# $\text{Rate} \times \text{Time} = \text{Distance}$ problem

To drive to the supermarket, Mable drives for $$m$$ miles, then drives $$12$$ miles per hour faster for the remaining $$\frac{4m}3$$ miles. The amount of time Mable spent driving at each of the two speeds was equal. What was Mable’s average speed during her drive to the supermarket, in miles per hour?

I get $$m=rt~\text{and}~\frac{4m}3=(r+12)t.$$ Next, WLOG let $$m=60.$$ I then find $$r=36$$ after some manipulation. Since the total distance is $$m+\frac{4m}{3}=60+80=140,$$ and the average speed is $$\frac{240}{140},$$ I get the answer $$\frac{120}{7}.$$ However, this is incorrect. I'm just a beginner at $$D=R\cdot T$$ problems, so please forgive me, since a lot fo my work is most likely incorrect. Help?

• Could you please explain how did you find 240?
– cgss
Sep 9 '20 at 21:09

Indicating with $$v$$ the uknown speed, we have that

$$t=\frac{m}{r}=\frac{\frac 43m}{r+12} \implies \frac13 mr=12m \implies r=36 \, (mph)$$

which corresponds to your result, then the average speed is given by

$$\bar r=\frac{m+\frac 43 m}{2 t}=\frac76 r=42\, (mph)$$

• Thanks! I do now understand this problem! So, I got the speed, but messed up the end. I've done some problems like these where it asks for the avg speed, and I always mess it up! Once again, thanks! Sep 9 '20 at 21:14
• @LaraLincy You are welcome! Probably you need some more practice to become more confident with it. Bye
– user
Sep 9 '20 at 21:17

WLOG suppose Mable's total journey time is 2 hours. Then in the 2nd hour she travels 12mi further than in the 1st hour. The distances travelled are in the ratio 3:4 so she travelled a total of 7 x 12mi in 2 hours making the average speed 42 mph.