Question

a construction worker is building a wall. the wall is 15.5 yd long. he needs to place a stud every 18 in. the first stud has already been placed at one end of the wall. how many additional studs are needed, if the last stud will be placed at the other end of the wall? first fill in the blanks on the left side of the equation using three of the ratios shown. then write your answer on the right side of the equation. ratios: \\(\frac{1\text{ ft}}{12\text{ in}}\\), \\(\frac{12\text{ in}}{1\text{ ft}}\\), \\(\frac{3\text{ ft}}{1\text{ yd}}\\), \\(\frac{1\text{ yd}}{3\text{ ft}}\\), \\(\frac{18\text{ in}}{1\text{ stud}}\\), \\(\frac{1\text{ stud}}{18\text{ in}}\\) \\(\frac{15.5\text{ yd}}{1} \times \square \times \square \times \square = \square \text{ studs}

Explanation:

Step1: Convert yards to feet

We use the ratio that cancels yards: $\frac{3\ \text{ft}}{1\ \text{yd}}$
$\frac{15.5\ \text{yd}}{1} \times \frac{3\ \text{ft}}{1\ \text{yd}}$

Step2: Convert feet to inches

We use the ratio that cancels feet: $\frac{12\ \text{in}}{1\ \text{ft}}$
$\frac{15.5\ \text{yd}}{1} \times \frac{3\ \text{ft}}{1\ \text{yd}} \times \frac{12\ \text{in}}{1\ \text{ft}}$

Step3: Convert inches to number of studs

We use the ratio that cancels inches: $\frac{1\ \text{stud}}{18\ \text{in}}$
$\frac{15.5\ \text{yd}}{1} \times \frac{3\ \text{ft}}{1\ \text{yd}} \times \frac{12\ \text{in}}{1\ \text{ft}} \times \frac{1\ \text{stud}}{18\ \text{in}}$

Step4: Calculate total studs, subtract 1

First compute the total studs:
$$15.5 \times 3 \times 12 \times \frac{1}{18} = 15.5 \times \frac{36}{18} = 15.5 \times 2 = 31\ \text{studs}$$
Subtract the already placed stud: $31 - 1 = 30$

Answer:

The filled equation is:
$\frac{15.5\ \text{yd}}{1} \times \frac{3\ \text{ft}}{1\ \text{yd}} \times \frac{12\ \text{in}}{1\ \text{ft}} \times \frac{1\ \text{stud}}{18\ \text{in}} = 31\ \text{total studs}$
Additional studs needed: 30