Problem Statement
Dr. Johnson allocated a total of $90 to his daughter and son. After the daughter transferred $10 from her portion to her brother, he ended up with twice the amount of money that she had.
Objective
Determine the original amounts given to each child by Dr. Johnson.
Solution
Let D represent the amount of money that Dr. Johnson’s daughter received, and S represent the amount received by his son. We can establish the following equations based on the information provided:
Equation 1
The total amount given to both children can be expressed as:
D + S = 90
From this, we can express S in terms of D:
S = 90 – D
Equation 2
After the daughter gives $10 to her brother, the following relationship holds:
2 × (D – 10) = S + 10
This can be simplified to:
2D – 20 = S + 10
Rearranging gives us:
2D = S + 30
Substituting S
Now, substituting the expression for S from Equation 1 into this equation:
2D = (90 – D) + 30
Simplifying
This leads to:
2D = 120 – D
Adding D to both sides results in:
3D = 120
Finding D
Solving for D gives:
D = 40
Finding S
Now substituting D back into Equation 1 to find S:
S = 90 – D
S = 90 – 40
S = 50
Conclusion
The original amounts received by each child were as follows: the daughter received $40, and the son received $50.