Respuesta :
Answer:
Case 1 = 13.33 years
Case 2 = 10.25 years
Explanation:
Case 1
Interest rate = 7.5% = 0.075
Future value = 2 × Present value
Now,
Future value = Present value × [ 1 + ( Rate × Time ) ]
or
2 × Present value = Present value × [ 1 + ( 0.075 × Time ) ]
or
2 = 1 + ( 0.075 × Time )
or
1 = 0.075 × Time
or
Time = 13.33 years
Case 2
Interest rate, r = 7% = 0.07
Future value = 2 × Present value
Now,
Future value = Present value × ( 1 + r )ⁿ
here,
n is the time
thus,
2 × Present value = Present value × ( 1 + 0.07 )ⁿ
or
2 = 1.07ⁿ
taking log both the sides
log(2) = log(1.07ⁿ)
also,
log(aᵇ) = b × log(a)
thus,
log(2) = n × log(1.07)
or
0.301 = n × 0.02938
or
n = 10.25 years
In Case 1, the minimum number of years to double the balance is 13.33 years, whereas Case 2 needs 10.25 years.
Step-by-step Computation
Case 1
[tex]\text{Interest rate} = 7.5[/tex]% or [tex]0.075[/tex]
[tex]\text{Future value} = 2[/tex] × [tex]\text{Present value}[/tex]
Now,
[tex]\text{Future value = Present value}[/tex] ×[tex][ 1 + ( \text{Rate}[/tex] ×[tex]\text{Time} ) ][/tex]
or
[tex]2[/tex] × [tex]\text{Present value = Present value}[/tex] ×[tex][ 1 + ( 0.075[/tex] × [tex]\text{Time ) ]}[/tex]
or
[tex]2 = 1 + ( 0.075[/tex] × [tex]\text{Time )}[/tex]
or
[tex]1 = 0.075[/tex] × [tex]\text{Time}[/tex]
or
[tex]\text{Time = 13.33 years}[/tex]
Case 2
[tex]\text{Interest rate, r = 7}[/tex]% [tex]= 0.07[/tex]
[tex]\text{Future value = 2}[/tex] × [tex]\text{Present value}[/tex]
Now,
[tex]\text{Future value = Present value}[/tex] × [tex](1+r)^{n}[/tex]
here,
[tex]\text{n = Time}[/tex]
thus,
[tex]2[/tex] × [tex]\text{Present value = Present value}[/tex] × [tex]( 1 + 0.07 )^{n}[/tex]
or
[tex]2 = 1.07^{n}[/tex]
[tex]\text{Taking log both the sides}[/tex]
[tex]\text{log(2)} = \text{log}(1.07^{n})[/tex]
also,
[tex]\text{log}(a^{b}) = b[/tex] × [tex]\text{log(a)}[/tex]
thus,
[tex]\text{log}(2) = n[/tex] × [tex]\text{log}(1.07)[/tex]
or
[tex]0.301 = n[/tex] × [tex]0.02938[/tex]
or
[tex]n = 10.25 \text{years}[/tex]
For more information about doubling interest, refer below
https://brainly.com/question/922796
