Compound Interest 2
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Question 1 of 5
1. Question
If you deposit $$$3{,}000$$ into an account earning `7%` interest compounded weekly, how much money will be in the account after `4` years?Round your answer to two decimal places- `\text(total amount)=$` (3968.64)
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Compound Interest Formula
$$A=\color\green{P}\left(1+\frac{\color{blue}{r}}{\color{#a300c1}{n}}\right)^{\large\color{#a300c1}{n}\color{#e85e00}{t}}$$Label Known Values
Principal Value $$= P = $3{,}000$$Interest Rate `= r = 7% \text(per annum) = 0.07 \text((as a decimal))`Time Elapsed `= t = 4` yearsNumber of times compounded per year `= n = 52 \text((Weekly))`Solve for the Amount `(A),` using the formula`A` `=` $$\color\green{P}\left(1+\frac{\color{blue}{r}}{\color{#a300c1}{n}}\right)^{\large\color{#a300c1}{n}\color{#e85e00}{t}}$$ Compound Interest Formula `=` $$\color\green{3{,}000}\left(1+\frac{\color{blue}{0.07}}{\color{#a300c1}{52}}\right)^{\large\color{#a300c1}{52}\times\color{#e85e00}{4}}$$ Substitute known values `=` $$(3{,}000)(1+0.00134615384)^{208}$$ `=` $$(3{,}000)(1.3228807)$$ Simplify `=` $$3{,}968.64$$ Round to two decimal places $$$3{,}968.64$$ total amount -
Question 2 of 5
2. Question
An amount of `$375` invested for `2` years grows to `$431.35`. The amount is compounded annually. What is the interest rate?Round your answer to two decimal places- `\text(Interest Rate )=` (7.25)`%`
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Compound Interest Formula
$$A=\color\green{P}(1+\color{blue}{r})^{\large\color{#a300c1}{n}}$$Label Known Values
Amount Earned`= A = $431.35`Principal `= P = $375`Number of Years `= n = 2` yearsSolve for the Interest Rate `(r)``A` `=` $$\color\green{P}(1+\color{blue}{r})^{\large\color{#a300c1}{n}}$$ Compound Interest Formula `431.35` `=` $$\color\green{375}(1+\color{blue}{r})^{\large\color{#a300c1}{2}}$$ Substitute known values `(431.35)/(375)` `=` `(1+r)^2` Divide both sides by `375` `1.15` `=` `(1+r)^2` `1.0725` `=` `1+r` Get square root of both sides `0.0725` `=` `r` Subtract `1` from both sides `r` `=` `0.0725` `r` `=` `0.0725``xx100%` Multiply by `100%` `r` `=` `7.25%` `7.25%` interest rate -
Question 3 of 5
3. Question
How much money would you need to deposit today at an interest rate of `7.5%` compounded annually to have $$$65{,}000$$ in the account after `8` years?Round your answer to two decimal places- `\text(Principal)=$` (36445.65)
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Compound Interest Formula
$$A=\color\green{P}(1+\color{blue}{r})^{\large\color{#a300c1}{n}}$$Label Known Values
Amount Earned$$= A = $65{,}000$$Rate `= r = 0.075`Number of Years `= n = 8 \text(years)`Solve for the Principal Amount `(P)``A` `=` $$\color\green{P}(1+\color{blue}{r})^{\large\color{#a300c1}{n}}$$ Compound Interest Formula $$65{,}000$$ `=` $$\color\green{P}(1+\color{blue}{0.075})^{\large\color{#a300c1}{8}}$$ Substitute known values $$65{,}000$$ `=` `P(1.075)^8 ` Evaluate $$\frac{65{,}000}{1.075^8}$$ `=` `P` Divide both sides by `1.075^8` $$\frac{65{,}000}{1.783477826}$$ `=` `P` $$36{,}445.65$$ `=` `P` `P` `=` $$36{,}445.65$$ $$\text{Principal}=$36{,}445.65$$ -
Question 4 of 5
4. Question
How much money would you need to deposit today at an interest rate of `6%` compounded annually to have $$$30{,}000$$ in the account after `5` years?Round your answer to two decimal places- `\text(Principal)=$` (22417.75)
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Compound Interest Formula
$$A=\color\green{P}(1+\color{blue}{r})^{\large\color{#a300c1}{n}}$$Label Known Values
Amount Earned$$= A = $30{,}000$$Rate `= r = 0.06`Number of years `= n = 5` yearsSolve for the Principal Amount`A` `=` $$\color\green{P}(1+\color{blue}{r})^{\large\color{#a300c1}{n}}$$ Compound Interest Formula $$30{,}000$$ `=` $$\color\green{P}(1+\color{blue}{0.06})^{\large\color{#a300c1}{5}}$$ Substitute known values $$30{,}000$$ `=` `P(1.06)^5 ` Evaluate $$\frac{30{,}000}{1.06^5}$$ `=` `P` Divide both sides by `1.06^5` $$\frac{30{,}000}{1.338225578}$$ `=` `P` $$22{,}417.75$$ `=` `P` `P` `=` $$22{,}417.75$$ $$\text{Principal}=$22{,}417.75$$ -
Question 5 of 5
5. Question
Two banks receive an investment of `$20 000` each with an interest rate of `7%` per annum. If one bank uses simple interest and the other bank uses compound interest, what will be the difference in the amount (`A`) after a period of `3` years?- `$` (300.86)
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Simple Interest Formula
`I = ``P``r``t`Compound Interest Formula
$$A=\color\green{P}\left(1+\frac{\color{blue}{r}}{\color{#a300c1}{n}}\right)^{\large\color{#a300c1}{n}\color{#e85e00}{t}}$$Label Known Values
Principal Value $$= P = $20{,}000$$Interest Rate `= r = 7%` per annum `= 0.07` as a decimalTime Elapsed `= t = 3` yearsNumber of times compounded per year `= n = 1` (Annually)To understand the concept Simple and Compound interest, let us first try to solve this problem without any formulas.Simple Interest:The principal value where the interest is applied stays the same every period.First year:`20 000+(0.07xx20 000)` `=` `20 000+1400` `=` `21 400` Second year:`21 400+(0.07xx20 000)` `=` `21 400+1400` `=` `22 800` Third year:`22 800+(0.07xx20 000)` `=` `22 800+1400` `=` `24 200` Compound Interest:The principal value where the interest is applied compounds every period.First year:`20 000+(0.07xx20 000)` `=` `20 000+1400` `=` `21 400` Second year:`21 400+(0.07xx21 400)` `=` `21 400+1498` `=` `22 898` Third year:`22 898+(0.07xx22 898)` `=` `22 898+1602.86` `=` `24 500.86` Therefore, the difference between the two amounts is `24 500.86-24 200=``300.86`This time, let’s see more of how Simple and Compound Interest work by using their formulas to solve the problem.Simple Interest:`I` `=` `P``r``t` `=` `20 000``xx``0.07``xx``3` `=` `4200` This is the interest earned over the `3` years, not the final amount.`A` (Final amount) `=` `P+I` `=` `20 000+4200` `=` `24 200` Compound Interest:`A` `=` $$\color\green{P}\left(1+\frac{\color{blue}{r}}{\color{#a300c1}{n}}\right)^{\large\color{#a300c1}{n}\color{#e85e00}{t}}$$ `=` $$\color\green{20\:000}\left(1+\frac{\color{blue}{(0.07)}}{\color{#a300c1}{(1)}}\right)^{\large\color{#a300c1}{(1)}\color{#e85e00}{(3)}}$$ `=` `20 000(1+0.07)^3` `=` `20 000(1.07^3)` `=` `20 000xx1.225043` `=` `24 500.86` We can confirm again, through the formulas, that the difference between the two amounts is `24 500.86-24 200=``300.86``$300.86`