Question

Alice gave Bob as many dollars as Bob had. Bob then gave Alice as many dollars Alice then had. At this point, each had 24 dollars. How much did Alice have at the beginning?

Please answer it step-by-step, and explained it very detailed. Thx.

Answer 1

Alice started with $30

Explanation:

Understanding the situation

There are 3 steps to the transactions:

1. the initial amounts, (before any transfers)
2. the amounts after Alice first gives Bob money, and
3. the amounts after Bob gives Alice some money back.

Organizing the variables

For organizational purposes, let's create variables for each person for each stage of the transaction, variables "a" for Alice, and "b" for Bob, with subscripts to denote which stage of the process we're talking about. For instance,
`a_1` is the amount Alice has at step 1 (before any transfers), and
`b_3` is the amount Bob has after Bob gives Alice some money back at the end.

Translating the sentences into equations

"Alice gave Bob as many dollars as Bob had."

This means that the magnitude of the change for Alice between her Step 1 and Step 2 was equal to the amount Bob had at the beginning (Step 1):
`a_1-a_2=b_1`

This also means that Bob's money in Step 2 increased to double what it used to be in Step 1:
`b_2=2b_1`

"Bob then gave Alice as many dollars Alice then had."

Similar to the last sentence, this means that the magnitude of the change for Bob between Step 2 and Step 3 was equal to the amount Alice had at Step 2:
`b_2-b_3=a_2`

It also means that Alice's money in Step 3 increased to double what it used to be in Step 2:
`a_3=2a_2`

"At this point, each had 24 dollars"

This means that both people had 24 dollars at Step 3:
`a_3=24` and
`b_3=24`

"How much did Alice have at the beginning?"

Solve for
`a_1`

Solving the system of equations

We effectively have a system of 6 equations and 6 unknowns. To solve this system, we'll be using substitution. For ease of explanation, I'll number the equations below:

Eq1:
`a_1-a_2=b_1`

Eq2:
`b_2=2b_1`

Eq3:
`b_2-b_3=a_2`

Eq4:
`a_3=2a_2`

Eq5:
`a_3=24`

Eq6:
`b_3=24`

Substitute Eq6 (
`b_3=24`) into Eq4

Eq4:
`a_3=2a_2`

`(24)=2a_2`

divide by 2

`12=a_2` ***(we'll need this later)

Substitute this (
`12=a_2`) and Eq5 (
`a_3=24`) into Eq3

Eq3:
`b_2-b_3=a_2`

`b_2-(24)=(12)`

add 24

`b_2=36`

Substitute this (
`b_2=36`) into Eq2

Eq2:
`b_2=2b_1`

`(36)=2b_1`

Divide by 2

`18=b_1`

Substitute this (
`18=b_1`) and ***(
`12=a_2`) into Eq1:

Eq1:
`a_1-a_2=b_1`

`a_1-(12)=(18)`

Add 12

`a_1=30`

So, Alice had 30 dollars at Step 1, before she gave any to Bob.