​Michael's bank contains only​ nickels, dimes, and quarters. There are 64 coins in​ all, valued at ​$5.30. The number of nickels is 8 short of being three times the sum of the number of dimes and quarters together. How many dimes are in the​ bank?

Answer:Number of dims in the bank are 10.                  Step-by-step explanation:Given : ​Michael's bank contains only​ nickels, dimes, and quarters. There are 64 coins in​ all, valued at ​$5.30. The number of nickels is 8 short of being three times the sum of the number of dimes and quarters together. To find : How many dimes are in the​ bank?Solution : Let number of nickels = xNumber of dims = yNumber of quarters = zTotal there are 64 coins. i.e. [tex]x+y+z=64[/tex] ....(1)The cost of nickels = $0.05The cost of dims = $0.10The cost of quarters = $0.25Total value at $5.30.i.e. [tex]0.05x+0.10y+0.25z=5.30[/tex] .....(2)The number of nickels is 8 short of being three times the sum of the number of dimes and quarters together. i.e. [tex]x=3(y+z)-8[/tex] .....(3)Substitute y+z from (1),[tex]x=3(64-x)-8[/tex] [tex]x=192-3x-8[/tex] [tex]4x=184[/tex] [tex]x=46[/tex] Substitute the value of x in (1) and (2), [tex]46+y+z=64[/tex] [tex]y+z=18[/tex] .....(4)[tex]0.05(46)+0.10y+0.25z=5.30[/tex] [tex]2.3+0.10y+0.25z=5.30[/tex] [tex]0.10y+0.25z=3[/tex] [tex]10y+25z=300[/tex]  .....(5)Solving equation (4) and (5),Substitute the value of y from (4) in (5),[tex]10(18-z)+25z=300[/tex][tex]180-10z+25z=300[/tex][tex]15z=120[/tex][tex]z=\frac{120}{15}[/tex][tex]z=8[/tex]Substitute the value of z in (4), [tex]y+8=18[/tex] [tex]y=10[/tex]So, Number of nickels are x=46Number of dims are y=10Number of quarters are z=8.Therefore, Number of dims in the bank are 10.