Answer:
94 cents
Step-by-step explanation:
Using eight coins, the maximum is [tex]2[/tex] dollars, so we need to find a value lower than that. If we imagine [tex]8[/tex] pennies, and add one to that, since we can make any value under [tex]8[/tex] cents, we will get [tex]9[/tex] cents. We can make this with a nickel, so we also need to be out of range of nickels, so we should try [tex]40+1[/tex]. We can make this with dimes, so we should try [tex]8\cdot10+1[/tex]. [tex]81[/tex] can be made with quarters, so we should try [tex]200+1[/tex] and get [tex]201[/tex] cents. But, we cannot forget about subtracting from these values. If we try [tex]40-1[/tex], we get [tex]39[/tex], which is made with dimes. Trying [tex]8\cdot10-1[/tex] makes [tex]79[/tex], but if we keep moving it down to the nearest multiple of [tex]25[/tex], or a quarter, we will get [tex]74[/tex]. [tex]74[/tex] can be made with [tex]8[/tex] coins though, but if we take it away again down to [tex]7\cdot10-1[/tex], we get [tex]69[/tex]. This looks hopeful. This has [tex]2[/tex] quarters, [tex]1[/tex] dime, [tex]1[/tex] nickel, and four pennies. If we keep trying and move it down to [tex]64[/tex], we can still make it. If we keep moving down till we get to the LCM of the values, we get [tex]50-1=49[/tex]. Making this does works, so now we know it is between [tex]75[/tex] cents and [tex]200[/tex] cents. Trying the nearest multiple of [tex]5[/tex] minus [tex]1[/tex], we get [tex]79[/tex]. This obviously works, so we need to get away from the multiple of [tex]25[/tex]. Trying [tex]84[/tex], we get [tex]3[/tex], [tex]1[/tex], and [tex]4[/tex] coin types, so this works. Now try [tex]89[/tex]. This is [tex]3[/tex] quarters, [tex]1[/tex] dime, and four pennies. Trying [tex]94[/tex], we get [tex]3[/tex] quarters, [tex]1[/tex] dime, [tex]1[/tex] nickel, and four pennies.There is nothing lower than this, so we have found it! :D This took me very long to do, and even I learned something from doing your super hard question! Thanks for the workout!
EDIT: Oops! I did less than or equal to! But you can use the same logic to do it again!
