Shirley measures the thickness of a piece of cardboard with a ruler that has centimeter and inch markings. which of the following is most likely the value that shirley measures?

Shirley is most probably going to use the side of the ruler with the centimeter markings because the cardboards are relatively thin. To have a more accurate measurement, it is more practical to use the markings which have smaller intervals or which are closer to each other. 

1 inch is around 2.54 centimeters which means that an inch is bigger than a centimeter. Measuring a relatively thin dimension of an object using large intervals would require the person to estimate "how much" the space is between the markings. This could lead to more errors.

A ruler with centimeter and inch markings are also typically divided into finer markings--the centimeters into millimeters (10 markings per cm), and the inch divided into 16 markings per inch. Even with these divisions, the millimeter markings are finer or closer to one another than the 1/16-interval markings of the inch. Using the closely-spaced markings would definitely give more accurate measurements than just estimating the spaces between the much larger divisions of the markings in the inches side. 

Shown below is an illustration of a ruler with both centimeter and inch markings (source: http://www.bvda.com/en/rulers-5-cm-2-inch)