## Important Shortcuts & Formulas on Problem Based on Ages:

**Ex. 1:** The age of the father 3 years ago was 7 times the age of his son. At present, the father’s age is five

times that of his son. What are the present ages of the father and the son?

**Ex. 2:** At present, the age of the father is five times the age of his son. Three years hence, the father’s age would be four times that of his son. Find the present ages of the father and the son.

**Ex. 3:** Three years earlier, the father was 7 times as old as his son. Three years hence, the father’s age would be four times of his son. What are the present ages of the father and the son?

**Ex. 4:** The sum of the ages of a mother and her daughter is 50 yrs. Also 5 yrs ago, the mother’s age was 7 times the age of the daughter. What are the present ages of the mother and the daughter?

**Ex. 5:** The sum of the ages of a son and father is 56 yrs. After 4 yrs, the age of the father will be three times that of the son. What is the age of the son?

**By the conventional method:-**

**Solution: 1.** Let the present age of son = x yrs

Then, the present age of father = 5xyr

3 years ago,

7(x – 3)= 5x – 3

Or, 7x – 21 =5x – 3

Or, 2x =18

x = 9 yrs

Therefore, son’s age = 9 years

Father’s age = 45 years

**Solution: 2.** Let the present age of son = x yrs

Then, the present age of father = 5x yrs

Or, 4x + 12=5x +3

x= 9yrs.

Therefore, son’s age = 9 yrs and father’s age = 45 yrs

**Solution: 3.** Let the present age of son = x yrs and the present age of father = y

yrs

3 yrs earlier, 7(x – 3) = y – 3 or, 7x – y =18………….(i)

3 yes hence, 4(x+3) = y +3

Or, 4x +12 = y + 3 or, 4x – y = – 9 …………(ii)

Solving (1) & (2) we get, x = 9 yrs & y =45 yrs

**Solution: 4.** Let the age of the daughter be x yrs.

Then, the age of the mother is (50x – x)yrs

5 yrs ago, 7(x – 5) = 50 – x – 5

Or, 8x = 50 – 5 +35 = 80

x =10

Therefore, daughter’s age = 10 yrs and mother’s age = 40yrs

**Solution: 5.** Let the age of the son be x yrs.

Then, the age of the father is (56 – x) yrs.

After 4 yrs, 3(x+4) = 56 – x +4

Or, 4x =56 +4 – 12 = 48

x = 12 yrs

Thus, son’s age = 12 yrs.

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