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QuestionSamir uses 2122 \frac{1}{2}" screws through a 78\frac{7}{8}" shelf and 12\frac{1}{2}" drywall. Find how deep each screw goes into the stud.

Studdy Solution

STEP 1

Assumptions1. The length of the screw is 1 \frac{1}{} " . The thickness of the shelf is 78\frac{7}{8} "
3. The thickness of the drywall is 1\frac{1}{} "
4. The screw will pass through both the shelf and the drywall before reaching the stud

STEP 2

First, we need to calculate the total thickness of the shelf and the drywall. We can do this by adding the thickness of the shelf and the drywall.
Total thickness=Thickness of shelf+Thickness of drywallTotal\, thickness = Thickness\, of\, shelf + Thickness\, of\, drywall

STEP 3

Now, plug in the given values for the thickness of the shelf and the drywall to calculate the total thickness.
Total thickness=78"+12"Total\, thickness = \frac{7}{8} " + \frac{1}{2} "

STEP 4

Calculate the total thickness.
Total thickness=78"+12"=1516"Total\, thickness = \frac{7}{8} " + \frac{1}{2} " = \frac{15}{16} "

STEP 5

Now that we have the total thickness, we can find how far each screw will rest in the stud. This can be found by subtracting the total thickness from the length of the screw.
Length in stud=Length of screw−Total thicknessLength\, in\, stud = Length\, of\, screw - Total\, thickness

STEP 6

Plug in the values for the length of the screw and the total thickness to calculate the length in the stud.
Length in stud=212"−1516"Length\, in\, stud =2 \frac{1}{2} " - \frac{15}{16} "

STEP 7

Convert the mixed number to an improper fraction to simplify the calculation.
212"=52"2 \frac{1}{2} " = \frac{5}{2} " Length in stud=52"−1516"Length\, in\, stud = \frac{5}{2} " - \frac{15}{16} "

STEP 8

To subtract these fractions, we need to have a common denominator. The least common denominator of2 and16 is16. So, we will convert the fraction with denominator2 to have a denominator of16.
52"=4016"\frac{5}{2} " = \frac{40}{16} " Length in stud=4016"−1516"Length\, in\, stud = \frac{40}{16} " - \frac{15}{16} "

STEP 9

Subtract the fractions to find the length in the stud.
Length in stud=4016"−1516"=2516"Length\, in\, stud = \frac{40}{16} " - \frac{15}{16} " = \frac{25}{16} "

STEP 10

Convert the improper fraction back to a mixed number to get the final answer.
Length in stud=916"Length\, in\, stud = \frac{9}{16} " Each screw will rest 916\frac{9}{16} " in the stud.

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