SOLUTION: Now, there are a different number of ordinary bicycles, tandem bikes, and tricycles in the shop. there are 135 seats,118 front handlebars, and 269 wheels. how many regular bikes, t

Every vehicle has a front handlebar.  There are 118 front handlebars so there
are 118 vehicles.  They all have two wheels except for the tricycles.  If they
all had 2 wheels there would be only 2×118 or 236 wheels.  But there are 269
wheels.  Therefore the extra 269-236 or 33 wheels are on the tricycles.  So
there are 33 tricycles.

Now we can eliminate the 33 tricycles. Each tricycle has 1 front handlebar and 1
seat.  We have already taken care of the handlebars and wheels.  So we subtract
those and the problem becomes: 

If all those 85 vehicles had just 1 seat there would be 85 seats.  But there
are 102 seats, so the extra 17 seats are on the tandem bikes.  

So there are 17 tandem bikes, and the remaining 68 seats are on regular
bicycles.

Answer: 68 regular bicycles, 17 tandem bikes, and 33 tricycles. 

Edwin

We could also do it using algebra:

Let x = the number of ordinary bicycles, 
Let y = the number of tandem bikes, and 
Let z = the number of tricycles


there are 135 seats,

There are x seats on the ordinary bicycles, 2y seats on the tandem bikes,
and z seats on the tricycles.  So that gives us the equation:

x + 2y + z = 135

118 front handlebars, 

There are x front handlebars on the ordinary bicycles, y front handlebars on the tandem bikes, and z front handlebars on the tricycles.  So that gives us the equation:

x + y + z = 118

and 269 wheels. 

There are 2x wheels on the ordinary bicycles, 2y wheels on the tandem bikes,
and 3z seats on the tricycles.  So that gives us the equation:

2x + 2y + 3z = 269

So the system of equations is

 x + 2y +  z = 135
 x +  y +  z = 118
2x + 2y + 3z = 269

Solve that and get x=68 regular bikes, y=17 tandem bikes, and z=33 tricycles.

Edwin