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Francois and Pierre each owe Claudine money. Today, Francois will make a payment equal to \(50\%\) of the amount he owes Claudine, and Pierre will make a payment equal to \(10\%\) of the amount he owes Claudine. Together, the two payments will be equal to \(40\%\) of the combined amount that Francois and Pierre owe Claudine.
Select for Francois and Pierre amounts that Francois and Pierre could owe Claudine that are jointly consistent with the given information. Make only two selections, one in each column.
€50
€250
€750
€3,750
€6,750
Let's use a simple table to track the debts and payments:
| Person | Amount Owed | Payment Rate | Payment Made |
| Francois | \(\mathrm{F}\) | 50% | \(0.50\mathrm{F}\) |
| Pierre | \(\mathrm{P}\) | 10% | \(0.10\mathrm{P}\) |
| Total | \(\mathrm{F} + \mathrm{P}\) | - | \(0.50\mathrm{F} + 0.10\mathrm{P}\) |
Key relationship: Combined payment = 40% of combined debt
The problem tells us:
Translating to an equation:
\(0.50\mathrm{F} + 0.10\mathrm{P} = 0.40(\mathrm{F} + \mathrm{P})\)
We need to find the mathematical relationship between \(\mathrm{F}\) and \(\mathrm{P}\), then identify which answer choices satisfy this relationship.
Starting with our equation:
\(0.50\mathrm{F} + 0.10\mathrm{P} = 0.40(\mathrm{F} + \mathrm{P})\)
Expanding the right side:
\(0.50\mathrm{F} + 0.10\mathrm{P} = 0.40\mathrm{F} + 0.40\mathrm{P}\)
Rearranging terms:
\(0.50\mathrm{F} - 0.40\mathrm{F} = 0.40\mathrm{P} - 0.10\mathrm{P}\)
\(0.10\mathrm{F} = 0.30\mathrm{P}\)
Simplifying:
\(\mathrm{F} = 3\mathrm{P}\)
Key finding: Francois owes exactly 3 times what Pierre owes.
Choices: €50, €250, €750, €3,750, €6,750
We need pairs where \(\mathrm{F} = 3\mathrm{P}\):
If Pierre owes €250:
Stop here - we found our answer.
Let's confirm this works:
These values satisfy our requirement that Francois owes 3 times what Pierre owes, and their partial payments together equal 40% of their combined debt.