In two alloys,gold and silver are in the ratios of $4:1$ and $1:3$. After alloying together $10 \text{ kg}$ of the first alloy,$16 \text{ kg}$ of the second,and several kilograms of pure gold,an alloy was obtained in which the ratio of gold to silver was $3:2$. Find the weight of the new alloy (in $\text{kg}$).

When $X$ is subtracted from the numbers $9, 15$ and $27$,the remainders are in continued proportion. What is the value of $X$?

₹ $900$ is divided among $A, B, C$; the division is such that $\frac{1}{2}$ of $A$'s money $= \frac{1}{3}$ of $B$'s money $= \frac{1}{4}$ of $C$'s money. Find the amount received by $A, B,$ and $C$.

If $Rs. 510$ is divided among $A, B, C$ in such a way that $A$ gets $\frac{2}{3}$ of what $B$ gets and $B$ gets $\frac{1}{4}$ of what $C$ gets,then their shares are respectively:

The ratio of two numbers is $4:5$. If both numbers are increased by $4$,the ratio becomes $5:6$. What is the sum of the two numbers?

The ratio of ages between $A$ and $B$ is $6: 5$ and the age of each $C$ and $D$ is $\frac{9}{10}$ times that of $B$. The age of $F$ is less than $A$ but greater than $B$. The ratio of ages between $B$ and $E$ is $2: 3$. Also,the age of $A$ is $3 \text{ years}$ less than $E$. What is the ratio of the ages of $A$ and $F$ if all the ages are integers?