the principal at lincoln middle school asked the seventh graders and eighth graders whether they would…

Explanation:

Step1: Find the number of eighth - graders who prefer the natural history museum

We know that the total number of eighth - graders is 68 and the number of eighth - graders who prefer the art museum is 29. Let the number of eighth - graders who prefer the natural history museum be $x$. Then $29 + x=68$. Solving for $x$, we get $x = 68 - 29=39$.

Step2: Find the total number of students who prefer the art museum

We know that the total number of students who prefer the natural history museum is 74 and the total number of students is 127. Let the total number of students who prefer the art museum be $y$. Then $y+74 = 127$, so $y=127 - 74 = 53$.

Step3: Find the number of seventh - graders who prefer the art museum

We know that the total number of students who prefer the art museum is 53 and the number of eighth - graders who prefer the art museum is 29. Let the number of seventh - graders who prefer the art museum be $z$. Then $z + 29=53$, so $z=53 - 29 = 24$.

Step4: Find the total number of seventh - graders

We know that the number of seventh - graders who prefer the art museum is 24 and the number of seventh - graders who prefer the natural history museum is 35. Let the total number of seventh - graders be $w$. Then $w=24 + 35=59$.

Step5: Calculate the percentage of eighth - graders who prefer the natural history museum

The number of eighth - graders who prefer the natural history museum is 39 and the total number of eighth - graders is 68. The percentage $p$ is given by the formula $p=\frac{39}{68}\times100%$. $p=\frac{3900}{68}% \approx 57.4%$

The completed table:

	Art museum	Natural history museum	Total
Seventh graders	24	35	59
Eighth graders	29	39	68
Total	53	74	127

Answer:

The percentage of eighth - graders who prefer the natural history museum is approximately 57.4%