The School Store Price List
At the school store, a volunteer writes two facts on the whiteboard but loses the price list. Fact 1: 3 notebooks and 2 pens cost $11. Fact 2: 2 notebooks and 3 pens cost $9. A student helper claims the prices cannot be found because there are two unknowns and only two facts, which is 'not enough.' Another student, Tessa, says the two facts are independent and can be combined. The store manager adds a constraint: every item price is a whole-dollar amount, and notebooks cost more than pens. The volunteer needs the exact prices before the lunch rush, and a wrong price will make the cash box short. The helpers must decide whether the facts determine a single price pair, solve for it, and explain how to check the answer against both facts.
1. Find the price of one notebook and one pen. Show how the two facts are combined and verify both equations.
Read the worked answer
Let n be the notebook price and p the pen price.
- 3n + 2p = 11
- 2n + 3p = 9
Multiply the first equation by 3 and the second by 2 so the p terms match:
- 9n + 6p = 33
- 4n + 6p = 18
Subtract: 5n = 15, so n = 3. Substitute into 3n + 2p = 11: 9 + 2p = 11, so 2p = 2 and p = 1.
Check both facts: 3(3) + 2(1) = 11; 2(3) + 3(1) = 9. The prices are whole dollars, and notebooks ($3) cost more than pens ($1), so the manager's constraint is satisfied.
2. The first student says two unknowns with two facts is 'not enough.' Explain the flaw in that claim and describe a case where the facts would not determine a unique price pair.
Read the worked answer
The claim confuses the number of equations with their independence. Two unknowns can be determined by two independent equations because each equation gives a different constraint; together they pin down a single intersection point. Here the equations are independent because their coefficients are not proportional: (3, 2) is not a multiple of (2, 3).
The facts would fail to determine a unique pair if one fact were a multiple of the other. For example, if Fact 2 were '6 notebooks and 4 pens cost $22,' it would be exactly twice Fact 1 and would add no new information. Then infinitely many price pairs could satisfy the single constraint, and the store would need another independent fact, or the whole-dollar and notebook-more-than-pen conditions might narrow but not necessarily uniquely determine the prices. Independence, not the raw count of equations, is what matters.