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Shivam invested some amount at SI and gets 45% more amount and the ratio between numerical value of rate of interest of Shivam's investment to time period of Shivam's investment is 5:1 . If Bhuvan invested Rs 6000 at the same rate and same period of time an CI , then find he will get How to find out how muchamount ?​

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Given, a certain sum was invested by a person on SI at 5% per annum.  

After 6 months, he again made an investment of the same amount on si at 6% p.A.  

After a certain period of time, the amount received from both the investments are equal which is rs.4600.  

We have to Find the sum he has invested in each investment?

Let the sum of amount invested be s.

We know that, simple interest =\frac{\mathrm{p} \times \mathrm{r} \times \mathrm{t}}{100}=100p×r×t

Where, p is sum of amount investe

be s.

We know that, simple interest =\frac{\mathrm{p} \times \mathrm{r} \times \mathrm{t}}{100}=100p×r×t

Where, p is sum of amount invested, r is rate percentage, t is time period of investment

Now, S.I at 5% for 6 months =\frac{s \times 5 \times 6 \text { months }}{100}=\frac{s \times 5 \times \frac{1}{2} \text { years }}{100}=\frac{s \times \frac{1}{2}}{20}=\frac{s}{20} \times \frac{1}{2}=\frac{s}{40}=100s×5×6 months =100s×5×21 years =20s×21=20s×21=40s

Let, the time period after 6 months be h.

Then S.I of 1st investment for time period h  =\frac{s \times 5 \times h}{100}=\frac{s \times h}{20}=100s×5×h=20s×h

Then S.I of 1st investment for time period h  =\frac{s \times 5 \times h}{100}=\frac{s \times h}{20}=100s×5×h=20s×h

And S.I of 2nd investment at 6% rate =\frac{s \times 6 \times h}{100}=100s×6×h

Now, according to given information,

1st investment + S.I of 1st investment for 6 months + S.I of 1st investment for time period h = 4600

\begin{gathered}\begin{array}{l}{S+\frac{s}{40}+\frac{s \times h}{20}=4600} \\\\ {S\left(1+\frac{1}{40}+\frac{2 \times h}{2 \times 20}\right)=4600} \\\\ {S\left(1+\frac{(1+2 h)}{40}\right)=4600 \rightarrow(1)}\end{array}\end{gathered}S+40s+20s×h=4600S(1+401+2×202×h)=4600S(1+40(1+2h))=4600→(1)

And, 2nd investment + S.I for time period h = 4600

\begin{gathered}\begin{array}{l}{S+\frac{s \times 6 \times h}{100}=4600} \\\\ {S\left(1+\frac{3 h}{50}\right)=4600 \rightarrow(2)}\end{array}\end{gathered}S+100s×6×h=4600S(1+503h)=4600→(2)

Now, equate (1) and (2)

\begin{gathered}\begin{array}{l}{S\left(1+\frac{1+2 h)}{40}\right)=S\left(1+\frac{3 h}{50}\right)} \\\\ {1+\frac{1+2 h}{40}=1+\frac{3 h}{50}} \\\\ {\frac{1+2 h}{40}=\frac{3 h}{50}}\end{array}\end{gathered}S(1+401+2h))=S(1+503h)1+401+2h=1+503h401+2h=503h

50(1 + 2h) = 40(3h)

5(1 + 2h) = 4 x 3h

5 + 10h = 12h

12h – 10h = 5

2h = 5

h = 2.5

now, substitute h value in (2)

\begin{gathered}\begin{array}{l}{S\left(1+\frac{3 \times 2.5}{50}\right)=4600} \\\\ {S\left(1+\frac{3}{20}\right)=4600} \\\\ {S \times \frac{23}{20}=4600} \\\\ {S=4600 \times \frac{20}{23}} \\\\ {S=200 \times 20=4000}\end{array}\end{gathered}S(1+503×2.5)=4600S(1+203)=4600S×2023=4600S=4600×2320S=200×20=4000

Hence, he invested Rs 4000 in each investment.

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