When Raj Tarun was born, his father was 32 years older than his brother and his mother was 25 years older than his sister. If Raj Tarun's brother is 6 years older than Raj Tarun and his mother is 3 years younger than his father, how old was Raj Tarun's sister when Raj Tarun was born ?

Area of equilateral triangle = ?3/4 a² = 4?3.
? a² = 16
? a = 4 cm

Here, n = 2, m =10, T₁ = 6 yr, T₂ = ?
? T₂ = [(m - 1) / (n - 1)] x T₁
= [(10 - 1)/(2 - 1)] x 6
= 54 yr

Given, SI₁ - SI₂ = 110,
T₁ = 2 Yr,
T₂ = 1 yr, R₁ = 8%, R₂ = 5%

According to the question,
[(P x 8 x 2)/100] - [(P x 5 x 1)/100] = 110
? 11P/100 = 110
? P = ? 1000

Let the positive number be N.
According to the question,
N + 10 = 200 / N
? N² + 10N = 200
? N² +10N - 200 =0
? (N - 10)(N + 20) = 0
? N = 10 & N= -20
But N not equal to -20 since N is a positive number
So, the required number is 10

Let the sum be p
Then, for R = 5%
SI = (4p - p) = 3p
? 3p = (P x 5 x T)/100 = PT/20
? T = 60 yr

Again, for another rate (R)
SI = (8P - P) = 7P
? 7P = (P x R x 60)/100
? R = (7 x 100)/60 = 35/3%
= 11²/₃ %

Here, P = ? 65800, R = 14% T = 4 yr
Hence, SI = (65800 x 14 x 4) / 100 =? 36848

? Required amount = P + SI
= 65800 + 36848
= ? 102648

Cost of 5 pens + 8 pencils = ?31
On multiplying by 3
15 pens + 24 pencils = 3 x 31 =
?15 pens + 24 pencils = 93

Given that, a = 8 cm, b = 10 cm and c = 12 cm

We know, that
s = (a + b + c)/2
= (8 + 10 + 12)/2
= 30/2 = 15 cm

? (s - a) = (15 - 8) = 7 cm
(s - b) = (15 - 10) = 5 cm
(s - c) = (15 - 12) = 3 cm

? Area = ?s(s - a) (s - b) (s - c)
= ?15 x 7 x 5 x 3
= ?1575
= ?25 x 63
= 5?63 cm

Let each sum be ? P then,
(P x 10 x 5)/100 - (P x 10 x 4)/100 = 220
? P/10 = 220
? P = 2200
Hence, each sum is ? 2200.

Let the number be N.
According to the question,
N - (N x 3/4) =163
? N/4 =163
? N=652