A consumer's utility function is given by:U(x,y) = 10xyCurrently, the prices of goods x and y are ...

a.   MRS = y/x
b.   The MRS=MRT condition says y/x = 3/5. Plug into the budget constraint yields 3x + 3x = 150, or simply x = 25, y = 15.
c.   Now the price of x is $6. To be able to purchase y=15 and x=25, this consumer will need an income of 225 which means we will need to increase her income by $75
d.   The original bundle gave this person a utility of: U(25,15)=3750. Now we wish to find how much income is needed to get this level of utility when the prices are now $6 and $5. So the first equation to consider is the indifference curve representing this utility level given by: 10xy = 3750

We also know that in order to be on this IC, the consumer will be at a point where the
MRS = MRT condition holds or: y/x = 6/5

And then plug this into the IC equation to get y = 21.2 and x = 17.7. To be able to purchase the bundle (17.7,21.2) the consumer will need an income of 212.2. Thus we need to give her $62.20.