Example 5: Five years ago, a man was seven times as old as his son. Five years hence, the father will be three times as old as his son. Find their present ages.
Solution: Suppose that son’s age five years ago be x years
Then father’s age five years ago = (7x) years
Son’s present age = (x + 5) years
Father’s present age = (7x + 5) years
Son’s age after five years = 7x + 5 + 5
= (7x + 10) years
According to the given condition,
Father’s age after five years = Three times son’s age after five years
Or, 7x + 10 = 3(x + 10)
Or, 7x + 10 =3x + 30
Or, 7x – 3x = 30 – 10
Or, 4x = 20
Or, x = 20/4
X = 5
Son’s age five years ago = 5 years
Son’s preset age = x + 5
= 5 + 5
= 10 years
Father’s present age = 7x ´ 5
= 7.5 ´5
= 35 + 5
= 40 years.
Check: Son’s age after five years = 10 + 5
= 15 years
Father’s age after five years = 40 + 5
= 45 years
Now, 45 = 3(15)
Hence, father’s age = three times son’s age
Therefore, the solution is correct.
Example 6: The present ages of Isha and Raghu are in the ratio 5 : 7. Four year later, their ages wil be in the ratio 3 : 4. Find their present ages.
Solution: Suppose that present ages of Isha and Raghu is 5x years and 7x respectively.
Isha’s age after four years = (5x + 4) years
Raghu’s age after four years = (7x + 4) years.
According to the given condition,
5x + 4/7x + 4 = 3/4
Cross-multiplying , we have
4(5x + 4) = 3(7x + 4)
Or, 20x + 16 = 21x + 12
or – 21x + 20x = 12 – 16
or – x = – 4
Therefore,
Isha’s present age = 5x = 5 ´ 4 = 20 years
Raghu’s present age = 7x = 7 ´ 4 = 28 years
Check: Isha’s ager after four years = 20 + 4
= 24 years
Raghu’s age after four years = 28 + 4
= 32 years
The ratio between Isha’s and Raghu’s age is 24 : 32 = 3 : 4
Hence, the solution is correct.
Example7: A motorboat covers a certain distanc downstream in a rive in five hours. It covers the same distance upstream in six hour. The speed of water is 2 km/h. Find the speed of the boat in still water.
Solution: Suppose that the speed of the boat in still water be x km/h
Speed of water = 2 km/h
Speed of the boat downstream = (x – 2) km/h
Distance covered in 5 hours = Speed ´ Time
= 5(x – 2) km
(The relative speed when the direction of the boat and the folw of water is the same = The sum of the speeds of the boat and water).
Distance covered in 6 hours = Speed ´ Distance = 6(x – 2) km
Speed of the boat upstream = (x – 2) km/h
(The relative speed when the boat travels opposite to the flow of water = The difference in speeds of the boat and water.
But the boat covers the same distance upstream and downstream.
Therefore,
6(x – 2) = 5(x – 2)
6x – 12 = 5x – 10
Or 6x – 5x = 12 + 10
Or x = 22
Hence, the speed of the boat in still water = 22 km/h
Check: Distance covered in 5 hours (downstream) = Speed ´ Time
= 5(22 + 2) = 120 km
Distance covered in 6 hours (upstream) = Speed ´ Time
= 6(22 – 2) = 120 km
In both cases, the distance is same. Hence the solution is correct.
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