Question

if a sum of money doubles itself in 8 years at simple interest the rate per annum

Answer 1

let the sum of money = 100
ACQ the sum of money double itself in 8 years
thus amount after 8years = 200
SI= 100
therefore rate of interest =
$100 \times 100 \div 100 \times 8$
=12.5%

Answer 2

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: SOLUTION \: \: } \mid}}}}}$

$\huge{\underline{ \mathfrak{ \pink{ \: \: Method \: \: \: 1 \: : \mapsto \: }}}}$

$\underline{ \mathfrak{ \: Given, \: }} \\ \\ \text{ \pink{A sum of money become double of itself}} \\ \text{ \pink{ in 8 years under simple interest.}} \\ \\ \underline{ \mathfrak{ \:Suppose, \: }} \\ \\ \text{ \red{Rate of interest = r }} \\ \\\text{ \red{ Principal = P }} \\ \\ \text{ \red{Amount, A = 2P}} \\ \\ \sf{ \blue{\therefore \: Simple \: \: Interest = Amount - Principal }} \\ \\ \sf{ \blue{\implies S.I. = 2P - P }} \\ \\ \: \: \: \: \: \: \: \: \: \sf{ \blue{\therefore \: S.I. = P}} \\ \\ \: \: \: \: \text{ \red{Time, n = 8 years.}}$

$\underline{ \mathfrak{ \: \: We \: \: know \: \: that, \: \: }} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \tt{S.I. = \frac{P \times r \times n}{100}} \\ \\ \tt{ \implies \:P = \frac{P \times r \times 8}{100} } \\ \\ \tt{ \implies \: 100 \times P = P \times r \times 8 }\\ \\ \tt{ \implies \:P \times r \times 8 = 100 \times P} \\ \\ \tt{ \implies \:r \times 8 = \frac{100 \times \cancel{P}}{ \cancel{P}} } \\ \\ \tt{ \implies \:r \times 8 = 100} \\ \\ \tt{ \implies \: r = \frac{100}{8} }\\ \\ \: \: \: \: \: \: \: \tt{ \therefore \: \: r = 12.5}$

$\: \: \: \: \sf{ \therefore \: \: \red{Rate \: \: of \: \: interest, r = \underline{ \: 12.5\% \: }}}$

$\underline{ \underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: }}$

$\huge{\underline{ \mathfrak{ \pink{ \: \: Method \: \: \: 2 \: : \mapsto \: }}}}$

$\underline{ \mathfrak{ \: Given, \: }} \\ \\ \text{ \pink{A sum of money become double of itself}} \\ \text{ \pink{ in 8 years under simple interest.}} \\ \\ \underline{ \mathfrak{ \:Suppose, \: }} \\ \\ \text{ \red{Rate of interest = r }} \\ \\ \: \: \: \: \text{ \red{Time, n = 8 years.}} \\ \\ \text{ \red{ Principal , P =Rs. 100 }} \\ \\ \text{ \red{Amount, A = Rs. 200}} \\ \\ \sf{ \blue{\therefore Simple \: \: Interest = Amount - Principal }} \\ \\ \sf{ \blue{\implies S.I. = Rs.(200- 100 )}} \\ \\ \: \: \: \: \: \: \: \: \: \sf{ \blue{\therefore \: \: S.I. = Rs. 100}}$

$\underline{ \mathfrak{ \: \: We \: \: know \: \: that, \: \: }} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \tt{S.I. = \frac{P \times r \times n}{100}} \\ \\ \tt{ \implies \:100 = \frac{ \cancel{100} \times r \times 8}{ \cancel{100}} } \\ \\ \tt{ \implies \:100 = r \times 8 } \\ \\ \tt{ \implies \: r = \frac{100}{8} }\\ \\ \: \: \: \: \tt{ \therefore \: \: r = 12.5}$

$\: \: \: \: \sf{ \therefore \: \: \red{Rate \: \: of \: \: interest, r = \underline{ \: 12.5\% \: }}}$