Problem Analysis:
The problem involves a mathematical puzzle where a man named David is trying to divide his estate among his three sons: Aaron, Bill, and Charlie. The will specifies the following conditions:
1.
Aaron's Share: He should receive half of the estate.
2.
Bill's Share: He should receive one-fourth of the estate.
3.
Charlie's Share: He should receive one-fifth of the estate.
However, there is a twist:
- After dividing the estate according to the will, David realizes that he has
one dollar left over.
- To resolve this issue, David decides to give the extra dollar to Aaron.
The task is to determine how much money David had in his estate initially.
---
Solution Approach:
Let the total amount of money in David's estate be denoted as \( x \) dollars.
#### Step 1: Express each son's share mathematically
- Aaron's share: \( \frac{1}{2}x \)
- Bill's share: \( \frac{1}{4}x \)
- Charlie's share: \( \frac{1}{5}x \)
#### Step 2: Sum of the shares
The sum of the shares given to Aaron, Bill, and Charlie is:
\[
\frac{1}{2}x + \frac{1}{4}x + \frac{1}{5}x
\]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 2, 4, and 5 is 20. Rewriting each fraction with a denominator of 20:
\[
\frac{1}{2}x = \frac{10}{20}x, \quad \frac{1}{4}x = \frac{5}{20}x, \quad \frac{1}{5}x = \frac{4}{20}x
\]
Adding these fractions:
\[
\frac{10}{20}x + \frac{5}{20}x + \frac{4}{20}x = \frac{19}{20}x
\]
So, the total amount distributed to the three sons is \( \frac{19}{20}x \).
#### Step 3: Account for the leftover dollar
After distributing \( \frac{19}{20}x \) of the estate, David has 1 dollar left over. This means:
\[
x - \frac{19}{20}x = 1
\]
Simplify the left-hand side:
\[
\frac{20}{20}x - \frac{19}{20}x = \frac{1}{20}x
\]
Thus, the equation becomes:
\[
\frac{1}{20}x = 1
\]
#### Step 4: Solve for \( x \)
Multiply both sides of the equation by 20:
\[
x = 20
\]
---
Final Answer:
David had
\$20 in his estate initially.
---
Verification:
- Aaron's share: \( \frac{1}{2} \times 20 = 10 \) dollars
- Bill's share: \( \frac{1}{4} \times 20 = 5 \) dollars
- Charlie's share: \( \frac{1}{5} \times 20 = 4 \) dollars
Total distributed:
\[
10 + 5 + 4 = 19 \text{ dollars}
\]
Leftover:
\[
20 - 19 = 1 \text{ dollar}
\]
This matches the condition in the problem. Therefore, the solution is correct.
\[
\boxed{20}
\]
Parent Tip: Review the logic above to help your child master the concept of reading comprehension grade 5 worksheet.