Solving the Mother and Daughter Age Difference Problem

Let $M$ represent the mother's present age and $D$ represent the daughter's present age.

Formulating the Equations

The problem provides two conditions relating their ages:

• Condition 1: "3 times a mother's present age is 24 years more than 9 times her daughter's present age." This translates to the equation:

  $3M = 9D + 24$

• Condition 2: "4 times the daughter's present age is 2 years less than the mother's present age." This translates to the equation:

  $4D = M - 2$

Simplifying and Solving the Equations

First, simplify the first equation by dividing both sides by 3:

$M = 3D + 8 \quad (1)$

Next, rearrange the second equation to express $M$ in terms of $D$:

$M = 4D + 2 \quad (2)$

Now, set the two expressions for $M$ equal to each other to solve for $D$:

$3D + 8 = 4D + 2$

Subtract $3D$ from both sides:

$8 = D + 2$

Subtract 2 from both sides:

$D = 6$

The daughter's present age is 6 years.

Calculating the Mother's Age

Substitute the value of $D$ (6) back into either equation (1) or (2). Using equation (2):

$M = 4(6) + 2$

$M = 24 + 2$

$M = 26$

The mother's present age is 26 years.

Finding the Age Difference

The question asks for the difference between the mother's and daughter's ages:

$Difference = M - D$

$Difference = 26 - 6$

$Difference = 20$

The difference between the mother's and daughter's ages is 20 years.