Malcolm’s mother is 27 years older than he is and their ages are both two digit numbers. If Malcolm swaps the digits in his age he gets his mother’s age.
a) How old is Malcolm if the sum o the digits in his age is 5?
b) What is the relationship between the digits in Malcolm’s age if the sum of the digits is unknown.
c) If the sum of the digits in Malcolm’s two-digit age is unknown, how many possible ages could he be? What are these ages?
All the solution has to be in algebraic form
To solve these questions algebraically, let's first represent Malcolm's age using variables. Let's say his age is in the form of AB, where A represents the tens digit, and B represents the ones digit. Similarly, his mother's age can be represented as BA.
a) How old is Malcolm if the sum of the digits in his age is 5?
If the sum of the digits is 5, we can form the equation A + B = 5.
Since Malcolm's mother is 27 years older than he is, we can set up another equation:
BA = AB + 27
To find the value of Malcolm's age, we can solve the system of equations by substitution or elimination. Let's use substitution:
From equation 1, we can express B in terms of A: B = 5 - A.
Substituting this value into equation 2:
(5 - A)A = A(5 - A) + 27
5A - A^2 = A(5 - A) + 27
5A - A^2 = 5A - A^2 + 27
0 = 27
This equation has no solution, meaning there is no valid age for Malcolm given the conditions stated.
b) What is the relationship between the digits in Malcolm’s age if the sum of the digits is unknown?
If we do not know the sum of the digits, we cannot determine a specific relationship between the digits. We can only state that the relationship between the tens digit (A) and the ones digit (B) can be represented by the equation A + B = unknown sum.
c) If the sum of the digits in Malcolm’s two-digit age is unknown, how many possible ages could he be? What are these ages?
If we do not have information about the sum of the digits, we can consider all possible combinations of two-digit numbers (excluding the digit 0). There are 90 possible two-digit numbers (10 to 99).
a: d+u=5
10(5-u)+u+27=10u+(5-u)
u=4
d=1
he is 14
b:9d+27=9u
c:from 14,25,.. to 69