The sum of the ages of a woman and her daughter is 46 years. In 4 years, the ratio of the woman's age to her daughter's age will be 7:2. The problem asks us to find their present ages.

1. Problem Description

The sum of the ages of a woman and her daughter is 46 years. In 4 years, the ratio of the woman's age to her daughter's age will be 7:

2. The problem asks us to find their present ages.

2. Solution Steps

Let $w$ be the woman's current age, and $d$ be the daughter's current age.

We are given that the sum of their ages is 46:

$w + d = 46$

In 4 years, the woman's age will be $w+4$ and the daughter's age will be $d+4$. The ratio of their ages at that time will be 7:2:

$\frac{w+4}{d+4} = \frac{7}{2}$

We now have a system of two equations with two variables:

(1) $w + d = 46$

(2) $\frac{w+4}{d+4} = \frac{7}{2}$

From equation (1), we can express $w$ in terms of $d$:

$w = 46 - d$

Substitute this into equation (2):

$\frac{(46 - d) + 4}{d+4} = \frac{7}{2}$

$\frac{50 - d}{d+4} = \frac{7}{2}$

Cross-multiply:

$2(50 - d) = 7(d+4)$

$100 - 2d = 7d + 28$

$100 - 28 = 7d + 2d$

$72 = 9d$

$d = \frac{72}{9}$

$d = 8$

Now substitute $d = 8$ back into $w = 46 - d$:

$w = 46 - 8$

$w = 38$

So, the woman's current age is 38 and the daughter's current age is

8.

3. Final Answer

The woman is currently 38 years old, and the daughter is currently 8 years old.