On his daily commute Bill always crosses a certain toll bridge

Let x be the number of days Bill commutes on the certain toll bridge. Since he
crosses the toll bridge twice daily, in order for the discount pass to become a
financial benefit for Bill

\(2x(1.5 - 0.8*1.5) > 50\) -> \(0.6x > 50\) -> \(x > \frac{50}{0.6}\) or \(83.33\)

Therefore, it will take 84(Option D) days for the discount pass to become a financial benefit for Bill.

Solution

Given:

• Bill crosses a certain toll bridge exactly twice a day
• Discount pass for $50 can decrease the cost of the toll by 20%
• Toll costs $1.50 per crossing

To find:

• After how many days of commuting will buying the discount pass result in a financial benefit for Bill

Approach and Working:
Let’s assume that it will take d days for Bill to get financial benefit from buying the pass

• The cost reduction in a day due to the pass = 20% of (1.5 * 2) = 0.6

As the investment in buying the pass is $50, to get financial benefit Bill must recover the investment totally

• We already know $0.6 is recovered in a day

Hence, if Bill gets financial benefits in d days, then we can write

• 0.6 * d > 50
Or, d > 83.3

As d denotes an integer, the minimum value of d will be 84

Hence, the correct answer is option D.

Answer: D

We can let d = the number of days to get a financial benefit (i.e. profit)and create the equation:

50 + 0.8(3) x d = 3d

50 + 2.4d = 3d

50 = 0.6d

d = 50/0.6 = 500/6 = 83.33

Since d must be a whole number, we see that 83 days still yields a slight loss for Bill. We must round up to 84 in order for the discount pass to result in a financial benefit for Bill.

Alternate Solution:

Each day, Bill normally pays $3.00. If he receives a 20% discount by buying the discount pass, he will pay 0.8 x $3.00 = $2.40 each day, thus saving $0.60 per day. Since the pass costs $50.00, we see that he will take 50/0.6 = 83.33 days to break even. If we round this value down to 83 days, he has not quite broken even. Therefore, in order to get a financial benefit, we round 83.33 up to 84 days.


Answer: D

I guess question is asking after how many days its after 83 days it will pass benefit. I marked C.

Am I wrong?

Before the pass,

Initial amount per crossing = $ 1.50

Initial amount in a single day = $ 1.50 * 2 = $ 3 per day

Let number of days for which Bill pays the toll is 'x'

Total amount that Bill pays without the pass for x days = $ 3 * x = 3x

Now suppose Bill buys the pass,

Pass amount = $ 50

Amount per crossing after buying the pass = $ 1.50 - [(20/100) * 1.50] = $ 1.20

Toll amount in a single day with pass = $ 1.20 * 2 = 2.40

Total amount that Bill pays with the pass for x days = 50 + 2.4x

For bill to profit, the amount that he pays with the pass should be less than the amount that he pays without the pass,

50 + 2.4x < 3x

=> 0.6x > 50

=> x > 250/3

=> x > 83.3333.....

Bill will profit when he buys the pass for more than 83 days approximately, and the minimum number of days more than this is 84 (days can only be integers), hence (D) is correct.

2.4*(d) + 50 = 3d ............(No benefit no loss)
d = 83 + 1/3

After 83 days of paying tolls, Bill goes back home and realises that buying a pass hasn't actually benefited him yet. Only on 84th day when he utilises the pass he gets benefited, and when he's home at night he thinks "Woah, after 84 days I see some benefit!"

Hence, D. 84