Question

Purse A which contains $1,000 today. If you leave it alone, it will contain $1,200 tomorrow (by magic). The next day, it will have $1,400. This pattern of $200 additional dollars per day will continue. Purse B which contains 1 penny today. Leave that penny in there, because tomorrow it will (magically) turn into 2 pennies. The next day, there will be 4 pennies. The amount in the purse will continue to double each day. How much money will be in each purse after a week? After two weeks? The genie later added that he will let the money in each purse grow for three weeks. How much money will be in each purse then? Which purse contains more money after 30 days?Purse A which contains $1,000 today. If you leave it alone, it will contain $1,200 tomorrow (by magic). The next day, it will have $1,400. This pattern of $200 additional dollars per day will continue.

Purse B which contains 1 penny today. Leave that penny in there, because tomorrow it will (magically) turn into 2 pennies. The next day, there will be 4 pennies. The amount in the purse will continue to double each day.
How much money will be in each purse after a week? After two weeks?
The genie later added that he will let the money in each purse grow for three weeks. How much money will be in each purse then?
Which purse contains more money after 30 days?

Answer 1

After calculating daily increases for Purses A and B over several time frames, it's evident that Purse B, which doubles its amount every day, holds a vastly greater sum after 30 days, illustrating the power of compound interest.

Step-by-step explanation:

Calculating Growth in Purses A and B

Let's start by determining the amount in each purse after one week (7 days) and after two weeks (14 days). Purse A gains $200 more each day, starting with $1,000 on day 1. Purse B's amount doubles every day, beginning with 1 penny on day 1.

For Purse A, we can calculate the total after one week as:

$1,000 + (6 days × $200/day) = $1,000 + $1,200 = $2,200

After two weeks (14 days):

$1,000 + (13 days × $200/day) = $1,000 + $2,600 = $3,600

For Purse B, since the amount doubles each day, we use the power of 2 to represent the growth:

After one week (7 days):

1 penny ×
`2^6` = 64 pennies (because we don't count the first day)

After two weeks (14 days):

1 penny ×
`2^13`= 8,192 pennies, or $81.92

After three weeks (21 days) for Purse A we get:

$1,000 + (20 days × $200/day) = $1,000 + $4,000 = $5,000

And for Purse B:

1 penny ×
`2^(20)` = 1,048,576 pennies, or $10,485.76

Now, let's see which purse contains more money after 30 days:

Purse A: $1,000 + (29 days × $200/day) = $1,000 + $5,800 = $6,800

Purse B:
`1 penny * 2^(29) = 536,870,912 \ pennies \, \ or\ \$\ 5,368,709.12`

After 30 days, Purse B contains significantly more money due to the power of compound interest.

Answer 2

After a Week(7 days)

`Purse \:A: \$2200\\Purse B: \$0.64`

After 2 Weeks(14 days)

`Purse \:A: \$3600\\Purse B: \$81.92`

After 3 Weeks(21 days)

`Purse \:A: \$5000\\Purse B: \$10485.76`

After 30 days

`Purse \:A: \$6800\\Purse B: \$5368709.12`

Therefore, Purse B contains more money after 30 days.

Step-by-step explanation:

Purse A

The amount of money($1000) in purse A for each consecutive day grows with $200. The sequence is written as:

1000,1200,1400,...

This is an arithmetic sequence with the first term being $1000 and the common difference $200.

Therefore, for any number of day, n, the amount of money in the purse,

`T_n=1000+200(n-1)`

Purse B

1 Penny=$0.01

The amount of money(1 Penny) in purse B for each consecutive day doubles. The sequence is written as:

0.01,0.02,0.04,...

This is a geometric sequence with the first term being $0.01 and the common ratio 2.

Therefore, for any number of day, n, the amount of money in purse B,

`T_n=0.01(2)^(n-1)`

After a Week(7 days)

`Purse \:A: T_7=1000+200(7-1)=\$2200\\Purse B: T_7=0.01(2)^(7-1)=\$0.64`

After 2 Weeks(14 days)

`Purse \:A: T_(14)=1000+200(14-1)=\$3600\\Purse B: T_(14)=0.01(2)^(14-1)=\$81.92`

After 3 Weeks(21 days)

`Purse \:A: T_(21)=1000+200(21-1)=\$5000\\Purse B: T_(21)=0.01(2)^(21-1)=\$10485.76`

After 30 days

`Purse \:A: T_(30)=1000+200(30-1)=\$6800\\Purse B: T_(30)=0.01(2)^(30-1)=\$5368709.12`