Q. A Question is given followed by two Statements I and II. Consider the Question and the Statements.

# A certain amount was distributed among X, Y and Z.

Question: Who received the least amount?

Statement-1: X received 4/5 of what Y and Z together received.

Statement-II: Y received 2/7 of what X and Z together received.

Which one of the following is correct in respect of the above Question and the Statements?

a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.

b) The Question can be answered by using either Statement alone.

c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

d) The Question cannot be answered even by using both the Statements together.

Correct answer: c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

##### Question from UPSC Prelims 2024 CSAT Paper

**Explanation:**

## Who received the least amount ?

**Let’s define variables:**

x = amount received by X

y = amount received by Y

z = amount received by Z

t = total amount distributed

From Statement I:

x = 4/5(y + z)

From Statement II:

y = 2/7(x + z)

**We can create a system of equations:**

1. x = 4/5(y + z)

2. y = 2/7(x + z)

3. x + y + z = t (total equation)

Substituting equation 1 into equation 2:

y = 2/7(4/5(y + z) + z)

y = 2/7(4/5y + 4/5z + z)

y = 8/35y + 8/35z + 2/7z

35y = 8y + 8z + 10z

27y = 18z

y = 2/3z

Substituting this back into equation 1:

x = 4/5(2/3z + z)

x = 4/5(5/3z)

x = 4/3z

Now we have the ratios:

x : y : z = 4/3 : 2/3 : 1 :: 4 : 2 : 3