I can't figure this one out thermal expansion of rod and ruler

[1MileCrash]

Homework Statement

At 19°C, a rod is exactly 20.01 cm long on a steel ruler. Both the rod and the ruler are placed in an oven at 219°C, where the rod now measures 20.18 cm on the same ruler. What is the coefficient of linear expansion for the material of which the rod is made?

Homework Equations

The Attempt at a Solution

The ruler is "normally" 20.01 cm.
Using the coefficient of thermal expansion for steel, I know that the oven caused it to lengthen to 20.054022 cm.

That means that the new ruler's inches are 1.0022 times longer than the "old ruler."

The rod measures 20.18 cm on the "new ruler," and since 20.18/1.0022 = 20.13570146, that's what the rod would have measured on the "old ruler."

Therefore the change in length of the rod is 20.1357 - 20.01 = .1257.

Therefore the rod material coefficient is

.1257 / (20.01)(200) = 3.140965e-5

This is wrong, and I've worked a few of them and my answers always seem to be off by a predictable amount (slightly more than half of the correct answer). What's going wrong here?

 

[ehild]

Think over the measurement with new ruler and old ruler.

You measure the length of a rod L with a ruler of inch scale.
You get that the rod is L=8 units long. One unit is 2.5 cm on your ruler. What is the length of the rod measured with a cm-scale ruler?

ehild