Dr. Johnson’s Lottery Win: A Breakdown of Expenses
Overview of Spending
Dr. Johnson recently won a substantial lottery prize. He allocated a significant portion of his winnings to various purchases, including a home, a yacht, and travel expenses. Ultimately, he donated the remaining funds to charity.
House Purchase
Initially, Dr. Johnson spent 4/5 of his total lottery winnings, denoted as X, on acquiring a large house.
Yacht Purchase
Next, he spent ¾ of the remaining amount on a yacht. The calculation for this expense is based on the funds left after the house purchase.
Travel Expenses
Following the yacht purchase, Dr. Johnson allocated 2/3 of his remaining money to travel. This decision further reduced his available funds.
Charitable Contribution
After these expenditures, Dr. Johnson had $50,000 left, which he generously donated to charity.
Calculating the Total Lottery Win
To determine the total amount Dr. Johnson won, we can set up the following equation:
1. Let X represent the total lottery winnings.
2. The amount spent on the house: 4/5 X.
3. Remaining after house purchase: X – 4/5 X = 1/5 X.
4. Amount spent on the yacht: ¾ of 1/5 X = 3/20 X.
5. Remaining after yacht purchase: 1/5 X – 3/20 X = 1/20 X.
6. Amount spent on travel: 2/3 of 1/20 X.
7. Remaining after travel expenses: 1/20 X – (2/3)(1/20 X) = 1/20 X – 1/30 X = 1/60 X.
According to the problem, this remaining amount equals $50,000:
1/3 (1/60 X) = 50,000.
By solving for X, we find:
X = 3,000,000.
Conclusion
Dr. Johnson won a total of $3,000,000 in the lottery, which he distributed across various expenses and charitable donations.