total of Rs 3,275, while the South canteen sold 32 samosas and 20 glasses of juice for Rs 2,620. The manager writes the sales as a matrix equation and tries to invert the coefficient matrix to recover the unit price of a samosa (s) and a glass of juice (j). A student volunteer claims the inverse exists because all entries are positive, and proceeds to compute prices. On Tuesday a third stall, East, sells 20 samosas and 15 glasses of juice for Rs 1,725. The manager wants to use all three records to settle the prices and also wants to know whether the Monday data alone could ever determine a unique price pair. Before publishing the menu, the manager must also test a proposal to raise both prices by 10% and check whether the new combined revenue from the Monday sales would exceed Rs 6,400.
1. Form the coefficient matrix for Monday and use its determinant to decide whether the Monday records alone determine unique prices. Explain what the determinant means in this situation.
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Monday's equations are 40s + 25j = 3275 and 32s + 20j = 2620. The coefficient matrix is M = [[40, 25], [32, 20]]. Its determinant is 40×20 − 25×32 = 800 − 800 = 0, so M has no inverse. Positive entries do not guarantee invertibility.
The rows are proportional because 40/32 = 25/20 = 5/4. The right-hand sides also satisfy 3275/2620 = 5/4, so the two equations are consistent and dependent. Dividing the first equation by 5 gives 8s + 5j = 655, which has infinitely many non-negative solutions, for example j = 35 gives s = 60, while j = 55 gives s = 47.5. Hence the Monday records alone cannot determine a unique price pair; the student's inverse calculation is invalid.
2. Use the East record together with the North record to find s and j, and verify the result against the South record.
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Use North and East: 40s + 25j = 3275 20s + 15j = 1725
Divide the first by 5 and the second by 5: 8s + 5j = 655 4s + 3j = 345
Twice the second equation is 8s + 6j = 690. Subtracting the first equation gives j = 35. Then 4s + 3(35) = 345, so 4s = 240 and s = 60.
Check with South: 32(60) + 20(35) = 1920 + 700 = 2620, which matches the recorded total. Therefore a samosa costs Rs 60 and a glass of juice costs Rs 35.
3. Test the 10% increase proposal. Would the combined revenue from the Monday sales exceed Rs 6,400?
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After a 10% rise, s = 66 and j = 38.5. Monday North revenue becomes 40(66) + 25(38.5) = 2640 + 962.5 = 3602.5. Monday South revenue becomes 32(66) + 20(38.5) = 2112 + 770 = 2882. Combined revenue is 3602.5 + 2882 = 6484.5.
Equivalently, 1.1(3275 + 2620) = 1.1(5895) = 6484.5. This exceeds 6400 by Rs 84.50, so the target is met, but only narrowly. The East record is not part of the Monday total and should not be included in this calculation.