# Q. When 70% of a number x is added to another number y, the sum becomes 165% of the value of y. When 60% of the number x is added to another number z, then the sum becomes 165% of the value of z.

Which one of the following is correct?

a. z < x < y

b. x < y < z

c. y < x < z

d. z < y < x

Correct Answer: a. z < x < y

#### Question from UPSC Prelims 2022 CSAT Paper

**Explanation : **

## When 70% of a number x is added to another number y, the sum becomes 165% of the value of y.

We can represent this information as an equation:

y + 0.70x = 1.65y

## When 60% of the number x is added to another number z, then the sum becomes 165% of the value of z.

We can represent this information as another equation:

z + 0.60x = 1.65z

Now we can solve the system of equations:

From the first equation, we can express x in terms of y:

0.70x = 1.65y – y

0.70x = 0.65y

x = 0.65y / 0.70

x = 13y/14

Now, substitute x in the second equation:

z + 0.60x = 1.65z

z + 0.60(13y/14) = 1.65z

z + 13y/23 = 1.65z

Now, let’s solve for z in terms of y:

13y/23 = 1.65z – z

13y/23 = 0.65z

z = 13y/23 / 0.65

z = 13y/15

Now we have:

x = 13y/14

z = 13y/15

Since 13/15 < 13/14, we can conclude that z < x.

Now let’s compare x and y. Since x = 13y/14, this means that x < y (because 13/14 < 1).

Thus, we have established the following relationship: z < x < y.

Therefore, the correct answer is (a) z < x < y.