In triangle ABC, DE is parallel to BC with D on AB and E on AC, and AD : DB = 1 : 4. Asked for area(ADE) : area(trapezium DBCE), a student answered 1 : 25.
What is the correct ratio, and what went wrong?

AD : AB = 1 : $(1+4)=1$ : 5, so area(ADE) : area(ABC) = ${\left(\frac{1}{5}\right)}^2=\frac{1}{25}$. Taking area(ABC) as 25 parts, trapezium DBCE is $25-1=24$ parts, so the required ratio is 1 : 24. The student's 1 : 25 is the comparison with the whole triangle; 1 : 5 skips the squaring and 1 : 4 quotes AD : DB.