**Q. Three years ago, Rashmi was thrice as old as Nazma. Ten years later, Rashmi will be twice as old as Nazma. How old are Rashmi and Nazma now?**

**Ans: **Let’s consider Rashmi’s present age be X and Nazma’s present age be Y

Hence, Rashmi’s age 3 years ago = X – 3

Nazma’s age 3 years ago = Y – 3

By given first condition, (X – 3) = 3 ( Y – 3)

∴ X – 3 = 3Y – 9

∴ X – 3 Y = – 9 + 3

∴ X – 3 Y = – 6 ………….. (i)

Rashmi’s age 10 years later = X + 10

and Nazma’s age 10 years later = Y + 10

By given second condition, (X + 10) = 2 ( Y + 10)

∴ X + 10 = 2( Y + 10)

∴ X + 10 = 2 Y + 20

∴ X – 2 Y = 20 – 10

∴ X – 2 Y = 10 ………….. (i)

Let’s subtract equation (i) from equation (ii), we get:

(X – 2 Y) – (X – 3 Y) = (10) – (- 6)

∴ X – 2 Y – X + 3 Y = 10 + 6

∴ Y = 16

By substituting value of Y in equation (ii), we get:

X – 2 Y = 10

∴ X – 2 (16) = 10

∴ X = 10 + 32 = 42

Therefore, present ages of Rashmi and Nazma are 42 years and 16 years.

**Check:**

*If Rashmi’s present age is 42 years, her age 3 years ago was 42 – 3 = 39 years *

*If Nazma’s present age is 16 years, and 5 yher age 3 years ago was 16 – 3 = 13 years*

*Hence, Rashmi was thrice as old as Nazma.*

*Rashmi’s age 10 years later : 42 + 10 = 52 years. **Nazma’s age 10 years later: 16 + 10 = 26 years. *

*Hence, Rashmi will be twice as old as Nazma.*

*Since it meets our given condition, hence our answer is correct.*

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