# At 9:00 on Saturday morning, two bicyclists heading in opposite directions pass each other on a bicycle path.The bicyclist heading north is riding 6 km/hour faster than the bicyclist heading south. At 10:15, they are 42.5 km apart. What are their rates?

Jun 17, 2017

$\text{North bicycle"=20" } \frac{k m}{h o u r}$

$\text{South bicycle"=14" } \frac{k m}{h o u r}$

#### Explanation:

We need to set up equations based on the known information. I always write down everything the question gives me and try to link the information with logic or established formulas. I automatically think of speed=distance/time in this case. Also, units are important, so decide what you want to use and stick with them for all calculations. I will use km and hours.

Okies, the total distance travelled by both cyclists is 42.5 km. This means that the sum of the distances will be 42.5 km. Let's use ${d}_{N}$ as the distance travelled by the cyclist going north, and ${d}_{S}$ for the cyclist going south:

${d}_{N} + {d}_{S} = 42.5 \text{ } k m$ (1)

Rearrange the speed formula to give the distance travelled as speed multiplied by time. We will use $N$ for the speed of the northward bound cyclist, and $S$ for the other cyclist. The time is from 9:00 to 10:15, so t = 1.25 hours. Now, we can write expressions for the distance travelled by each cyclist:

${d}_{S} = 1.25 S$ (2)

Remember that the northward bound cyclist is going 6 km/hour faster, so:

$N = S + 6$

$\Rightarrow {d}_{N} = 1.25 N = 1.25 \left(S + 6\right) = 1.25 S + 7.5$ (3)

At this point, we want to solve equation (1), so we need to replace either ${d}_{N}$ with ${d}_{S}$, or vice versa, so that there is only one variable to solve. This means we need a second expression. We do this by evaluating (3) - (2):

${d}_{N} = 1.25 S + 7.5$ (3)
- ${d}_{S} = 1.25 S$ (2)

$\Rightarrow {d}_{N} - {d}_{S} = 1.25 S - 1.25 S + 7.5$

$\Rightarrow {d}_{N} - {d}_{S} = 7.5 \Rightarrow {d}_{N} = {d}_{S} + 7.5$ (4)

Now substitute (4) into (1) to solve for ${d}_{S}$:

${d}_{N} + {d}_{S} = 42.5 \Rightarrow \left({d}_{S} + 7.5\right) + {d}_{S} = 42.5$

$\Rightarrow 2 {d}_{S} + 7.5 = 42.5 \Rightarrow 2 {d}_{S} = 35 \Rightarrow {d}_{S} = 17.5 \text{ } k m$

${d}_{N} = 42.5 - 17.5 = 25 \text{ } k m$

Now we know the distances travelled by both so we use speed = distance/time to get the rates:

$N = \frac{25}{1.25} = 20 \text{ } \frac{k m}{h o u r}$

$S = \frac{17.5}{1.25} = 14 \text{ } \frac{k m}{h o u r}$