Answer:
The total number of the payments is 18 ⇒ answer A
Step-by-step explanation:
* Lets revise the rule of compounded monthly payment
→ [tex]EMI=\frac{(A)(r)}{[1-\frac{1}{(1+r)^{n}}]}[/tex], 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 = [tex]\frac{19.2}{12*100}=0.016[/tex]
- Substitute these values in the rule to find n
∴ 71.5 = [tex]\frac{1100(0.016)}{[1-\frac{1}{(1+0.016)^{n}}]}=\frac{17.6}{[1-\frac{1}{(1.016)^{n}}]}[/tex]
- By using cross multiplication
∴ 71.5[1 - [tex]\frac{1}{(1.016)^{n}}[/tex] ] = 17.6
- Divide both sides by 71.5
∴ 1 - [tex]\frac{1}{(1.016)^{n}}[/tex] = [tex]\frac{16}{65}[/tex]
- Subtract 1 from both sides
∴ - [tex]\frac{1}{(1.016)^{n}}[/tex] = - [tex]\frac{49}{65}[/tex]
- Multiply both sides by -1
∴ [tex]\frac{1}{(1.016)^{n}}[/tex] = [tex]\frac{49}{65}[/tex]
- By using cross multiplication
∴ 49[ [tex](1.016)^{n}[/tex] ] = 65
- Divide both sides by 49
∴ [tex](1.016)^{n}[/tex] = [tex]\frac{65}{49}[/tex]
- Insert log for both sides
∴ ㏒ [tex](1.016)^{n}[/tex] = log( [tex]\frac{65}{49}[/tex] )
- Put n in-front of the ㏒
∴ n㏒(1.016) = ㏒( [tex]\frac{65}{49}[/tex] )
- Divide both sides by ㏒(1.016)
∴ n = 17.8 ≅ 18
* The total number of the payments is 18
