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 (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 (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: 38 years Daughter: 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 (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 × 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 (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 × 20 F = 40 Step 4: State the father's age. The father's present age is 40 years. The father's age is: 40 years 5. A fraction becomes (9)/(11) if 2 is added to both numerator and the denominator. If 3 is added to both numerator and denominator it becomes (5)/(6). Find the fraction. Step 1: Define variables. Let the fraction be (x)/(y), where x is the numerator and y is the denominator. Step 2: Formulate equations based on the given information. "A fraction becomes (9)/(11) if 2 is added to both numerator and the denominator": (x+2)/(y+2) = (9)/(11) Cross-multiply: 11(x+2) = 9(y+2) 11x + 22 = 9y + 18 11x - 9y = 18 - 22 11x - 9y = -4 (1) "If 3 is added to both numerator and denominator it becomes (5)/(6)": (x+3)/(y+3) = (5)/(6) Cross-multiply: 6(x+3) = 5(y+3) 6x + 18 = 5y + 15 6x - 5y = 15 - 18 6x - 5y = -3 (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) 55x - 45y = -20 (3) 9(6x - 5y) = 9(-3) 54x - 45y = -27 (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 = (-45)/(-5) y = 9 Step 4: State the fraction. The fraction is (x)/(y) = (7)/(9). The fraction is: (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 (1)/(2). Find the fraction. Step 1: Define variables. Let the fraction be (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 (1) "If the denominator is increased by 3 it fraction become (1)/(2)": (x)/(y+3) = (1)/(2) Cross-multiply: 2x = 1(y+3) 2x = y+3 (