If a sum doubles in 6 years, how much will it be in 8 years?

### Concept & Simple Interest Let sum = $P$. Doubles in 6 years means $S.I. = P$ in 6 years. We need to find the Amount after 8 years. Amount = Principal + Interest after 8 years. ### Step-by-Step Solution * $S.I. = P$ in 6 years. * $S.I.$ in 1 year = $\frac{P}{6}$. * $S.I.$ in 8 years = $8 \times \left(\frac{P}{6}\right) = \frac{4P}{3}$. * Amount in 8 years = $P + \frac{4P}{3} = \frac{7P}{3} = 2\frac{1}{3} P$. * Wait, the options are in terms of times the principal, but the answer I got is not an option. Let me re-read. The question might mean what is the interest multiplier. * Let's re-read: "how much will it be". This usually means Amount. * Let me re-check. Maybe the options are just the multiplier for the sum. Let's re-calculate. * Amount = $P + \frac{4P}{3} = \frac{7P}{3} = 2\frac{1}{3}$ times. Still not there. * Let's re-read the options. Oh, maybe it means what is the multiplier of the principal for the interest. Let's assume it asks for what is the interest multiplier. * Let's try another interpretation. Is it compound interest? If $A = P(1+r)^n$, $2P = P(1+r)^6 \Rightarrow (1+r) = 2^{1/6}$. Amount in 8 yrs = $P(1+r)^8 = P(2^{8/6}) = P(2^{4/3}) = P \cdot 2 \cdot 2^{1/3}$. No, usually these are simple interest unless stated. * Let me re-read the options carefully from the image: (a) $1\frac{1}{2}$ times (b) $1\frac{1}{3}$ times (c) $1\frac{1}{4}$ times (d) $1\frac{3}{4}$ times * Let's re-read the question. "how much will it be in 8 years". * Wait, if $S.I = P$ in 6 years, rate $R = \frac{100 \times P}{P \times 6} = \frac{100}{6} = \frac{50}{3}$%. * Interest in 8 years $= \frac{P \times \frac{50}{3} \times 8}{100} = \frac{400P}{300} = \frac{4P}{3} = 1\frac{1}{3}P$. * Ah! The question implies "How much will the interest be". "it" refers to the interest. Let's assume "it" is interest. Then interest is $1\frac{1}{3}$ times the principal. * Or maybe "a sum" means just the simple interest. Let's read context. Usually "sum" means principal. * Let's assume the question is asking for the ratio of new amount to original amount. New Amount $= P + \frac{4}{3}P = \frac{7}{3}P = 2\frac{1}{3}P$. Not in options. * Let's assume it's asking for Simple Interest in 8 years. $SI = \frac{4}{3}P = 1\frac{1}{3}P$. This matches option (b). So the question is asking for the interest. ### Exam Strategy & Shortcut Interest is directly proportional to time. Interest for 6 years = $P$ (since it doubles) Interest for 8 years = $P \times \frac{8}{6} = \frac{4}{3}P = 1\frac{1}{3}P$. ### Common Pitfall Assuming "how much will it be" means Amount. In this specific question, based on the options, it asks for the Simple Interest. ### Final Answer Therefore, the correct answer is **$1\\frac{1}{3}$ times**.