Suppose that omar's marginal utility for cups of coffee is constant at 2.5 utils per cup no matter how many cups he drinks. on the other hand, his marginal utility per doughnut is 11 for the first doughnut he eats, 10 for the second he eats, 9 for the third he eats, and so on (that is, declining by 1 util per additional doughnut). in addition, suppose that coffee costs $1 per cup, doughnuts cost $1 each, and omar has a budget that he can spend only on doughnuts, coffee, or both. how big would that budget have to be before he would spend a dollar buying a first cup of coffee?

Calculating the marginal utility per dolor for doughnut which is marginal utility divided by unit price. So that will be 11 for first, 10 for second, 9 for third, 8 for fourth, 7 for fifth, 6 for sixth, 5 for seventh, 4 for eighth, 3 for ninth, 2 for tenth.  
Now the same for coffee = 2.5 / $1 = 2.5.  
Compare the MU per dollar of both goods to determine the amount of money he needs to buy the first cup of coffee.  
So obviously the MU value will be higher for doughnut until he reaches his ninth that 3>2.5, for the tenth doughnut coffee MU is higher so he will buy 9 doughnuts for $1 and then buy his first coffee. 
Total would be $9 on doughnut and $1 on coffee bringing it to 9 + 1 that is $10.