Question

At what rate of simple interest any some amounts to 5/4 of the principal in 2.5 years​

Answer 1

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To determine the rate of simple interest at which an amount grows to
`\displaystyle\sf (5)/(4)` of the principal in 2.5 years, we can use the formula for simple interest:

`\displaystyle\sf I= P\cdot R\cdot T`

where:

`\displaystyle\sf I` is the interest earned,

`\displaystyle\sf P` is the principal amount,

`\displaystyle\sf R` is the rate of interest, and

`\displaystyle\sf T` is the time period.

Given that the amount after 2.5 years is
`\displaystyle\sf (5)/(4)` of the principal, we can set up the equation:

`\displaystyle\sf P+ I= P+\left((P\cdot R\cdot T)/(100)\right) =(5)/(4)\cdot P`

Simplifying the equation, we get:

`\displaystyle\sf (5P)/(4) =(P)/(1) +(P\cdot R\cdot T)/(100)`

Now, let's solve for the rate of interest,
`\displaystyle\sf R`. We can rearrange the equation as follows:

`\displaystyle\sf (5P)/(4) -(P)/(1) =(P\cdot R\cdot T)/(100)`

`\displaystyle\sf (5P-4P)/(4) =(P\cdot R\cdot T)/(100)`

`\displaystyle\sf (P)/(4) =(P\cdot R\cdot T)/(100)`

`\displaystyle\sf 100P =4P\cdot R\cdot T`

`\displaystyle\sf R =(100P)/(4P\cdot T)`

Simplifying further, we find:

`\displaystyle\sf R =(100)/(4\cdot T)`

Substituting the given time period of 2.5 years, we get:

`\displaystyle\sf R =(100)/(4\cdot 2.5)`

`\displaystyle\sf R =(100)/(10)`

`\displaystyle\sf R =10`

Therefore, the rate of simple interest required for the amount to grow to
`\displaystyle\sf (5)/(4)` of the principal in 2.5 years is 10%.

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