# *The difference between simple interest and compound interest at the rate of 10% for two years on an amount is Rs. 20. Find the pr

*The difference between simple interest and compound interest at the rate of 10% for two years on an amount is Rs. 20. Find the principal value?*

1️⃣ ₹.100
2️⃣ ₹ 200
3️⃣ ₹. 20000
4️⃣ ₹ 2000​

1. ### Given :

• Difference between simple Interest and Compound Interest = Rs. 20
• Rate = 10%
• Time = 2 years

• Principal

### Concept:

→ Formula to calculate Simple Interest :-

$$\star \quad\boxed{\pmb{ \sf{Simple \: \: Interest = \dfrac{P \times R \times T}{100}}}}$$

→ Formula to calculate Compound Interest :-

$$\\ \star \quad \boxed{\pmb{ \sf Compound \: \: Interest = P\Bigg[1 + \dfrac{r}{100}\Bigg]^n – P}}$$

where,

• P = Principal
• R = Rate
• T and n = Time

### Solution :

Let the Sum [Principal] as P.

Simple Interest :-

$$\\ \dashrightarrow\sf \quad Simple \: \: Interest = \dfrac{P \times R \times T}{100}$$

$$\\ \dashrightarrow\sf \quad Simple \: \: Interest = \dfrac{P \times 10 \times 2}{100}$$

$$\\ \dashrightarrow\sf \quad Simple \: \: Interest = \dfrac{P \times 1 \not0 \times 2}{10 \not0}$$

$$\\ \dashrightarrow\sf \quad Simple \: \: Interest = \dfrac{P \times 2}{10}$$

$$\\ \dashrightarrow\sf \quad Simple \: \: Interest = \dfrac{2P}{10}$$

$$\\ \dashrightarrow\sf \quad \pmb{ \quad Simple \: \: Interest =Rs. \: \dfrac{2P}{10}}$$

Compound Interest :-

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = P\Bigg[1 + \dfrac{r}{100}\Bigg]^n – P$$

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = P\Bigg[1 + \dfrac{10}{100}\Bigg]^2 – P$$

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = P\Bigg[1 + \dfrac{1 \not0}{10 \not0}\Bigg]^2 – P$$

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = P\Bigg[1 + \dfrac{1}{10}\Bigg]^2 – P$$

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = P\Bigg[\dfrac{10 + 1}{10}\Bigg]^2 – P$$

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = P\Bigg[\dfrac{11}{10}\Bigg]^2 – P$$

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = P\Bigg[\dfrac{11}{10} \times \dfrac{11}{10} \Bigg] – P$$

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = P\Bigg[\dfrac{121}{100} \Bigg] – P$$

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = \dfrac{121}{100}P – P$$

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = \dfrac{121P – 100P}{100}$$

$$\\ \dashrightarrow\sf \quad Compound \: \: Interest = \dfrac{21P}{100}$$

$$\\ \dashrightarrow \quad \sf \pmb{ Compound \: \: Interest = \dfrac{21P}{100}}$$

According to the question,

★ Compound Interest – Simple Interest= Rs. 20

$$\\ \dashrightarrow\sf \quad\dfrac{21P}{100} – \dfrac{2P}{10} = 20$$

$$\\ \dashrightarrow\sf \quad\dfrac{21P – 20P}{100} = 20$$

$$\\ \dashrightarrow\sf \quad P = 20 \times 100$$

$$\\ \dashrightarrow\sf \quad P = 2000$$

• Principal = ₹ 2000
2. Given : Difference between Compound Interest and Simple interest is Rs.20 , The rate is 10% p.a & the time is 2 yrs

Need To Find : The Principal.

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❍ Let’s Consider the Principal be P .

⠀⠀⠀⠀⠀Finding Simple interest :

$$\dag\:\:\it{ As,\:We\:know\:that\::}\\$$

$$\qquad \dag\:\:\bigg\lgroup \sf{ Simple \:Interest \:: \dfrac{P\times R \times T }{100} }\bigg\rgroup \\\\$$

⠀⠀⠀⠀⠀Here P is the Principal , R is the Rate of Interest & T is the Time.

⠀⠀⠀⠀⠀⠀$$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\$$

$$\qquad \longmapsto\sf Simple \:Interest \:= \dfrac{ P \times 10 \times 2 }{100} \\$$

$$\qquad \longmapsto\sf Simple \:Interest \:= \dfrac{ P \times \cancel {10} \times 2 }{10\cancel {0}} \\$$

$$\qquad \longmapsto\sf Simple \:Interest \:= \dfrac{P \times 2 }{10} \\$$

$$\qquad \longmapsto\sf Simple \:Interest \:= \dfrac{2P }{10} \\$$

$$\qquad \longmapsto\bf \bigg( 0.2P \bigg) \qquad\: \longrightarrow\:\: Simple \:Interest \\$$

⠀⠀⠀⠀⠀Finding Compound interest :

$$\dag\:\:\it{ As,\:We\:know\:that\::}\\$$

$$\qquad \dag\:\:\bigg\lgroup \sf{ Compound \:Interest \:: P \bigg( 1 +\dfrac{R }{100}\bigg) ^T – P }\bigg\rgroup \\\\$$

⠀⠀⠀⠀⠀Here P is the Principal , R is the Rate of Interest & T is the Time.

⠀⠀⠀⠀⠀⠀$$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\$$

$$\qquad \longmapsto\sf Compound \:Interest \:= P\bigg( 1 + \dfrac{ 10 }{100}\bigg)^2 – P \\$$

$$\qquad \longmapsto\sf Compound \:Interest \:= P\bigg( 1 + \dfrac{ \cancel{10} }{10\cancel{0}}\bigg)^2 -P \\$$

$$\qquad \longmapsto\sf Compound \:Interest \:= P\bigg( 1 + \dfrac{ 1 }{10}\bigg)^2 – P \\$$

$$\qquad \longmapsto\sf Compound \:Interest \:= P\bigg( \dfrac{ 10 +1 }{10}\bigg)^2- P \\$$

$$\qquad \longmapsto\sf Compound \:Interest \:= P\bigg( \dfrac{ 11 }{10}\bigg)^2 – P \\$$

$$\qquad \longmapsto\sf Compound \:Interest \:= P\bigg( \cancel {\dfrac{ 11 }{10}}\bigg)^2 – P \\$$

$$\qquad \longmapsto\sf Compound \:Interest \:= P\bigg( 1.1\bigg)^2 \\$$

$$\qquad \longmapsto\sf Compound \:Interest \:= P\bigg( 1.21\bigg) – P \\$$

$$\qquad \longmapsto\sf Compound \:Interest \:= 1.21P – P \\$$

$$\qquad \longmapsto\bf \bigg( Rs. \:0.21P\bigg) \qquad\: \longrightarrow\:\: Compound \:Interest \\$$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

⠀⠀⠀⠀⠀⠀$$\underline {\boldsymbol{\star\:According \: To \: Question \: \: \::}}\\$$

• Difference between Compound Interest and Simple interest is Rs.20 .

$$\qquad \longmapsto \sf 0.21P – 0.2P =20 \\$$

$$\qquad \longmapsto \sf 0.01P =20 \\$$

$$\qquad \longmapsto \sf P =\cancel {\dfrac{20}{0.01}} \\$$

$$\qquad \longmapsto \frak{\underline{\purple{\:P = Rs.2,000 }} }\bigstar \\$$

Therefore,

⠀⠀⠀⠀⠀$$\therefore {\underline{ \mathrm {\:Principal \:is\:\bf{Rs.2,000}}}}\\$$

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