If a milkman wishes to earn a profit of 16.66% by selling milk at cost

Say 1 litre costed 100 ..
Now with profit of 16.66 or 50/3 %, 1 litre should cost 100+50/3=350/3..
So for 350/3, we get 1 litre.
And for 1, we get 3/350 litre
Finally for 100, we get 100*3/350=300/350
So milk is 300/350 or 6/7 and water will be 1-6/7=1/7=14.28%

B

Let the Cost Price of Milk(1 liter) = 1
C.P of water = 0
Then, Selling price of the Milk = 1

Profit = 16.6%
C.P of 1 liter Mixture = \(100/116.66\)


C.P= 0.8571
Mean Price = C.P of Mixture
Water/Milk = \((C.P of Milk - Mean Price)/(Mean Price - C.O of Water)\)
= \((1 - 0.8571)/(0.8570 - 0)\)
= \(0.1429/0.857\)
So, out of one liter 0.1429*100 = 14.29 % is water

Hi chetan2u,
Not able to understand this solution , can you make this a little easier to understand?

My reasoning if it helps anyone:

Let's assume we were originally selling 100ml of milk for $100.

To get a profit of 16.667% or \(\frac{1}{6}\). we need:

\(x * \frac{7}{6} = 100\)
\(x = \frac{600}{7}\)

Since our milk is priced at $1/ml, it's fair to say we have \(\frac{600}{7}\) ml of milk in our mixture and the rest is water. Water is free so by replacing part of our milk with water we can make an additional 16.667% profit.

Total amount in our mixture is 100ml or \(\frac{700}{7}\).

\(\frac{700}{7} - \frac{600}{7} = \frac{100}{7}ml of water\).

\(% of water in our solution = total ml of water : total ml of solution\)

\(\frac{100}{7}/\frac{700}{7}\)

or 1/7 = 14.28%

Knowing fractions comes handy. 16.66%=1/6
So The milkman actually sell 6 rupees milk at (6*1/6 +1)=7 rupees.
So amount of water added was 1 litre.
So fraction of water added 1/7=14.28%

Let the quantity be 100 and the CP is 100. => 1 unit costs 1
Now SP is also 100 after a profit of 16.66%. i.e. SP of 100 has arrived after a profit of 16.66%.
Therefore, CP = 100/1.166 = 85.76.
Now, Quantity for an amount of 85.76 = 85.76 i.e. pure milk = 85.76 => Water in 100 => 100-85.76 = 14.23.
14.23/100 = 14.23%.

Here percent profit = 16.66% = 1/6

So , let the original milk be x then
water has to be added = the profit which he is expected to earn = 1/6 of original mixture ( here it is milk , x litre)

So total mixture = x+x/6 = 7x/6 ( after adding the water to milk ).

Now percent of water = (x/6)/(7x/6) = 1/7 = 14.28%

IMO B

Suppose that the milkman has 100 litres of milk costing $1 per liter

i.e. his revenue = $116.66

Since seller professes to sell at cost price which is $1 per liter therefore quantity that he sells = 116.66 litres

So he is making 100 litres of milk into a 116.66 litres liters of Milk water solution by adding pure water to it

i.e. QUantity of Water added = 116.66 - 100 = 16.66 litres

Water as % of Mixtures = (16.66/116.66)*100 = 14.28%

Hint: 16.66 = 100/6

Answer Option B

Solution:

To determine the percentage of water that needs to be mixed with milk to achieve a 16.66\% profit by selling the mixture at the cost price of pure milk, follow these steps:

Given:
- Profit Desired: 16.66\%
- Selling Price of Mixture: Same as the Cost Price of pure milk.

Assumptions:
- Let’s assume the Cost Price (CP) of pure milk is \$1 per liter.
- Water is free (i.e., no cost).

Let:
\[
m = \text{Quantity of pure milk used}
\]
\[
w = \text{Quantity of water added}
\]
\[
x = m + w = \text{Total quantity of the mixture}
\]

Profit Calculation:

Profit Percentage Formula:
\[
\text{Profit \%} = \left( \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \right) \times 100
\]

Since the selling price is the same as the cost price of pure milk, the profit comes from using less milk and adding water.

Profit from Mixing:
\[
\text{Profit} = \frac{x - m}{m} \times 100 = 16.66\%
\]

Solving for \( x \):
\[
\frac{x - m}{m} = 0.1666
\]
\[
x = m + 0.1666m = 1.1666m
\]

This implies that for every 6 parts of milk, there are 1 part of water:
\[
\frac{w}{m} = \frac{1}{6}
\]

Total Parts in Mixture: \( 6 + 1 = 7 \)

Percentage of Water in the Mixture:
\[
\text{Percentage of Water} = \left( \frac{1}{7} \right) \times 100 \approx 14.28\%
\]

Answer:
B) 14.28\%