John and Miles each had some money. First Miles gave John a quarter. Then John gave Miles one-third of his money. Then Miles gave John one-third of his money. At the end, John had $2.50 and Miles had $2.00. How much did they each start with?
To solve this problem, we can work backwards step by step. Let's start with the final situation:
At the end, John had $2.50 and Miles had $2.00.
Step 1: John received $2.50 and Miles received $2.00.
Step 2: John gave Miles one-third (1/3) of his money.
Step 3: After John gave Miles one-third (1/3) of his money, Miles had $2.00.
Now, let's go backwards to the previous step:
Step 3: After John gave Miles one-third (1/3) of his money, Miles had $2.00.
Step 2: John gave Miles one-third (1/3) of his money.
Step 1: John received $2.50 and Miles received $2.00.
At this point, we can calculate the amount that John originally had by reversing the process. Since John received $2.50, we need to find out what the total amount was before Miles gave him one-third (1/3) of his money.
To find the original amount, we can divide $2.50 by 2/3 (since Miles gave John one-third (1/3) of his money) and multiply the result by 3/2 (to get back to the original amount before Miles gave John the one-third (1/3) of his money).
($2.50) ÷ (2/3) × (3/2) = $3.75
So, initially, John had $3.75.
Next, let's find out how much Miles originally had. We can subtract the amount that John initially had from the total amount they both started with:
Miles' original amount = Total amount − John's original amount
Miles' original amount = ($2.00 + $3.75) − $3.75
Miles' original amount = $2.00
Therefore, originally, John had $3.75 and Miles had $2.00.