Shape sums
Given some relationships amongst these shapes, how many triangles equal one diamond?
Age
11 to 14
Challenge level
Problem
In the set of equations below, how many $\bigtriangleup$s are equal to $\Diamond$?
$\bigtriangleup + \bigtriangleup = \Box$
$\Box + \bigtriangleup = \bigcirc$
$\Diamond = \bigcirc + \Box + \bigtriangleup$
$\Diamond = \bigcirc + \Box + \bigtriangleup$
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: six
$\Box = 2 \bigtriangleup$, from the first equation.
Then, $\bigcirc = \Box + \bigtriangleup = 2\bigtriangleup + \bigtriangleup = 3\bigtriangleup$.
Then, $\Diamond = \bigcirc + \Box + \bigtriangleup = 3\bigtriangleup + 2\bigtriangleup + \bigtriangleup = 6 \bigtriangleup$.
Therefore, $\Diamond$ is worth $6 \bigtriangleup$.