The sum of the ages of a woman and her daughter is 58years. In six years time the ratio of their ages will be 27 :8. Find their present ages.

w + d = 58

(w+6) / (d+6) = 27/8
Solve for w and d. Just to get you started these simplify to
w + d = 58
8w - 27d = 114
so have at it.

w + d = 58 ... w = 58 - d

(w + 6) / (d + 6) = 27 / 8 ... 8 w + 48 = 27 d + 162 ... 8 w = 27 d + 114

8(58 - d) = 27 d + 114

solve for d , then substitute back to find w

To solve this problem, let's break it down into steps:

Step 1: Define variables
Let's define the variables:
- Let x be the age of the woman
- Let y be the age of the daughter

Step 2: Set up equations based on the information given
From the problem description, we can form two equations:
- Equation 1: The sum of their ages is 58: x + y = 58
- Equation 2: In six years, the ratio of their ages will be 27:8. This means that (x + 6)/(y + 6) = 27/8

Step 3: Solve the equations simultaneously
To solve the equations simultaneously, we can use the substitution method.

Using Equation 1, solve for y:
y = 58 - x

Substitute this value of y into Equation 2:
(x + 6)/((58 - x) + 6) = 27/8

Simplify the equation:
(8x + 48)/(64 - x) = 27/8

Cross-multiply:
8(8x + 48) = 27(64 - x)

Expand and simplify:
64x + 384 = 1728 - 27x

Combine like terms:
91x = 1344

Divide both sides by 91:
x = 14.769

Since age cannot be in fractions, we need to round to the nearest whole number, which gives us:
x ≈ 15

Step 4: Find the daughter's age
Substitute the value of x back into Equation 1:
15 + y = 58

Subtract 15 from both sides:
y = 43

Therefore, the present ages of the woman and her daughter are approximately 15 years and 43 years, respectively.