PUZZLE

Abena is 54 years old and her mother is 80. How many years ago was Abena’s mother three times her age?

ANSWER

Preambles: The ages 54 and 80 represent the present ages of Abena and her mother respectively. We are seeking a number of years (say m years) ago when Abena’s mother was three times as old as Sally. Thus m is our subject of interest. Now let’s follow the steps below to solve the puzzle:

Step 1: Denote the ages of Abena and her mother today by x and y respectively. Thus x=54; y=80.

Step 2: m years ago, Abena was (x-m) years old; and her mother was (y-m) years old.

Step 3: From the preamble, (y-m) must be equal to 3*(x-m). Therefore, (y-m) = 3(x-m).

Step 4: Expand the brackets: y-m = 3x – 3m.

Step 5: Make m the subject of the equation in step 4: 2m = 3x – y; m = (3x -y)/2.

Step 6: Substitute the values of x and y into the equation for m: m = [(3*54) -80]/2 = (162 -80)/2 = 82/2 = 41.

Therefore, 41 years ago, Abena’s mother was (80 -41) = 39 years old; and Abena was (54 -41) = 13 years old. Thus Abena’s mother was 3 times Abena’s age: 39 = 3*13!

CHALLENGE YOURSELF

Abena’s mother is 26 years older than Abena today. How many years older was Abena’s mother 41 years ago?