Let's solve this problem step-by-step.
1. Step 1: Define the variables.
- Let the rate of the faster cyclist be [tex]\( r \)[/tex] km/h.
- Therefore, the rate of the slower cyclist is [tex]\( r - 9 \)[/tex] km/h.
2. Step 2: Write the relationship for distance traveled.
- Distance = Rate × Time.
- The faster cyclist travels in 3 hours at a rate of [tex]\( r \)[/tex] km/h, so the distance covered by the faster cyclist is [tex]\( 3r \)[/tex] kilometers.
- The slower cyclist travels in 3 hours at a rate of [tex]\( r - 9 \)[/tex] km/h, so the distance covered by the slower cyclist is [tex]\( 3(r - 9) \)[/tex] kilometers.
3. Step 3: Set up the equation.
- The total distance apart after 3 hours is the sum of the distances traveled by both cyclists:
[tex]\[ 3r + 3(r - 9) = 159 \][/tex]
4. Step 4: Simplify the equation.
[tex]\[ 3r + 3r - 27 = 159 \][/tex]
[tex]\[ 6r - 27 = 159 \][/tex]
5. Step 5: Solve for [tex]\( r \)[/tex].
[tex]\[ 6r = 159 + 27 \][/tex]
[tex]\[ 6r = 186 \][/tex]
[tex]\[ r = \frac{186}{6} \][/tex]
[tex]\[ r = 31 \][/tex]
So, the rate of the faster cyclist is [tex]\( 31 \)[/tex] km/h.
6. Step 6: Find the rate of the slower cyclist.
- The slower cyclist's rate is [tex]\( r - 9 \)[/tex]:
[tex]\[ r - 9 = 31 - 9 \][/tex]
[tex]\[ \text{Rate of the slower cyclist} = 22 \text{ km/h} \][/tex]
Therefore, the rates of the cyclists are:
- Faster cyclist: 31 km/h
- Slower cyclist: 22 km/h
