Operational Decision
What symbol should be inserted to make the equation true?
Age
11 to 14
Challenge level
Problem
Which symbol ($+$, $-$, $\div$ or $\times$) should replace $\oplus$ to make the following equation true?
$$ 1\times 2\times \left(3\oplus 4 + 5\right) \times \left(6\times 7 + 8+ 9\right) = 2006 $$
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: $\times$
Guessing
$3\div4$ is not a whole number so it's probably not $\div$
$1\times2\times(3-4+5)\times(6\times7+8+9)=2\times4\times59=8\times59\approx240$ too small
$1\times2\times(3+4+5)\times(6\times7+8+9)=2\times11\times59=22\times59\approx 1200$ too small
$1\times2\times(3\times4+5)\times(6\times7+8+9)=2\times17\times59=34\times59\approx 2100$ about right
So it must be $\times$
Working backwards
From the original equation
$$ 1\times 2\times \left(3\oplus 4 + 5\right) \times \left(6\times 7 + 8+ 9\right) = 2006$$it follows that
$$ 2\times \left(3\oplus 4 + 5\right) \times \left(6\times 7 + 8+ 9\right) = 2006, $$that is $\left(3\oplus 4 + 5\right)\times 59 = 1003$, that is $3\oplus 4 + 5=17$, that is $3\oplus 4=12$. Therefore, $\oplus$ should be replaced by $\times$.