Question

use the following equalities to derive conversion factors for the next two problems
2.54 cm = 1 inch
10 mm = 1 cm
100 cm = 1 m
1000 m = 1 km

Explanation:

Since the problem statement only provides the equalities to derive conversion factors but doesn't present a specific question (like converting a certain quantity from one unit to another), it's incomplete.

If you want to convert a length from one unit to another, for example, let's say converting inches to millimeters:

Step1: Convert inches to cm

Given \( 2.54 \, \text{cm} = 1 \, \text{inch} \), so to convert \( x \) inches to cm, we use the conversion factor \( \frac{2.54 \, \text{cm}}{1 \, \text{inch}} \). So \( x \) inches \( = x\times2.54 \, \text{cm} \).

Step2: Convert cm to mm

Given \( 10 \, \text{mm} = 1 \, \text{cm} \), so to convert \( y \) cm to mm, we use the conversion factor \( \frac{10 \, \text{mm}}{1 \, \text{cm}} \). So \( y \) cm \( = y\times10 \, \text{mm} \).

Step3: Combine the two steps

If we have \( x \) inches, first convert to cm: \( x\times2.54 \, \text{cm} \), then convert to mm: \( (x\times2.54)\times10 \, \text{mm}=x\times25.4 \, \text{mm} \). So the conversion factor from inches to mm is \( 25.4 \, \text{mm/inch} \).

But since your problem doesn't have a specific conversion task, please provide the specific question (like convert 5 inches to km, or 10 mm to inches etc.) so that I can solve it properly.

Answer:

Step1: Convert inches to cm

Given \( 2.54 \, \text{cm} = 1 \, \text{inch} \), so to convert \( x \) inches to cm, we use the conversion factor \( \frac{2.54 \, \text{cm}}{1 \, \text{inch}} \). So \( x \) inches \( = x\times2.54 \, \text{cm} \).

Step2: Convert cm to mm

Given \( 10 \, \text{mm} = 1 \, \text{cm} \), so to convert \( y \) cm to mm, we use the conversion factor \( \frac{10 \, \text{mm}}{1 \, \text{cm}} \). So \( y \) cm \( = y\times10 \, \text{mm} \).

Step3: Combine the two steps

If we have \( x \) inches, first convert to cm: \( x\times2.54 \, \text{cm} \), then convert to mm: \( (x\times2.54)\times10 \, \text{mm}=x\times25.4 \, \text{mm} \). So the conversion factor from inches to mm is \( 25.4 \, \text{mm/inch} \).

But since your problem doesn't have a specific conversion task, please provide the specific question (like convert 5 inches to km, or 10 mm to inches etc.) so that I can solve it properly.