Question

the compound interest on 30000at 7% per anum is rs 4374 the period is
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Answer 1

Answer:

The time period for the given compound interest is 2 years.

Step-by-step-explanation:

We have given that,

• Principal ( P ) = 30000
• Rate of interest ( R ) = 7 %
• Compound interest ( CI ) = Rs. 4374

We have to find the time period ( N ).

We know that,

Compound interest = Amount - Principal

$\displaystyle{\implies\sf\:4374\:=\:P\:\left(\:1\:+\:\dfrac{R}{100}\:\right)^N\:-\:P}$

$\displaystyle{\implies\sf\:4374\:=\:P\:\left[\:\left(\:1\:+\:\dfrac{R}{100}\:\right)^N\:-\:1\:\right]}$

$\displaystyle{\implies\sf\:4374\:=\:30000\:\left[\:\left(\:1\:+\:\dfrac{7}{100}\:\right)^N\:-\:1\:\right]}$

$\displaystyle{\implies\sf\:\left(\:\dfrac{100\:+\:7}{100}\:\right)^N\:-\:1\:=\:\dfrac{4374}{30000}}$

$\displaystyle{\implies\sf\:\left(\:\dfrac{107}{100}\:\right)^N\:=\:\dfrac{4374}{30000}\:+\:1}$

$\displaystyle{\implies\sf\:\left(\:\dfrac{107}{100}\:\right)^N\:=\:\dfrac{4374\:+\:30000}{30000}}$

$\displaystyle{\implies\sf\:\left(\:\dfrac{107}{100}\:\right)^N\:=\:\cancel{\dfrac{34374}{30000}}}$

$\displaystyle{\implies\sf\:\left(\:\dfrac{107}{100}\:\right)^N\:=\:\dfrac{11458}{10000}}$

$\displaystyle{\implies\sf\:\left(\:\dfrac{107}{100}\:\right)^N\:=\:1.1458}$

$\displaystyle{\implies\sf\:(\:1.07\:)^N\:=\:1.1458}$

Taking log to the base 1.07 on both sides,

$\displaystyle{\implies\sf\:\log_{1.07}\:(\:1.07\:)^N\:=\:\log_{1.07}\:(\:1.1458\:)}$

We know that,

$\displaystyle{\boxed{\pink{\sf\:\log_b\:(\:b^k\:)\:=\:k\:}}}$

$\displaystyle{\implies\sf\:N\:=\:\log_{1.07}\:(\:1.1458\:)}$

We know that,

$\displaystyle{\boxed{\blue{\sf\:\log_b\:(\:a\:)\:=\:\dfrac{\log_c\:(\:a\:)}{\log_c\:(\:b\:)}\:}}}$

$\displaystyle{\implies\sf\:N\:=\:\dfrac{\log_e\:(\:1.1458\:)}{\log_e\:(\:1.07\:)}}$

$\displaystyle{\implies\sf\:N\:=\:\dfrac{\ln\:(\:1.1458\:)}{\ln\:(\:1.07\:)}}$

$\displaystyle{\implies\sf\:N\:=\:\cancel{\dfrac{0.1361}{0.0676}}}$

$\displaystyle{\implies\sf\:N\:=\:2.013}$

$\displaystyle{\implies\sf\:N\:\approx\:2.01}$

$\displaystyle{\implies\:\underline{\boxed{\red{\sf\:N\:\approx\:2\:years\:}}}}$

∴ The time period for the given compound interest is 2 years.