Father is four times the age of his daughter. If after 5 years, He would be three times of daughter’s age, Then further after 5 years, How many times he would be of his daughter’s age? 

Father is four times the age of his daughter. If after 5 years, He would be three times of daughter’s age, Then further after 5 years, How many times he would be of his daughter’s age?

The father will be 2.5 times the daughter's age.

Let's solve this step-by-step:

1. Define variables:

   • Let D be the daughter's current age.
   • The father's current age is then 4D (as he is four times older).

2. Translate the information into equations:

   • After 5 years:
     • Daughter's age: D + 5
     • Father's age: 4D + 5 (adding 5 years to their current ages)
   • Given information: After 5 years, the father will be three times the daughter's age.
     • Equation: 4D + 5 = 3(D + 5)

3. Solve the equation:

   • Expand the right side of the equation: 4D + 5 = 3D + 15
   • Combine like terms: D = 10

4. Find the father's current age:

   • Father's age = 4D = 4 * 10 = 40

5. Calculate the age difference after 10 years:

   • Daughter's age after 10 years: D + 10 = 10 + 10 = 20
   • Father's age after 10 years: 4D + 10 = 40 + 10 = 50

6. Determine the ratio of their ages after 10 years:

   • Divide the father's age by the daughter's age: 50 / 20 = 2.5

Therefore, after 10 years (5 years from now and another 5 years after that), the father will be 2.5 times the daughter's age.

Linear Equations in Algebra

Linear equations are fundamental equations in algebra that represent straight lines on a graph. They are equations where each term is either a constant or the product of a constant and a single variable raised to the first power. The general form of a linear equation in one variable, x, is:

ax + b = 0

Where a and b are constants, and a ≠ 0. The solution to this equation is a single value for x, which represents the point where the line intersects the x-axis on a graph.

In two variables, x and y, a linear equation takes the form:

ax + by + c = 0

Where a, b, and c are constants, and at least one of a or b is not zero. The solution to this equation is a line in the xy-plane.

Linear equations can also be written in slope-intercept form:

y = mx + b

Where m is the slope of the line and b is the y-intercept (the point where the line intersects the y-axis).

Solving linear equations involves finding the values of the variables that make the equation true. This typically involves isolating the variable of interest on one side of the equation. This can be done through various algebraic manipulations such as addition, subtraction, multiplication, and division.

Linear equations are widely used in various fields of mathematics, science, engineering, economics, and more for modeling relationships between variables that exhibit a linear trend.