In what ratio must a person mix three kinds of tea costing $60/kg, $75

[quote="Bunuel"]In what ratio must a person mix three kinds of tea costing $60/kg, $75/kg and $100 /kg so that the resultant mixture when sold at $96/kg yields a profit of 20%?

A. 1 : 2 : 4
B. 3 : 7 : 6
C. 1 : 4 : 2
D. 1 : 4 : 3
E. 1 : 4 : 4

Can someone please provide a solution to this question?
thanks in advance

Bunuel, can you please explain the wording here? "yields a profit of 20%?" and how that is derived into a formula??

By checking options one by one
I got the answer c
As Cost price of mixture =96/1.20= 80

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We can have multiple ratio to satisfy the situation. Solving through options is the right way, I guess.

I solved this problem using 3 way ratio as follows:

Firstly, assuming x = cost price
1.2x = 96
x=96/1.2 = 80

Using the number line method:

60-----------75---80---------100

Distance between 60 to 80 = 20
Distance between 75 to 80 = 5
Distance between 100 to 80 = 20
The distance between 60 to 80 and 80 to 100 are same, however, the 75/kg price pulls down the average, also, the minimum quantity for 100/kg is 20,
So, 5 X 4 =20
Total on LHS = 20 +20 =40
RHS = 20 X2 =40
So final ratio = 1 : 4 : 2

Please comment whether this is an acceptable way to solve the problem.

If the selling price is $96 and profit is 20%, it means the cost price is $80. 20% of $80 will be $16. Familiarity with numbers will help you see it quickly. If not, use the formula
\(\frac{Profit}{100} = \frac{SP}{CP} - 1\)

Avg of some ratio of 60, 75 and 100 should be 80. One simple case is 50% $60 tea and 50% $100 tea but that is not in the options. We must mix $75 tea too. Since 3 of the options have $60 and $75 in the ratio 1:4, I will check that first. 

Visualise the scale method. The distance of 15 between them in split in the ratio 4:1 i.e. 12 and 3. So average is 12 away from $60 i.e. their avg is $72. 

Now the avg of $72 and $100 should be $80. So again visualise the scale or use the formula

\(\frac{w1}{w2} = \frac{A2 - Avg}{Avg - A1} = \frac{100 - 80}{80 - 72} = \frac{5}{2} \)

For 5 parts of $72, we need 2 parts of $100. Hence 1:4:2 works. (1 part of $60 and 4 parts of $75 give 5 parts of $72. So we need 2 parts of $100 now)

Answer (C)

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A Quick and reasonably simple method is to rely on 3 foundations:
1. What is cost price based on profit.
120/100 (Cost) = 96 since 96 accommodates 20% Profit.
Therefore: Cost is 80

2. Since cost is 80, the answer would be easier to find as it will be a multiple of 80, easier to discern than some other value.

3. Trial and error with ratios. Since the percentages are 60/75/100: Question will be in a format:
60x + 75y + 100z/x+y+z = x+y+z(80)

Here use trial and error and it will be easy to rule out what is not a multiple of 80 and what is.
C forms the answer sequence and fits in it, answer cannot be anything else.

- Selling the mix at $96/kg yielded a profit of 20%

This means that Selling Price of the mixture = 1.2 Cost Price of the mixture.

So,

Cost Price of the mixture = 96/1.2 = $80/kg.

Let the 3 types of tea be mixed in the ratio of a:b:c.

Overall Cost Price of the mix = \(\frac{(60a + 75b + 100c) }{ (a + b + c)}\) = $80/kg

Simplifying this,

60a + 75b + 100c = 80a + 80b + 80c
=> 20c = 20a + 5b
=> 4c = 4a + b

We can use the choices here. For which of the five choices of a:b:c is 4c = 4a+b?

We can see that only choice C fits.

1:4:2 ->

-> 4c = 8
-> 4a + b = 4(1) + 4 = 8

Answer: Choice C. 1:4:2.

---
Harsha

Identify that cost is 96/1.2 = 80.

Find the answer choice that cost total is 80.

Checking each answer choices,

A. 1 : 2 : 4
= 1/760 + 2/775 + 4/7*100
= (60 + 150 + 400)/7
= 610/7 ≈ 87.14 → Incorrect

B. 3 : 7 : 6
= 3/1660 + 7/1675 + 6/16*100
= (180 + 525 + 600)/16
= 1305/16 ≈ 81.56 → Incorrect

C. 1 : 4 : 2
= 1/760 + 4/775 + 2/7*100
= (60 + 300 + 200)/7
= 560/7 = 80 → Correct

D. 1 : 4 : 3
= 1/860 + 4/875 + 3/8*100
= (60 + 300 + 300)/8
= 660/8 = 82.5 → Incorrect

E. 1 : 4 : 4
= 1/960 + 4/975 + 4/9*100
= (60 + 300 + 400)/9
= 760/9 ≈ 84.44 → Incorrect