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A business permits its customers to pay with a credit card or to receive a percentage discount of r for paying cash. For credit card purchases, the business receives 97% of the purchase price one-half month later. At an annual effective rate of discount of 22%, the two payments are equivalent. Find r.

Correct answer: 0.04

My work: We want to compare present values of two payments, and they must be set equal.

$x(1-0.22)^{1.5/12} = x(1-\frac{r}{100})$, what is wrong with this formulation? I'm confused about what the problem means when it says "two payments are equivalent". Is it saying the payment amount in cash is same as the 97% the business receives or the discounted value of the 97%? If so, what do we multiply $(1-0.22)^{1.5/12}$ with to find said discounted value of the 97%?

MinYoung Kim
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1 Answers1

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Your idea to equate the present values is correct. In the cash discount scenario, an item that costs $1$ would be paid immediately in cash at a value of $1(1-r)$, so for example, if $r = 5\%$, the business would accept a cash payment of $0.95$.

In the credit card scenario, the business receives the amount $0.97$ half a month later (note: "one-half month" here means $0.5$ months, not $1.5$ months). If the annual effective rate of discount is $d = 0.22$, then the effective rate of discount over this period is $$1 - (1 - d)^{1/24} \approx 0.010299.$$ So the present value of this payment is $$0.97 (1 - 0.010299) \approx 0.96001.$$ Equating this to $1-r$ yields $r \approx 0.04$ as claimed.

Note that the equation you set up is missing two things: first, that the left-hand side should include a factor of $0.97$. This is because the amount the business receives is not only delayed in time, but also reduced in size: the credit card company presumably keeps the other $3\%$. Second, the exponent is not $1.5/12$ but rather $0.5/12 = 1/24$, due to an apparent misinterpretation of the wording of the question as I have explained above. As you can also see, the use of $x$ is unnecessary for this question, since it cancels out. Finally, you used $r/100$ which is acceptable if $r$ is expressed as a percentage, whereas I leave the conversion until the end.

So, your corrected equation would read $$0.97 x (1 - 0.22)^{0.5/12} = x (1 - r/100).$$ Solving for $r$ yields the $4\%$ answer we obtained earlier.

heropup
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