The sum of the numerator and denominator of a fraction is $12$. If $1$ is added to both the numerator and the denominator the fraction becomes $3 \over 4$. Find the fraction.
$4$ men and $6$ boys can finish a piece of work in $5$ days while $3$ men and $4$ boys can finish it in $7$ days. Find the time taken by $1$ man alone or that by $1$ boy alone.
If one diagonal of a trapezium divides the other diagonal in the ratio $1 : 3$. Prove that one of the parallel sides is three times the other.
The diagonals of a trapezium $ABCD$ in which ADBC intersect at $O$ and $BC2AD$. Find the ratio of the areas of $\triangle AOD$ to $\triangle BOC$.
If $\sin(a - b) = \frac 1 2$ and $\cos(a+b) = \frac 1 2$, $0 < a + b \leq 90$ $A > B$, find $a$ and $b$.
Also, this isn't homework. Its from the last year question paper. These ones I failed to solve.