Computing your liability $-$ An auto insurance example

Although car insurance is legally required by all states, the coverage provided by different policies can vary dramatically, such that even an "insured" motorist can end up paying large amounts of money out$-$of$-$pocket. Consider the following example involving Shen, who has a personal auto policy which covers him, any family member using the automobile, and anyone using the automobile with permission. This policy provides liability coverage with limits of $50/100/50,$ no medical payment insurance, and uninsured and underinsured motorist coverage with limits of $\frac{1}{3}.$ He has collision and comprehensive auto insurance, both with a deductible of $$500.$

Suppose one day Shen swerves to avoid a cat on the freeway and smashes into the car on his right. He breaks multiple ribs and is taken to the emergency room. Although he suffered no long$-$term damage, his medical bills totaled $$1,938.$ The driver of the other car, a woman named Megan, was struck directly by Shen's car and suffered from multiple severe injuries that rendered her unable to work for nearly a year. Her medical bills, plus compensation for being unable to work, totaled $$135,947.$ Megan's insurance company successfully filed a claim against Shen, exercising subrogation rights.

Coverage $A$ of Shen's plan will reimburse him with for Megan's medical bills. Shen's medical bills are

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