Brian's number
Brian chooses an integer and operates on it. Work out the largest integer that he could have chosen.
Age
14 to 16
Challenge level
Problem
Brian chooses an integer, multiplies it by 4 then subtracts 30. He then multiplies his answer by 2 and finally subtracts 10. His answer is a two-digit number.
What is the largest integer he could have chosen?
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
Answer: $21$
Working backwards
$\times 4 \rightarrow -30\rightarrow \times2 \rightarrow -10\rightarrow$ answer 2 digits
$\div4\leftarrow +30 \leftarrow \div2 \leftarrow + 10 \leftarrow$ answer 2 digits
Answer 2 digits, largest possible $99$
| Answer | $+10$ | $\div2$ | $+30$ | $\div4$ |
| $99$ | $109$ | $54.5$ | $84.5$ | Not an integer |
| $98$ | $198$ | $54$ | $84$ | $21$ good |
So the largest possible choice is 21.
Using algebra
Brian chooses be $b$
Multiply by $4$ and subtract $30$: $4b-30$
multiply this by $2$ and subtract $10$: $2(4b-30)-10 = 8b-70$
Answer two digits, largest possible $99$
If Brian got $99$:
$8b-70=99\\
\Rightarrow 8b=169\\
\Rightarrow b = 21.125$
So $b\le21.125$ and $b$ is an integer so $b=21$