he has 50 rock songs.
5 + 6 = 11
so the total number of songs is 101 < 11*10 < 120, that is 110.
so he has 50 rock songs and 60 dance songs.
the diagonal of a rectangle splits the rectangle into two equal right triangles. the legs of the triangle are 6cm and 8cm. this means the pythagorean theorem can be applied to find the third side of the triangle.
the pythagorean theorem is .
here a= 6 and b=8.
using eight coins, the maximum is dollars, so we need to find a value lower than that. if we imagine pennies, and add one to that, since we can make any value under cents, we will get cents. we can make this with a nickel, so we also need to be out of range of nickels, so we should try . we can make this with dimes, so we should try . can be made with quarters, so we should try and get cents. but, we cannot forget about subtracting from these values. if we try , we get , which is made with dimes. trying makes , but if we keep moving it down to the nearest multiple of , or a quarter, we will get . can be made with coins though, but if we take it away again down to , we get . this looks hopeful. this has quarters, dime, nickel, and four pennies. if we keep trying and move it down to , we can still make it. if we keep moving down till we get to the lcm of the values, we get . making this does works, so now we know it is between cents and cents. trying the nearest multiple of minus , we get . this obviously works, so we need to get away from the multiple of . trying , we get , , and coin types, so this works. now try . this is quarters, dime, and four pennies. trying , we get quarters, dime, 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! for the workout!
edit: oops! i did less than or equal to! but you can use the same logic to do it again!