1170.24 – Dimes and Pennies

You have been saving dimes and pennies in a jar on your dresser. One day you dump them all out. Your little brother barges in without knocking and wants to know how much money you have. You have counted up, so you tell him that you have $10.54 and that there are 181 coins. Further, if he can tell you how many dimes there are without looking, you say you'll give him ten of them as a reward for good thinking (even though he didn't knock). We can't send you the ten dimes but you can figure it out anyway and feel proud of yourself.

Solution

One way:

p+10d=1054p+d=181So 9d=873→d=97 and p=84.\begin{aligned} p + 10d &= 1054 \\ p + d &= 181 \\ \text{So } 9d &= 873 \rightarrow d = 97 \text{ and } p = 84. \end{aligned}

Another way (cleverer!)

If all 181 coins are pennies, you'd have $1.81. Each time a penny is replaced by a dime, the total amount goes up by 9¢. If there are x dimes, the amount goes up by 0.09x.

So

1.81+0.09x=10.54→0.09x=8.73→x=97.1.81 + 0.09x = 10.54 \rightarrow 0.09x = 8.73 \rightarrow x = 97.