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This problem trains for: SAT-I, AMC-8, GMAT.

The Three Musketeers are off to adventures together but they like to pay their fair share of expenses. On a trip to England, each spends on meals an average of 78 ducats per week. Porthos eats 6 times as much as the average man, Athos eats 3 times less than Porthos and Aramis eats 5 times as much as the average man. What is the difference between the largest amount and the smallest amount one of them spends for food?

(characters from Alexandre Dumas - "The Three Musketeers")

If Aramis eats 3 times less than Porthos, this means he eats 2 times as much as the average man. Therefore, the amounts they eat are in the ratio: 6:2:5.

Denote these amounts by a, b, c. We have:

equation

Applying the properties of equivalent ratios:

equation

But:

equation

equation

equation

Porthos pays 108 ducats.

equation

equation

Athos pays 36 ducats and Aramis pays 90 ducats.

The difference between the largest amount paid and the smallest amount paid is:

equation