find the two speeds

Question

A driver drove 70 miles at a slower speed and 300 miles at a speed 40 miles per hour faster. if the time spent driving the faster speed was twice that spent on the slower speed, find the two speeds during the trip.

Answer 1

Hi Ndndnd,

 

Sanhita answered this question correctly, but I thought I would explain it in a different way, so that you have an alternate explanation. There are several steps to this interesting question. The first is to represent each of the two speed scenarios with an equation. Here is the list of terms that I will use in my algebraic expressions/equations, with their respective meanings:

s=slower speed.

 

Speed is a unit of distance/unit time. For instance, two kinds of speed measurements are kilometers/hour (km/hr) and miles/hour (mi/hr)

 

t= time spent driving the slower speed (in my example, I will use hours)

 

Here are the two expressions:

 

1) (70 mi)/t = s

 

On the left-hand side of the equal sign is distance traveled (70 mi)/ time (t), which is an expression for speed (s). I used brackets for clarity. The expression should be read as 70 miles over t (the whole expression) is equal to speed (s). I will use this method for the rest of the expressions in this answer.

 

2) (300 mi)/(2t) = (s + 40)

 

In your question, it says that the faster speed is 40 miles per hour (mph), which is a speed) faster than the slower speed, which I designated as "s." The question also says that the amount of time spent driving the faster speed was twice (2 times) the time spent traveling the slower speed.

 

Step 2:

 

You have two unknowns. The best thing to do when you have two algebraic unknowns is to make both equations equal to one of the variables and then to set the equations equal to each other. To do this. you will subtract 40 from both sides of the second equation to get, [(300 mi)/2t]-40 = s . Again, the brackets are to prevent algebraic errors and the different brackets do not have special meanings for this expression. You should read this expression as 30 miles over 2t, minus 40 mph. Not 30 mi/(2t-40)!!! Please make sure that you are reading this carefully and ask questions if you have them.

Step 3:

 

Now that you have two expressions, both equal to s,

 

(70 mi)/t = s    and    [(300 mi)/2t]-40 = s   

 

you can set the left sides of both expressions equal to each other, since they are both equal to the same thing (s), like so:

 

(70 mi)/t= [(300 mi)/2t]-40 

 

Now you have just ONE unknown! The magic of algebra!

 

Step 4:

Now you need to solve the expression for t. When you do this, you will get:

 

80 mi/ t = 40 mph

 

t= 2 hours

 

Step 5:

Now, you plug t back into equation 1 in step 1 and solve for speed, like so:

(70 mi)/ 2 hrs = s

 

35 mph = s

 

Since we know that the faster speed is equal to the slower speed plus 40 mph, the faster speed is 35 mph + 40 mph, 75 mph.

 

Hence, the two speeds are 35 miles per hour and 75 miles per hour.

Answer 2

The slower speed being Smiles per hour and time spent at slower speed being T hours,

ST= 70 .... (1)

(S+40)*2T=300

=>ST + 40T=150.... Dividing both sides by 2

=>70+40T=150 ..... Substituting ST from (1)

=>40T=80 .... Subtracting 70 on from both sides

=>T=2.... Dividing both sides by 40

Substituting T=2 in (1), S*2= 70

=>S=35 .... Dividing both sides by 2

Hence both the speeds are 35 miles per hour and 75 miles per hour.