Given below are two quantities named I and II. Based on the given information, determine the relation between the two quantities.

**Quantity I:** The present ages of three business colleagues, Sumit, Anil, and Deepak, are in the ratio 5:7:8 respectively. Seven years ago, the sum of their ages was 79. Find the present age of Deepak in years.

**Quantity II:** A daughter's present age is three-fifth of the age of her mother. After seven years, she will be two-thirds of the age of her mother. Find the mother's present age in years.

Question

Given below are two quantities named I and II. Based on the given information, determine the relation between the two quantities.

**Quantity I:** The present ages of three business colleagues, Sumit, Anil, and Deepak, are in the ratio 5:7:8 respectively. Seven years ago, the sum of their ages was 79. Find the present age of Deepak in years.

**Quantity II:** A daughter's present age is three-fifth of the age of her mother. After seven years, she will be two-thirds of the age of her mother. Find the mother's present age in years.

Accepted Answer

**Solution:**

**Quantity I:**

1. Let the present ages of Sumit, Anil, and Deepak be $5x$, $7x$, and $8x$ years respectively.

2. Seven years ago, their ages were $(5x - 7)$, $(7x - 7)$, and $(8x - 7)$.

3. The sum of their ages seven years ago is given as:
   $$ (5x - 7) + (7x - 7) + (8x - 7) = 79 $$

4. Simplify the equation:
   $$ 5x + 7x + 8x - 21 = 79 $$
   $$ 20x - 21 = 79 $$
   $$ 20x = 100 $$
   $$ x = 5 $$

5. Present age of Deepak:
   $$ 8x = 8 	imes 5 = 40 	ext{ years} $$

**Quantity II:**

1. Let the mother's present age be $x$ years.

2. Daughter's present age is three-fifth of mother's age:
   $$ 	ext{Daughter's age} = rac{3x}{5} $$

3. After seven years, daughter's age will be:
   $$ rac{3x}{5} + 7 $$

4. After seven years, mother's age will be:
   $$ x + 7 $$

5. Given that after seven years, daughter's age is two-thirds of mother's age:
   $$ rac{3x}{5} + 7 = rac{2}{3}(x + 7) $$

6. Multiply both sides by 15 to clear denominators:
   $$ 15 	imes igg(rac{3x}{5} + 7igg) = 15 	imes rac{2}{3}(x + 7) $$
   $$ 3 	imes (3x + 35) = 10 	imes (x + 7) $$

7. Simplify:
   $$ 9x + 105 = 10x + 70 $$

8. Rearranging:
   $$ 105 - 70 = 10x - 9x $$
   $$ 35 = x $$

9. Mother's present age:
   $$ 35 	ext{ years} $$

**Comparison:**

- Quantity I (Deepak's age) = 40 years
- Quantity II (Mother's age) = 35 years

Since $40 > 35$, **Quantity I > Quantity II**.

**Answer:** Quantity I > Quantity II

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