To solve this problem, we'll start by denoting the incomes and expenditures of A and B using two variables, and then use the given conditions to find their savings.
Let's denote:
- Income of A as [tex]\( 5x \)[/tex]
- Income of B as [tex]\( 3x \)[/tex]
- Expenditure of A as [tex]\( 9y \)[/tex]
- Expenditure of B as [tex]\( 5y \)[/tex]
We are given that the income of A is twice the expenditure of B. So:
[tex]\[ 5x = 2 \times 5y \][/tex]
[tex]\[ 5x = 10y \][/tex]
[tex]\[ x = 2y \][/tex]
Now that we have [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], we can substitute [tex]\( x \)[/tex] back into the expressions for incomes and expenditures to find their savings.
Income of A:
[tex]\[ 5x = 5 \times 2y = 10y \][/tex]
Income of B:
[tex]\[ 3x = 3 \times 2y = 6y \][/tex]
Expenditure of A:
[tex]\[ 9y \][/tex]
Expenditure of B:
[tex]\[ 5y \][/tex]
Now we calculate the savings of each:
Savings of A:
[tex]\[ \text{Savings}_A = \text{Income}_A - \text{Expenditure}_A \][/tex]
[tex]\[ \text{Savings}_A = 10y - 9y \][/tex]
[tex]\[ \text{Savings}_A = y \][/tex]
Savings of B:
[tex]\[ \text{Savings}_B = \text{Income}_B - \text{Expenditure}_B \][/tex]
[tex]\[ \text{Savings}_B = 6y - 5y \][/tex]
[tex]\[ \text{Savings}_B = y \][/tex]
Therefore, the ratio of the savings of A to the savings of B is:
[tex]\[ \text{Savings}_A : \text{Savings}_B = y : y = 1 : 1 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{3. \, 1:1} \][/tex]
