a mason can build a wall in 6 hours less than an apprentice. Together they can build the wall in 4 hours. How long would it take the apprentice to build the wall?

time for apprentice -- x

time for mason -- x-6

rate of apprentice = 1/x
rate of mason = 1/(x-6)
combined rate = 1l/x + 1/(x-6)
= (2x - 6)/(x(x-6))

then

1/[(2x - 6)/(x(x-6))] = 4
x(x-6)/2x-6) = 4
x^2 - 6x = 8x - 24
x^2 - 14x + 24=0
(x-12)(x-2) = 0
x = 12 or x = 2 , the last makes no sense since x-6 would be negative

so x = 12
the apprentice could build the wall in 12 hours

Let's assume that the apprentice takes X hours to build the wall.

Since the mason can build the wall in 6 hours less than the apprentice, the mason takes (X - 6) hours to build the wall.

When they work together, their combined work rate is 1/4 of the wall per hour since they can finish the wall in 4 hours.

The mason's work rate is 1/(X - 6) of the wall per hour and the apprentice's work rate is 1/X of the wall per hour.

Given this information, we can create the equation: 1/(X - 6) + 1/X = 1/4

To solve this equation, we can multiply both sides by 4X(X - 6) to remove the fractions:

4X + 4(X - 6) = X(X - 6)

Simplifying this equation:

4X + 4X - 24 = X^2 - 6X

8X - 24 = X^2 - 6X

Rearranging the equation:

X^2 - 14X + 24 = 0

Now we can solve this quadratic equation using factoring or the quadratic formula.

Factoring:

(X - 2)(X - 12) = 0

So either X - 2 = 0 or X - 12 = 0

If X - 2 = 0, then X = 2

If X - 12 = 0, then X = 12

Since we are looking for the time it takes for the apprentice to build the wall, we discard the solution X = 2.

Therefore, the apprentice takes 12 hours to build the wall.

To solve this problem, we can assign variables and use algebraic equations. Let's denote the time it takes the apprentice to build the wall as "A" hours, and the time it takes the mason to build the wall as "M" hours.

According to the given information, the mason can build the wall in 6 hours less than the apprentice. This can be expressed as M = A - 6.

Also, when they work together, they can complete the wall in 4 hours. Using the concept of work rates, we know that when working together, their rates of work are additive. So, their combined work rate is 1 wall per 4 hours.

Using the formula: Rate × Time = Work, we can create two equations to represent their work rates:

1/A + 1/M = 1/4 ... (Equation 1) (This represents their combined work rate)
M = A - 6 ... (Equation 2) (This represents the mason's work rate in terms of the apprentice's time)

To solve this system of equations, we will substitute Equation 2 into Equation 1:

1/A + 1/(A - 6) = 1/4

Now, let's simplify and solve for A:

Multiply both sides of the equation by 4A(A - 6) to eliminate the denominators:

4(A - 6) + 4A = A(A - 6)

4A - 24 + 4A = A^2 - 6A

8A - 24 = A^2 - 6A

Rearrange the equation to a quadratic form:

A^2 - 14A + 24 = 0

Factoring or using the quadratic formula yields:

(A - 2)(A - 12) = 0

So, we have A = 2 or A = 12.

Since we are looking for the time taken by the apprentice, the solution A = 12 satisfies the problem's condition. Therefore, it would take the apprentice 12 hours to build the wall.