A jeweler has bars of $18-$carat gold and $12-$carat  gold. How much of each must be melted together to obtain a bar of $16-$carat gold weighing$120$grams ? (pure gold is $24-$carat).

$A$ and $B$ together can do a piece of work in $15$ days. If $A$’s one day work is $1\frac{1}{2}$ time the one day’s work of $B$, find in how many days can each do the work.

$2 \text{men and} 5 \text{women}$ can do a piece of work in$4$ days,while one man and one woman can finish it in $12$days. How long would it take for $1$man to do the work ?

A train covered a certain distance at a uniform speed. If the train had been $30$ km/h faster, it would have taken $2$ hours less than the scheduled time. If the train were slower by $15$ km/hr, it would have taken $2$ hours more than the scheduled time. Find the length of the jouney.

A boat takes $2$ hours to go $40$ km down the stream and it returns in $4$ hours. Find the speed of the boat in still water and the speed of the stream.

An aeroplane flies $1680$ km with a head wind in $3.5$ hours. On the return trip with same wind blowing, the plane takes $3$ hours. Find the plane’s air speed and the wind speed.

A part of monthly hostel charges is fixed and the remaining depend on the number of days one has taken food in the mess. When Bhawana takes food for$20$ days, she has to pay rs. $2600$. As hostel charges; whereas Divya takes food for$26$days, she pays rs.$3020$ as hostel charges. Find the fixed charges and the cost of food per day.