A year ago Bo was five times as old as Jo, who today is under 20 years old.
Bo is now three times as old as Mo will be in four years, and four times as old as Flo was three years ago.
How old are Bo, Jo, Mo, and Flo?
Solution to the Problem:
Bo's current age is 36, Jo's current age is 8, Mo's current age is 8, and Flo's current age is 12.
Let B = Bo's current age
Let J = Jo's current age
Let M = Mo's current age
Let F = Flo's current age
Then you can write three equations and one inequality from the clues given:
B - 1 = 5 (J - 1)
J < 20
B = 3 (M + 4)
B = 4 (F - 3)
The use the distribute property to get:
B = 5J - 4
B = 3M + 12
B = 4F - 12
I rewrote the three equations so that I got the ages of the other three people in terms of Bo's current age:
J = (B + 4) / 5
M = (B - 12) / 3
F = (B + 12) / 4
Since we have more variables than equations, there is an infinite number of solutions to this problem (and that is why Ms. McCarthy gave the inequality).
To get whole numbers, (B + 4) must be divisible by 5, (B - 12) must be divisible by 3, and (B + 12) must be divisible by 4.
So, I began listing the possible ages of Bo that will make Jo's, Mo's and Flo's ages a whole number. This may sound like trial and error, but I am using this approach so that I can find a general solution for all the possible answers. I needed to look for a pattern.
For Jo's age to be a whole number, Bo must be:
1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 86, 91, 96, 101, 106, 111, 116, 121, 126, ...
For Mo's age to be a whole number, Bo must be:
12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, ...
For Flo's age to be a whole number, Bo must be:
0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, ...
Now, we look for numbers that appear in all three lists, and we find 36, 96, 156, 216, ...
There is an infinite number of solutions until we look at the inequality.
Looking at the equations above for Jo, Mo, and Flo, I noticed that the least common multiple of 5, 3, and 4 is 60.
Then I can write Bo's age as 36 + 60n where n = 0, 1, 2, 3, 4, ...
Here is a table of answers for the three equations:
| Name | n = 0 | n = 1 | n = 2 | n = 3 |
|---|---|---|---|---|
| Bo | 36 | 96 | 156 | 216 |
| Jo | 8 | 20 | 32 | 44 |
| Mo | 8 | 28 | 48 | 68 |
| Flo | 12 | 27 | 42 | 57 |
From the table, we can complete our general solution:
Bo's age is 36 + 60n where n = 0, 1, 2, 3, 4, ...
Jo's age is 8 + 12n where n = 0, 1, 2, 3, 4, ...
Mo's age is 8 + 20n where n = 0, 1, 2, 3, 4, ...
Flo's age is 12 + 15n where n = 0, 1, 2, 3, 4, ...
Also, note from the table that the only age for Jo that is less than 20 is 8, so the column of the table where n = 0 becomes your answer.
Correctly solved by:
| 1. K. Sengupta | Calcutta, India |
| 2. Davit Banana | Istanbul, Turkey |
| 3. Ivy Joseph | Pune, Maharashtra, India |
| 4. Dr. Hari Kishan |
D.N. College, Meerut, Uttar Pradesh, India |
| 5. Sudhir Bavdekar | Mumbai, India |
| 6. Colin (Yowie) Bowey | Beechworth, Victoria, Australia |
| 7. Kelly Stubblefield | Mobile, Alabama, USA |
| 8. Seth Cohen | Concord, New Hampshire, USA |