Solving Mother and Daughter Age Difference Problem

This problem requires us to find the age difference between a mother and her daughter based on two given conditions relating their ages. We can solve this using algebra by setting up and solving a system of equations.

Step 1: Define Variables

Let's represent the mother's current age as '$M$' years and the daughter's current age as '$D$' years.

Step 2: Formulate Equations

We are given two conditions. Let's translate them into mathematical equations:

• Condition 1: "2 times the mother's age is 26 years more than 4 times her daughter's age." This translates to:

  $2M = 4D + 26$

• Condition 2: "3 times the daughter's age is 7 years less than the mother's age." This translates to:

  $3D = M - 7$

Step 3: Solve the System of Equations

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

1. $2M = 4D + 26$

2. $3D = M - 7$

Let's simplify the first equation by dividing both sides by 2:

$M = 2D + 13 \quad (\text{Equation 1 simplified})$

Now, we can use substitution. Substitute the expression for '$M$' from the simplified Equation 1 into Equation 2:

$3D = (2D + 13) - 7$

Simplify the equation:

$3D = 2D + 6$

Subtract '$2D$' from both sides to solve for '$D$':

$3D - 2D = 6$

$D = 6$

So, the daughter's age is 6 years.

Now, substitute the value of '$D$' back into the simplified Equation 1 to find the mother's age:

$M = 2(6) + 13$

$M = 12 + 13$

$M = 25$

So, the mother's age is 25 years.

Step 4: Calculate the Age Difference

The question asks for the difference between the mother's age and the daughter's age.

Age Difference = Mother's Age - Daughter's Age

Age Difference = $M - D$

Age Difference = $25 - 6$

Age Difference = $19$

Conclusion

The difference in age between the mother and the daughter is 19 years.