Answer:
The difference between the principal and the compound interest in three years is Rs 17,994
Step-by-step explanation:
The compound interest is given according to the following formula;
![C.I. = P \cdot \left ( 1 + \dfrac{r}{n} \right ) ^{n\cdot t} - P](https://tex.z-dn.net/?f=C.I.%20%3D%20P%20%5Ccdot%20%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7Bn%7D%20%5Cright%20%29%20%5E%7Bn%5Ccdot%20t%7D%20-%20P)
The given amount of the compound after 2 years = Rs 5,460
The given amount of the compound after 4 years = Rs 12,066.60
Therefore, we have;
...(1)
...(2)
Dividing equation (2) by (1), we have;
![\dfrac{12,066.60}{5,460} = \dfrac{P \cdot \left ( \left ( 1 + \dfrac{r}{100} \right ) ^{4} - 1\right )}{P \cdot \left (\left ( 1 + \dfrac{r}{100} \right ) ^{2} -1 \right ) } =\dfrac{\left ( 1 + \dfrac{r}{100} \right ) ^{4} - 1}{\left ( 1 + \dfrac{r}{100} \right ) ^{2} -1 }](https://tex.z-dn.net/?f=%5Cdfrac%7B12%2C066.60%7D%7B5%2C460%7D%20%3D%20%5Cdfrac%7BP%20%5Ccdot%20%5Cleft%20%28%20%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%5Cright%20%29%20%5E%7B4%7D%20-%201%5Cright%20%29%7D%7BP%20%5Ccdot%20%5Cleft%20%28%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%5Cright%20%29%20%5E%7B2%7D%20-1%20%5Cright%20%29%20%7D%20%3D%5Cdfrac%7B%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%5Cright%20%29%20%5E%7B4%7D%20-%201%7D%7B%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%5Cright%20%29%20%5E%7B2%7D%20-1%20%20%7D)
![Let \ \left ( 1 + \dfrac{r}{100} \right ) ^{2} = x, we \ get;](https://tex.z-dn.net/?f=Let%20%5C%20%20%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%5Cright%20%29%20%5E%7B2%7D%20%3D%20x%2C%20we%20%5C%20get%3B)
![\dfrac{12,066.60}{5,460} =\dfrac{\left ( 1 + \dfrac{r}{100} \right ) ^{4} - 1}{\left ( 1 + \dfrac{r}{100} \right ) ^{2} -1 } = \dfrac{x^2 - 1}{x - 1}](https://tex.z-dn.net/?f=%5Cdfrac%7B12%2C066.60%7D%7B5%2C460%7D%20%3D%5Cdfrac%7B%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%5Cright%20%29%20%5E%7B4%7D%20-%201%7D%7B%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%5Cright%20%29%20%5E%7B2%7D%20-1%20%20%7D%20%3D%20%5Cdfrac%7Bx%5E2%20-%201%7D%7Bx%20-%201%7D)
∴ 12,066.60 × (x - 1) = 5,460 × (x² - 1) = 5,460 × (x - 1) ×(x + 1)
∴ 12,066.60 × (x - 1)/(x - 1) = 5,460 × (x + 1)
12,066.60/5,460 = x + 1
x = 12,066.60/5,460 - 1 = 1.21 = 121/100
x = 121/100
![\left ( 1 + \dfrac{r}{100} \right ) ^{2} = x = \dfrac{121}{100}](https://tex.z-dn.net/?f=%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%5Cright%20%29%20%5E%7B2%7D%20%3D%20x%20%3D%20%5Cdfrac%7B121%7D%7B100%7D)
![1 + \dfrac{r}{100} =\sqrt{ \dfrac{121}{100}} = \dfrac{11}{10}](https://tex.z-dn.net/?f=1%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%20%20%3D%5Csqrt%7B%20%5Cdfrac%7B121%7D%7B100%7D%7D%20%3D%20%5Cdfrac%7B11%7D%7B10%7D)
We get
![\dfrac{12,066.60}{5,460} =\dfrac{221}{100}](https://tex.z-dn.net/?f=%5Cdfrac%7B12%2C066.60%7D%7B5%2C460%7D%20%3D%5Cdfrac%7B221%7D%7B100%7D)
![\therefore \dfrac{12,066.60}{5,460} =\dfrac{221}{100} = \left ( 1 + \dfrac{r}{100} \right ) ^{2}](https://tex.z-dn.net/?f=%5Ctherefore%20%5Cdfrac%7B12%2C066.60%7D%7B5%2C460%7D%20%3D%5Cdfrac%7B221%7D%7B100%7D%20%3D%20%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%5Cright%20%29%20%5E%7B2%7D)
![1 + \dfrac{r}{100} = \sqrt{ \dfrac{221}{100} } = \dfrac{\sqrt{221} }{10}](https://tex.z-dn.net/?f=1%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%3D%20%5Csqrt%7B%20%5Cdfrac%7B221%7D%7B100%7D%20%7D%20%3D%20%5Cdfrac%7B%5Csqrt%7B221%7D%20%7D%7B10%7D)
![\dfrac{r}{100} = \dfrac{\sqrt{221} }{10} - 1](https://tex.z-dn.net/?f=%5Cdfrac%7Br%7D%7B100%7D%20%3D%20%5Cdfrac%7B%5Csqrt%7B221%7D%20%7D%7B10%7D%20-%201)
![\dfrac{r}{100} = \dfrac{11}{10} - 1 = \dfrac{1}{10} = 0.1](https://tex.z-dn.net/?f=%5Cdfrac%7Br%7D%7B100%7D%20%20%20%3D%20%5Cdfrac%7B11%7D%7B10%7D%20-%201%20%3D%20%5Cdfrac%7B1%7D%7B10%7D%20%3D%200.1)
r = 100 × 0.1 = 10%
r = 10%
Therefore, we have;
![5,460 = P \cdot \left ( 1 + \dfrac{r}{100} \right ) ^{2} - P = P \times \left ( 1 + 0.1\right ) ^{2} - P](https://tex.z-dn.net/?f=5%2C460%20%3D%20P%20%5Ccdot%20%5Cleft%20%28%201%20%2B%20%5Cdfrac%7Br%7D%7B100%7D%20%5Cright%20%29%20%5E%7B2%7D%20-%20P%20%3D%20P%20%5Ctimes%20%5Cleft%20%28%201%20%2B%200.1%5Cright%20%29%20%5E%7B2%7D%20-%20P)
![5,460 = P \times \left ( 1 + 0.1\right ) ^{2} - P = P \times \left (\left ( 1 + 0.1\right ) ^{2} - 1\right) = P \times \dfrac{21}{100}](https://tex.z-dn.net/?f=5%2C460%20%20%3D%20P%20%5Ctimes%20%5Cleft%20%28%201%20%2B%200.1%5Cright%20%29%20%5E%7B2%7D%20-%20P%20%3D%20P%20%5Ctimes%20%5Cleft%20%28%5Cleft%20%28%201%20%2B%200.1%5Cright%20%29%20%5E%7B2%7D%20-%201%5Cright%29%20%3D%20P%20%5Ctimes%20%5Cdfrac%7B21%7D%7B100%7D)
![P = \dfrac{100}{21} \times 5,460 = 26,000](https://tex.z-dn.net/?f=P%20%3D%20%5Cdfrac%7B100%7D%7B21%7D%20%5Ctimes%205%2C460%20%3D%2026%2C000)
The principal = Rs. 26,000
The compound interest in 3 years is therefore;
![CI_3 = 26,000 \times \left ( 1 + \dfrac{10}{100} \right ) ^{3} - 26,000= 8606](https://tex.z-dn.net/?f=CI_3%20%3D%2026%2C000%20%5Ctimes%20%5Cleft%20%28%201%20%2B%20%5Cdfrac%7B10%7D%7B100%7D%20%5Cright%20%29%20%5E%7B3%7D%20-%2026%2C000%3D%208606)
The difference, 'd', between the principal and the compound interest in three years, is given as follows;
d = P - CI₃
d = 26,600 - 8606 = 17994
The difference between the principal and the compound interest in three years, d = Rs 17,994.