Hi William,

I can get you started.

Suppose she arranges her quarters, nickels and dimes in piles with one quarter, one nickel and one dime in each pile. Each pile has three coins and $\large \frac{177}{3} \normalsize = 59$ so the largest number of piles she can have is 59. The value of each pile is $0.25 + 0.10 + 0.05= 0.40$ dollars. Thus if she has 59 piles the value of the coins she would have is $59 \times \$0.40 = \$23.60.$ But we know the total value of her coins is $\$19.90$ so 59 piles is too many.

Can you compete the solution?
Penny