In a collection of nickels, dimes, and quarters worth $6.90, the ratio of the number of nickels to dimes is 3: 8. the ratio of the number of dimes to quarters is 4: 5. find the number of each type of coin.

N=number of nickels
d=number of dimes
q=number of quarters

example
if n:d=3:8
lets say n=3 and d=8
3*8=8*3
so
8n=3d
slightly confusing but go with it, it migh make sense or it might not

n:d=3:8
8n=3d

d:q=4:5
5d=4q

solve for one of them
solve for nickels in first and quarters in second

8n=3d, n=(3/8)d
5d=4q, (4/5)d=q

5n+10d+25q=690
divide by 5 to make it easeir
n+2d+4q=138
sub (3/8)d for n
sub (4/5)d for q
(3/8)d+2d+5(5/4)d=138
(3/8)d+2d+25/4d=138

3/8+2+25/4=
3/8+16/2+50/8=
69/8

(69/8)d=138
times both sides by 8
69d=1104
divide both sides by 69
d=16

16 dimes
sub back

(5/4)d=q
(5/4)16=q
80/4=q
20=q

(3/8)d=n
(3/8)16=n
48/8=n
6=n

6 nickles
16 dimes
20 quarters
yah