(b) Rs 2000, 3.5 years and 4 years
Let the sum be Rs x.
∴ Rs x are lent at 8% for t years and Rs x are lent at 7% for \(\bigg(t+\frac{1}{2}\bigg)\)years.
∴ \(\frac{x\times{t}\times8}{100}\) + x = 2560
⇒ 8tx + 100x = 256000 .......(i)
and \(\frac{x\times{(2t+1)}\times7}{2\times100}\) + x = 2560
⇒ \(\frac{14tx+7x+200x}{200}\) = 2560
⇒ 14tx + 207x = 512000 .....(ii)
Performing (i) × 7 – (ii) × 4, we get
(56tx + 700x) – (56tx + 828x) = 256000 × 7 – 512000 × 4
⇒ 700x – 828x = 1792000 – 2048000
⇒ 128x = 256000 ⇒ x = 2000
Putting the value of x in (i), we get
8 × t × 2000 + 100 × 2000 = 256000
⇒ 16000t = 256000 – 200000
⇒ 16000t = 56000 ⇒ t = \(\frac{56000}{16000} = \) 3.5 years.