A 

44one fourth14​" 

drywall screw is countersunk​ (i.e., screwed until it is below the surface of the​ drywall) by 

two sixteenths216​" 

to fasten a 

four eighths48​" 

sheet of drywall to the ceiling joist.​ (a) How much of the screw extends into the​ joist? (b) How much of the screw is within the​ drywall?

Question content area bottom

Part 1

​(a) 

enter your response here​" 

​(Type an​ integer, fraction, or mixed​ number.)

Let's break down the problem step by step:

### Given:
- The total length of the screw is \( 4 \frac{1}{4} \) inches.
- The screw is countersunk by \( \frac{2}{16} \) inches into the drywall.
- The drywall thickness is \( \frac{4}{8} \) inches.

### Part (a): How much of the screw extends into the joist?

To find out how much of the screw extends into the joist, subtract the amount of screw that is within the drywall (including the countersink) from the total length of the screw.

1. Convert \( 4 \frac{1}{4} \) into an improper fraction:
   \[
   4 \frac{1}{4} = \frac{17}{4} \text{ inches}
   \]

2. Add the thickness of the drywall and the countersunk portion:
   \[
   \text{Drywall thickness} = \frac{4}{8} = \frac{1}{2} \text{ inches}
   \]
   \[
   \text{Countersink depth} = \frac{2}{16} = \frac{1}{8} \text{ inches}
   \]
   \[
   \text{Total depth in drywall} = \frac{1}{2} + \frac{1}{8} = \frac{4}{8} + \frac{1}{8} = \frac{5}{8} \text{ inches}
   \]

3. Subtract the depth in drywall from the total length of the screw:
   \[
   \text{Length in joist} = \frac{17}{4} - \frac{5}{8}
   \]

4. Convert \(\frac{17}{4}\) to have a common denominator with \(\frac{5}{8}\):
   \[
   \frac{17}{4} = \frac{34}{8}
   \]

5. Perform the subtraction:
   \[
   \frac{34}{8} - \frac{5}{8} = \frac{29}{8}
   \]

Thus, the length of the screw that extends into the joist is \( \frac{29}{8} \) inches or \( 3 \frac{5}{8} \) inches.

### Part (b): How much of the screw is within the drywall?

This is simply the length of the screw that is in the drywall, including the countersink.

From above, we already calculated:
   \[
   \text{Total depth in drywall} = \frac{5}{8} \text{ inches}
   \]

So, the length of the screw within the drywall is \( \frac{5}{8} \) inches.

### Summary:
(a) The length of the screw extending into the joist is \( \frac{29}{8} \) inches or \( 3 \frac{5}{8} \) inches.

(b) The length of the screw within the drywall is \( \frac{5}{8} \) inches.

Would you like further details or explanations?

Here are five related questions:
1. How do you convert a mixed number to an improper fraction?
2. What is the process for adding and subtracting fractions with different denominators?
3. How do you interpret the physical meaning of a countersunk screw in a practical situation?
4. What is the significance of knowing how much of a screw extends into a material?
5. How do you simplify fractions after performing arithmetic operations?

**Tip:** Always check your calculations to ensure that the measurements are accurate and reflect the real-world scenario described.

Explore how to calculate the length of a drywall screw extending into a joist and within drywall. This problem involves fractions and practical applications of construction math. Suitable for Grades 6-8 students.

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