2 years ago, Salim was 3 times as old as his daughter and 6 years later he will be four years older than twice her age. How old are they now?

Here are the solutions to the problems:

3. 2 years ago, Salim was 3 times as old as his daughter and 6 years later he will be four years older than twice her age. How old are they now?

Step 1: Define variables for their present ages.
Let Salim's present age be $S$ years.
Let his daughter's present age be $D$ years.

Step 2: Formulate equations based on the given information.

• 2 years ago: Salim's age was $S-2$. Daughter's age was $D-2$. "Salim was 3 times as old as his daughter": $$S-2 = 3(D-2)$$ $$S-2 = 3D-6$$ $$S = 3D-4 \quad (1)$$
• 6 years later (from now): Salim's age will be $S+6$. Daughter's age will be $D+6$. "he will be four years older than twice her age": $$S+6 = 2(D+6) + 4$$ $$S+6 = 2D+12+4$$ $$S+6 = 2D+16$$ $$S = 2D+10 \quad (2)$$

Step 3: Solve the system of equations.
Set equation (1) equal to equation (2) since both equal $S$:

$$3D-4 = 2D+10$$

Subtract $2D$ from both sides:

$$3D-2D-4 = 10$$ $$D-4 = 10$$

Add 4 to both sides:

$$D = 10+4$$ $$D = 14$$

Substitute the value of $D$ into equation (1) to find $S$:

$$S = 3(14)-4$$ $$S = 42-4$$ $$S = 38$$

Step 4: State their present ages.
Salim's present age is 38 years.
His daughter's present age is 14 years.

Their present ages are:
Salim: $\boxed{\text{38 years}}$
Daughter: $\boxed{\text{14 years}}$

4. The age of the father is twice the sum of the ages of his 2 children. After 20 years, his age will be equal to the sum of the ages of the children. Find the age of his father.

Step 1: Define variables.
Let the father's present age be $F$ years.
Let the sum of the present ages of his 2 children be $S$ years.

Step 2: Formulate equations based on the given information.

• Present ages: "The age of the father is twice the sum of the ages of his 2 children": $$F = 2S \quad (1)$$
• After 20 years: Father's age will be $F+20$. Each child's age will increase by 20 years. Since there are 2 children, the sum of their ages will increase by $2 \times 20 = 40$ years. So, the sum of the children's ages will be $S+40$. "his age will be equal to the sum of the ages of the children": $$F+20 = S+40 \quad (2)$$

Step 3: Solve the system of equations.
Substitute equation (1) into equation (2):

$$2S+20 = S+40$$

Subtract $S$ from both sides:

$$2S-S+20 = 40$$ $$S+20 = 40$$

Subtract 20 from both sides:

$$S = 40-20$$ $$S = 20$$

Now substitute the value of $S$ back into equation (1) to find $F$:

$$F = 2S$$ $$F = 2 \times 20$$ $$F = 40$$

Step 4: State the father's age.
The father's present age is 40 years.

The father's age is: $\boxed{\text{40 years}}$

5. A fraction becomes $\frac{9}{11}$ if 2 is added to both numerator and the denominator. If 3 is added to both numerator and denominator it becomes $\frac{5}{6}$. Find the fraction.

Step 1: Define variables.
Let the fraction be $\frac{x}{y}$, where $x$ is the numerator and $y$ is the denominator.

Step 2: Formulate equations based on the given information.

• "A fraction becomes $\frac{9}{11}$ if 2 is added to both numerator and the denominator": $$\frac{x+2}{y+2} = \frac{9}{11}$$ Cross-multiply: $$11(x+2) = 9(y+2)$$ $$11x + 22 = 9y + 18$$ $$11x - 9y = 18 - 22$$ $$11x - 9y = -4 \quad (1)$$
• "If 3 is added to both numerator and denominator it becomes $\frac{5}{6}$": $$\frac{x+3}{y+3} = \frac{5}{6}$$ Cross-multiply: $$6(x+3) = 5(y+3)$$ $$6x + 18 = 5y + 15$$ $$6x - 5y = 15 - 18$$ $$6x - 5y = -3 \quad (2)$$

Step 3: Solve the system of equations.
We have the system:

1. $11x - 9y = -4$
2. $6x - 5y = -3$ Multiply equation (1) by 5 and equation (2) by 9 to eliminate $y$:

$$5(11x - 9y) = 5(-4) \implies 55x - 45y = -20 \quad (3)$$ $$9(6x - 5y) = 9(-3) \implies 54x - 45y = -27 \quad (4)$$

Subtract equation (4) from equation (3):

$$(55x - 45y) - (54x - 45y) = -20 - (-27)$$ $$55x - 54x - 45y + 45y = -20 + 27$$ $$x = 7$$

Substitute $x=7$ into equation (2):

$$6(7) - 5y = -3$$ $$42 - 5y = -3$$ $$-5y = -3 - 42$$ $$-5y = -45$$ $$y = \frac{-45}{-5}$$ $$y = 9$$

Step 4: State the fraction.
The fraction is $\frac{x}{y} = \frac{7}{9}$.

The fraction is: $\boxed{\frac{7}{9}}$

6. The sum of the numerator and denominator of the fraction is 12. If the denominator is increased by 3 it fraction become $\frac{1}{2}$. Find the fraction.

Step 1: Define variables.
Let the fraction be $\frac{x}{y}$, where $x$ is the numerator and $y$ is the denominator.

Step 2: Formulate equations based on the given information.

• "The sum of the numerator and denominator of the fraction is 12": $$x + y = 12 \quad (1)$$
• "If the denominator is increased by 3 it fraction become $\frac{1}{2}$": $$\frac{x}{y+3} = \frac{1}{2}$$ Cross-multiply: $$2x = 1(y+3)$$ $$$$