An extensive survey of gym users in Metropolis indicates that 68% of the current

Question

An extensive survey of gym users in Metropolis indicates that 68% of the current members of Silver's will continue their annual membership when it expires, 13% will quit and join Fitness Lab, and the rest will quit and join ThinNFit. Fitness Lab has been unable to keep its equipment in good shape, and as a result 32% of its members will defect to Silver's, and 34% will leave for ThinNFit. ThinNFit's members are quite happy, and as a result, 96% plan on renewing their annual membership, with half of the rest planning on moving to Silver's and half to Fitness Lab. Make a long-range prediction of the ultimate market shares of the three health clubs, assuming that current trends continue. (Round the answers to one decimal places.)

Silver's
? %

Fitness Lab
? %

ThinNFit
? %

Explanation / Answer

.68x+.32*y+.02*(1-x-y)=x

.13*x+.36*y+.02*(1-x-y)=y

-.34x+.3y=-.02

.11x-.66y=-.02

-.68x+.66y=-.044

-.57x=-.064

x=.112

then y=(-.044+.68*.112)/.66=.049

so z= 1-.112-.049=.839

Thus 11.2% are at silvers, 4.9% are at Fitness Lab, and 83.9 are at ThinNFit

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