The Price of Darjeeling Tea (in rupees per kilogram) is 100+0.10n GMAT Problem Solving

Question: The price of Darjeeling tea (in rupees per kilogram) is 100+0.10n, on the nth day of 2007 (n = 1, 2,... 100), and then remains constant. On the other hand, the price of Ooty tea (in rupees per kilogram) is 89 + 0.15n, on the nth day of 2007 (n = 1, 2, ..., 365). On what date in 2007 will the prices of these two varieties of tea be equal? (Note: 2007 was not a leap year)

1. May 21
2. May 20
3. April 20
4. April 11
5. April 10

Answer: B
Solution and Explanation:
Approach Solution 1:

We can put those two formulas on an equal footing because they both give us the price on Day n.

We get: 100+0.10n=89+0.15n

Subtract 89 from both sides of the equation to get: 11+0.10n=0.15n
Subtract 0.10n from both sides of the equation: 11=0.05n
Solve: n=11/0.05=220
Therefore, it appears that on Day 220, the two prices will be equal.
The Darjeeling tea price formula, however, only lasts for the first 100 days after which point it is always the same.

On Day 100, the price of Darjeeling tea = 100+0.10(100)=100+10=110
So, after Day 100, the price of Darjeeling tea = 110 rupees
So let's now determine which day the price of Ooty tea is 110 rupees

We can write the equation: 89+0.15n=110
Subtract 89 from both sides to get: 0.15n=21
Solve: n=21/0.15=140

So, on Day 140, the two teas will each sell for 140 rupees per kilogram.

Our job now is to determine what day of the year is Day 140

January has 31 days
February has 28 days (in a non-leap year)
March has 31 days
April has 30 days

31 + 28 + 31 + 30 = 120 days.
In other words, Day 120 is on April 30th

Thus, Day 140 will occur on May 20th, 20 days from now.

Correct Answer: B

Approach Solution 2:

We shall first determine whether this equation is valid. If the value of n is more than 100, it will not be.

N = 220, which is more than 100, because 100+0.10n = 89+0.15n 11 = 0.05*n.

Darjeeling tea's value will therefore be at a price that it will be on day 100 because it will be constant after that.

100+0.10n = 100+0.10∗100 = 110

89+0.15n = 110 => 0.15n = 21 => n = 140

January 31^st
February: 28 days
March lasts 31 days.
thirty days in April
20th day of May

The total time is 140 days.

Answer B: May 20th

Correct Answer: B

Approach Solution 3:

Darjeeling tea = 100 + 0.1n
After n>=100, price = 110
Ooty Tea = 89 + 0.15n
Date at which both are equal
Seeing options April 10 is the 100th day of 2007 and all are after that only
So 110 = 89 + 0.15n
21 = 0.15n
700 = 5n
N = 140
31 + 28 +31 + 30 = 120
May 20 is the correct answer. It is option B.

Correct Answer: B

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