A Milkman Cheats his Customers by Adding Water to the Milk He Sells GMAT Problem Solving

Question: A milkman cheats his customers by adding water to the milk he sells. He starts the day with 1000 litres of milk. However, after selling every 500 litres of milk, he adds and equal quantity of water to what he already has. The milkman can get away by cheating as long as the milk to water ratio does not fall below 1:7. He sells off all the milk if any addition of water will make the ratio fall below 1:7. Each customer buys exactly 2 litres of milk a day from him. What is the total number of customers the milkman has cheated if he ends upselling all the milk ?

7550
7500
5500
1500
1000

Correct Answer: E

Solution and Explanation
Approach Solution 1:

Because we refill 500 L water (or after taking away 500 L of the liquid), the milk to total volume ratio is decreased by

\(\frac{500}{1000}\)

=\(\frac{1}{2}\)every time we refill (hence multiply by 50% every time).

After 1st refill, we have Milk:Total = 1:2, and after the 2nd refill we have Milk:Total = 1:4 etc.
We do not want to fall below Milk:Total of = 1:8. After the 3rd refill we hit 1:8 exactly.
Therefore we can refill 3 times, which means we can use up 500*4 = 2000 L of milk.

Then we can get away with cheating \(\frac{2000}{2}\)(considering ½ every time we refill)

= 1000 customers.
Hence, E is the correct answer.

Approach Solution 2:

Given:

The day starts with 1000L of milk.
Sells 500L to 250 customers, as each buy 2L.

The Milkman has 500L of pure milk remaining, which he tops up with 500L of water.
The ratio of the Milk:Water mixture is now 1:1 (within the realm of what he can sell) and the total volume is back up to 1000L.
He sells 500L to 250 customers.
Total of diluted milk remaining is 500 L, consisting of 250L of milk and 250L of water.
He tops up with 500L of water to a total of 1000L,
As per that, at present there are 250L of milk and 750L of water
The ratio becomes 1:3 of Milk:Water (still within the realm of what he can sell).
He sells 500L to 250 customers.
Milkman has 500L of diluted milk remaining, consisting of 125L of milk and 375L of water.
He tops up with 500L of water to a total of 1000L
Hence, there's 125L of milk and 875L of water
The ratio is 1:7 of Milk:Water (just within the realm of what he can sell, the 1:7 limit).
Since he won't be able to dilute it any further,
Now the milkman sells all remaining 1000L of diluted milk, to 500 customers.
Now comes the trap to avoid
We add all the customers in bold and we get 1250... which is not one of the answers.
The question asks how many customers were cheated, not how many the milk was sold to.
The milkman didn't actually cheat the first 250 as they bought pure milk.
So the correct answer is 1250-250
This comes at 1000 customers.
Hence, the correct answer is E.

Approach Solution 3:

1000lM=100 customers buy 5l each=500lM
Rest 500lM+500lW=1000l
Ration=1:1
Rest 500l=250lW+250lM
500l+500l(W)=250l(W)+500l(W)+250(M)
Ration=1:3 Rest 500l=125l(M)+375l(W)
adding 500l water 1000(l)=125l(M)+875l(W)
Ratio=1:7
Total cheated customers=500+250+250 =1000 customers were cheated.

“A milkman cheats his customers by adding water to the milk he sells.”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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A Milkman Cheats his Customers by Adding Water to the Milk He Sells GMAT Problem Solving

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