I am 25 years younger than my oldest brother. When the digits of his age are multiplied together, the result is my age. And when the digits of my age are added together, the result is the first digit of his age. How old am I? Thanks.
I don't think I can solve this one using Algebra! So, here goes some "Logic!"
First, my understanding is that by "first digit", I take it to mean the "left" digit of his age, i.e., 7 in 75.
If my brother is in his 30s, then I would have to be 12, since the sum =3
If my brother is in his 40s, then I would have to be 22, since the sum =4.........and so on up to his 90s, in which case I would to be 72. In each case, the last digit of my age is 2. Therefore, the last digit of his age must be a 7.
And now it should be easy to see that the only instance where the digits of his age, when multiplied together, equal my age is when he is 67 and I am 42!.
That is my attempt!. Let us see if somebody can solve it using Algebra!.
Let your broher's age = 10a + b where a and b are the digits of his age
And your age is 10a + b - 25 = 10(a - 2) + (b- 5) where (a -2) and (b- 5) are the digits of your age
And....the digits of his age multiplied togeher = your age
ab = 10a + b - 25 (1)
And.....the sum of he digits of your age = the first digit of his age
(a - 2) + (b - 5) = a → a + b - 7 = a → b - 7 = 0 → b = 7
So ......using (1) we have
a7 = 10a + 7 - 25 →
-3a = - 18 →
a = 6
Your brother is 67 and you are 67 - 25 = 42
I don't think I can solve this one using Algebra! So, here goes some "Logic!"
First, my understanding is that by "first digit", I take it to mean the "left" digit of his age, i.e., 7 in 75.
If my brother is in his 30s, then I would have to be 12, since the sum =3
If my brother is in his 40s, then I would have to be 22, since the sum =4.........and so on up to his 90s, in which case I would to be 72. In each case, the last digit of my age is 2. Therefore, the last digit of his age must be a 7.
And now it should be easy to see that the only instance where the digits of his age, when multiplied together, equal my age is when he is 67 and I am 42!.
That is my attempt!. Let us see if somebody can solve it using Algebra!.
Thank you CPhill. Very kind and too modest of you!! I can see myself attempting to solve it using Algebra, but from your solution I don't think I could have factored "10a + b - 25" into "10(a - 2) + (b- 5)" !!!. Believe or not, I plugged it into Wolfram/Alpha and it CANNOT factor it!!!??.