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This problem trains for: SAT-I, GMAT, AMC-8, Math Kangaroo 5-6, Math Kangaroo 7-8.

The Sirius Cybernetics Corporation sells teleporting equipment. In order to increase sales, they sell their basic service at a 15% loss. They know, however, that few people teleport economy and they sell the upgrades at a hefty profit of 35%. What is the ratio between the total sales of basic services and the total sales of upgrades at which the company breaks even?

(entity from Douglas Adams' "Hitchhiker's Guide to the Universe")

The break even occurs when the losses equal the gains to result in zero profit.

Denote the total sales of basic services during a time period by S.

The loss incurred as a result of these sales is:

equation

Denote the total sales of upgrades during the same time period by U.

The gain incurred as a result of selling upgrades is:

equation

If losses equal gains (break even):

equation

The desired ratio is:

equation

equation