Question

Mitchell took out a loan for $1100 at a 19.2% APR, compounded monthly, to

Answer 1

Answer:

The total number of the payments is 18 ⇒ answer A

Step-by-step explanation:

* Lets revise the rule of compounded monthly payment

→ $EMI=\frac{(A)(r)}{[1-\frac{1}{(1+r)^{n}}]}$, where

- A is the loan amount

- r is monthly interest in decimal (R/12*100))

- n the total number of payments

∵ A = $1100

∵ EMI = $71.5

- Interest rate is 19.2% APR

∵ r = $\frac{19.2}{12*100}=0.016$

- Substitute these values in the rule to find n

∴ 71.5 = $\frac{1100(0.016)}{[1-\frac{1}{(1+0.016)^{n}}]}=\frac{17.6}{[1-\frac{1}{(1.016)^{n}}]}$

- By using cross multiplication

∴ 71.5[1 - $\frac{1}{(1.016)^{n}}$ ] = 17.6

- Divide both sides by 71.5

∴ 1 - $\frac{1}{(1.016)^{n}}$ = $\frac{16}{65}$

- Subtract 1 from both sides

∴ - $\frac{1}{(1.016)^{n}}$ = - $\frac{49}{65}$

- Multiply both sides by -1

∴ $\frac{1}{(1.016)^{n}}$ = $\frac{49}{65}$

- By using cross multiplication

∴ 49[ $(1.016)^{n}$ ] = 65

- Divide both sides by 49

∴ $(1.016)^{n}$ = $\frac{65}{49}$

- Insert log for both sides

∴ ㏒ $(1.016)^{n}$ = log( $\frac{65}{49}$ )

- Put n in-front of the ㏒

∴ n㏒(1.016) = ㏒( $\frac{65}{49}$ )

- Divide both sides by ㏒(1.016)

∴ n = 17.8 ≅ 18

* The total number of the payments is 18