Triangular algebra
Weekly Problem 26 - 2017
The angles in the triangle are shown in the diagram in terms of x and y. If x and y are positive integers, what is the value of x+y?
The angles in the triangle are shown in the diagram in terms of x and y. If x and y are positive integers, what is the value of x+y?
Age
14 to 16
Challenge level
Problem
Image
The interior angles of a triangle are $\left(5x+3y\right)^\circ$, $\left(10y+30\right)^\circ$ and $\left(3x+20\right)^\circ$, where $x$ and $y$ are positive integers.
What is the value of $x+y$?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: $15$
$$\begin{align}(5x+3y)+(10y+30)+(3x+20)&=180\\
8x+13y&=130\\
8\times0+13\times10 &=130\\
\uparrow \hspace{18mm}\uparrow\hspace{2mm} &\\
x\hspace {18mm} y\hspace{2mm}&\end{align}$$
but $x$ shouldn't be $0$
$$\begin{align}13\times10&=130\\8\times13+13\times2&=130\\
\uparrow \hspace{18mm}\uparrow \hspace{1mm}&\\
x\hspace {18mm} y\hspace{1mm}&\end{align}$$ So $x+y=13+2=15$