Stationery bundles and a pricing claim that may not add information
At a stationery shop near a school in Salem, notebooks and pens are sold in bundle offers. Bundle A contains 3 notebooks and 2 pens and costs ₹190. Bundle B contains 5 notebooks and 3 pens and costs ₹310. A teacher buys bundles for class prizes and records the totals. Later the shopkeeper introduces Bundle C: 4 notebooks and 3 pens for ₹260, and claims that this third offer proves the prices are fair because it is consistent with the first two. The teacher writes the three equations in matrix form and wants to know whether the third equation is independent information or a consequence of the first two. She also wants the actual price of one notebook and one pen, and she wants to know what would change if Bundle C had been priced at ₹265 instead. The shopkeeper says the determinant of the first two equations is not zero, so there must be a unique price pair; the teacher wants that determinant computed and interpreted.
1. Solve 3x+2y=190 and 5x+3y=310 using a determinant method, where x is the price of a notebook and y is the price of a pen. Check the solution.
Read the worked answer
The coefficient determinant is D = 3(3) - 2(5) = 9 - 10 = -1. Since D is not zero, the system has a unique solution. D_x = 190(3) - 2(310) = 570 - 620 = -50, so x = D_x/D = (-50)/(-1) = 50. D_y = 3(310) - 190(5) = 930 - 950 = -20, so y = D_y/D = (-20)/(-1) = 20. Check: 3(50)+2(20)=150+40=190 and 5(50)+3(20)=250+60=310. A notebook costs ₹50 and a pen costs ₹20.
2. Write the equation for Bundle C and decide whether it is independent of the first two equations. Interpret the result for the shopkeeper's claim.
Read the worked answer
Bundle C gives 4x+3y=260. Using x=50 and y=20, the left side is 4(50)+3(20)=200+60=260, so the equation is satisfied by the unique solution of the first two equations. It is not independent: it is a linear combination of the first two. In fact, 3(3x+2y) - (5x+3y) = 4x+3y, and 3(190) - 310 = 260. So Bundle C is a consequence of A and B, not new evidence. The shopkeeper's claim that it 'proves' fairness is misleading; it merely repeats information already contained in the first two bundles.
3. If Bundle C were priced at ₹265 instead, what would the three-equation system imply? Decide consistency and explain.
Read the worked answer
The first two equations still give the unique solution x=50, y=20. For Bundle C at ₹265, the required equation is 4x+3y=265. But at the solution, 4(50)+3(20)=260, not 265. Therefore the third equation is not satisfied by the only pair that satisfies the first two. The full system is inconsistent: no single notebook-and-pen price pair can satisfy all three offers. The ₹265 price is not a harmless extra equation; it contradicts the first two unless one of the bundle contents or prices is recorded incorrectly.