simple equations STD VI

Present ages of Anu and Raj are in the ratio 4:5. Eight years from now the ratio of their ages will be 5:6. Find their present ages.

Answer:

Let the present ages of Anu and Raj be 4x years and 5x years respectively.

After eight years. Anu’s age = (4x + 8) years;

After eight years, Raj’s age = (5x + 8) years.

Therefore, the ratio of their ages after eight years = (4x + 8)/(5x + 8)

But this is given as 5: 6

Therefore, (4x + 8)/(5x + 8) = 5/6

Cross-multiplication gives 6 (4x + 8) = 5 (5x + 8)

Or, 24x + 48 = 25x + 40

Or, 24x + 48 – 40 = 25x

Or, 24x + 8 = 25x

Or, 8 = 25x – 24x

Or, 8 = x

Therefore, Anu’s present age = 4x = 4 × 8 = 32 years

Raj’s present age = 5x = 5 × 8 = 40 years

The ages of Hari and Harry are in the ratio 5:7. Four years from now the ratio of their ages will be 3:4. Find their present ages.

Answer:

Let the present ages of Hari and Harry be 5x years and 7x years respectively.

After eight years. Hari’s age = (5x + 4) years;

After eight years, Harry’s age = (7x + 4) years.

Therefore, the ratio of their ages after eight years =

(5x + 4)/(7x + 4)

This is given to be 3: 4

Therefore, (5x + 4)/(7x + 4) = 3/4

Cross-multiplication gives  4(5x + 4)=  3(7x + 4)

Or, 20x + 16 = 21x + 12

Or, 16 – 12 = 21x – 20x

Or, 4 = x

Or, x = 4

Therefore, Hari’s present age = 5x = 20 years

Harry’s present age = 7x = 28 years

The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2. Find the rational number.

Answer:

Let the numerator be y

So, the denominator will be 8 + y

Given, ( y + 17)/ (8 + y -1) = 3/2

Or, 2(y + 17) = 3( 7 + y)

Or, 2y + 34 = 21 + 3y

Or, 34 – 21 = 3y – 2y

Or, y = 13

So, the rational number is = 13/21