Alien Currency
The planet Zog has both green and blue bank notes. Can you work out how many zogs two green and three blue bank notes are worth?
Age
14 to 16
Challenge level
Problem
On planet Zog, they use green and blue bank notes.
Three green notes and eight blue notes are worth 46 zogs.
Eight green notes and three blue notes are worth 31 zogs.
How many zogs are two green notes and three blue notes worth?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: $19$
Elimination multiplying equations by $3$ and $8$
$\begin{align}3g+8b=46\quad\Rightarrow\quad&24g+64b=368\\
8g+3b=31\quad\Rightarrow\quad&\ \ \underline{24g+9b=\ \ 93}\\
\text{subtracting gives:}\quad&\ \ \ \qquad 55b=275\\
&\ \ \ \Rightarrow\ \ 11b=55\\
& \ \ \ \Rightarrow\quad\ \ b=5\end{align}$
And so $3g+40=46\Rightarrow g=2$
$\therefore 2g+3b= 4+15=19$
Elimination by combining equations more than once
$$\begin{align} 3g+8b=46&\\
\underline{+\qquad 8g+3b=31}&\\
11g+11b=77&\\
\Rightarrow \quad g+b=\ \ 7&\\
\Rightarrow 3g+3b=21\end{align}$$
$\begin{align} 3g+8b=46&\\
\underline{-\qquad 3g+3b=21}&\\
5b=25&\\
\Rightarrow \quad b=\ 5\ &\end{align}$ $\begin{align} 8g+83=31&\\
\underline{-\qquad 3g+3b=21}&\\
5g=10&\\
\Rightarrow \quad g=\ 2\ &\end{align}$
Therefore $2g + 3b = 19$.
Getting $2g+3b$ without finding $g$ and $b$
$\begin{align} 3g+8b=46&\\
\underline{+\qquad 8g+3b=31}&\\
11g+11b=77&\\
\Rightarrow \quad g+b=\ \ 7& \end{align}$
$3g+8b$
$+ \quad (\ g\ +\ b\ )\times$ some to get blues and greens in ratio $2:3$
difference will be $5$ since $8$ and $3$ have difference of $5$
$10:15 = 3+7:8+7$
$\begin{align} 3g+8b=46&\\
\underline{+\qquad 7g+7b=49}&\\
10g+15b=95&\\
\Rightarrow \ \ 2g+3b=19& \end{align}$