Problem solving - Ratio / Proportion

[guerrero]

Hello All, Please asssit me understanding & solving this problem.

A tea merchant buys 2 varieties of tea - the price of the first being twice of the second. He sells the mixture at 17.50$ per kg, thereby making a profit of 25%. If the ratio of the amounts of the first tea and the second tea in the mixture is 2:3, then find the cost of each tea?

thank you in anticipation !

[Mike@Magoosh]

A tea merchant buys 2 varieties of tea - the price of the first being twice of the second. He sells the mixture at 17.50$ per kg, thereby making a profit of 25%. If the ratio of the amounts of the first tea and the second tea in the mixture is 2:3, then find the cost of each tea?

Hi, there. I'm happy to help with this one.

Let me first say -- holy schnikies! This is a challenging question: I would say in the 800+ region.

Let's start with a little backwards calculation. The price of $17.50/kg represents a 25% profit, so it's a 25% increase over the cost.

C*1.25 = 17.25

C = 17.50/1.25 = 14

The mixture costs the tea merchant $14/kg.

He mixes the teas in a 2:3 ratio, and gets one kilogram. 2 + 3 = 5, so he uses 2/5 kg = 0.4 kg of Tea #1 and 3/5 kg = 0.6 kg of Tea #2.

Now, Tea #1 has a price of 2P/kg, and Tea #2 has a price of P/kg.
The 2/5 kg of Tea #1 costs 4P/5.
The 3/5 kg of Tea #2 costs 3P/5
The combined cost is 4P/5 + 3P/5 = 7P/5 for the full kilogram, and this should equal $14.

14 = 7P/5 ---> 70 = 7P, so P = 10 and 2P = 20

The first tea costs $20 and the second tea costs $10.

Check: Suppose he marks up the price to a profit first. The cost of Tea #1 with a 25% profit is $25/kg. The cost of Tea #2 with a 25% profit is $12.50/kg. Now, make a mixture with 2 kg of the first and 3 kg of the second. That would cost 2*(25) + 3*(12.50) = $87.50 for the entire 5 kg mixture. It would have a per-kilogram price of 87.50/5 = $17.50, which is the price the merchant was charging, so our answers work.

Does all that make sense?

Here's another practice question with percent increases:
https://gmat.magoosh.com/questions/30
When you submit your answer to that, the next page will have the complete video explanation.

Let me know if you have any further questions.

Mike

[GMATGuruNY]

Hello All, Please asssit me understanding & solving this problem.

A tea merchant buys 2 varieties of tea - the price of the first being twice of the second. He sells the mixture at 17.50$ per kg, thereby making a profit of 25%. If the ratio of the amounts of the first tea and the second tea in the mixture is 2:3, then find the cost of each tea?

thank you in anticipation !

Cost of the mixture:
A profit of 25% implies that the selling price is equal to 5/4 of the cost:
17.5 = (5/4)c
c = (4/5)(17.5) = $14 per kilogram.

To determine what fraction of the cost is spent on each tea, plug in dummy values that satisfy the given conditions.
Since the first tea costs twice as much as the second, let the cost of the first tea = 2 and the cost of the second tea = 1.
Since the ratio of the teas is 2:3, every 5 kilograms of the mixture is composed of 2 kilograms of the first tea and 3 kilograms of the second tea.
The cost of 2 kilograms of the first tea = 2*2 = 4.
The cost of 3 kilograms of the second tea = 3*1 = 3.
Since 4+3=7, of every $7 spent on the mixture, $4 is spent on the first tea.
Thus, the fraction spent on the first tea = 4/7.

The actual cost of 5 kilograms of the mixture = 5*14 = 70.

Thus:
The total cost of 2 kilograms of the first tea = (4/7)70 = 40.
The cost per kilogram of the first tea = 40/2 = 20.

The total cost of 3 kilograms of the second tea = 70-40 = 30.
The cost per kilogram of the second tea = 30/3 = 10.

[1947]

Once we find the actual price of tea is $14. we can also use weighted mean equation
2/3 = (14-x)/2x-14
x= 10
2x =20

tough question indeed....in first go I used the figure 17.5 instead of finding 14.

[Stuart@KaplanGMAT]

Hello All, Please asssit me understanding & solving this problem.

A tea merchant buys 2 varieties of tea - the price of the first being twice of the second. He sells the mixture at 17.50$ per kg, thereby making a profit of 25%. If the ratio of the amounts of the first tea and the second tea in the mixture is 2:3, then find the cost of each tea?

thank you in anticipation !

Hi!

As the other experts noted, this is a really tough math problem - which is why, almost certainly, the best way to approach it is strategically.

If you had provided answer choices (please always provide the choices!), we could have backsolved. When you have a very complicated word problem with a concrete question and information, and the answer choices are numbers, backsolving is frequently the quickest way to solve.

In fact, the better you're doing on the test the more you benefit from alternative approaches like backsolving and picking numbers.

Stuart

[guerrero]

Sweet! thank you very much for taking time in solving this problem . The explanation here helped me build my foundation in Ratio/ Propotion .

[kevincanspain]

You could call the price of the two varieties 2p and p.
The amount of the two varieties 2x and 3x

Thus the average purchase cost is (4xp + 3xp)/5x = 7x/5

The selling price would have to be 5/4 of the average purchase cost i.e. 7p/4 = 1.75p

Thus p= 10