Think Of Two Numbers Poster
Think Of Two Numbers Poster
Age
11 to 14
Challenge level
Problem
This poster is based on the problem Think of Two Numbers.
The poster is available as a PDF, or the image below can be clicked on to enlarge it.
Student Solutions
I subtract 9 from your answer, and the digits of the number that I'm left with, are your two numbers.
For example, if your final answer is 74, when I subtract 9, I get 65, so I know your numbers were 6 and 5.
How does it work?
If the two numbers are $a$ and $b$,
| Take one of your numbers |
$a$ |
| Add $1$ |
$a+1$ |
| Multiply by $5$ |
$5(a+1) = 5a+5$ |
| Add $1$ again |
$5a+5+1 = 5a+6$ |
| Double your answer |
$2(5a+6) = 10a+12$ |
| Subtract $1$ |
$10a+12-1 = 10a+11$ |
| Add your second number |
$10a + 11 + b$ |
| Add $2$ |
$10a + 11 + b + 2 = 10a + b + 13$ |
| Double again |
$2(10a + b + 13) = 20a + 2b + 26$ |
| Subtract $8$ |
$20a + 2b + 26 - 9 = 20a + 2b + 18$ |
| Halve this number |
$\frac12(20a + 2b + 18) = 10a + b + 9$ |
Final answer is $10a + b + 9$, so I subtract $9$ to leave $10a+b$
$10a + b$ is $a$ tens and $b$ ones
| Tens | Ones |
| $a$ | $b$ |
Since $a$ and $b$ are both less than $10$, they fit in the tens and ones columns and so they are the digits of the number.