word problem

Question

toby is running a 26 mile marathon. He runs the first part of the marathon at an average speed of 7 mph. He then decides to increase his speed to an average of 7.8 mph. In doing this, it takes him three hours and 30 minutes to run the marathon. How many miles did he run at each speed?

Answer 1

We know rate * time = distance

So if t₁ is the time of the initial run,

and t₂ is the time of the faster run:

 

7t₁   = d₁ (d₁ being distance ran at first speed)

And (7.8)t₂  = d₂ (d₂ being the distance ran at second speed)

So if total distance is 26 miles,

and total time is 3.5 hours, then we know:

(formula 1) d₁ + d₂ =26

(formula 2) t₁ + t₂  = 3.5

 

Substituting rate * time for the distances  d1 and d2 of formula 1, we get:

 

(formula 3) 7t₁ + (7.8)t₂ = 26

 

As we learned in Algebra, if we have two formulas with the same variables,  we can find the place where both are true. We do this by combining the formulas to eliminate the extra variable and solve the problem:

In this set, we can do that easily by multiply both sides of formula 2 by -7

formula 2 : t₁ + t₂ = 3.5
formula 2 * -7 :   -7(t₁ + t₂)  = -7 (3.5)

-7t₁ -7t₂ = -24.5 (new formula 2)

7t₁ + (7.8)t₂ = 26 (formula 3)

 

Combining these (adding them together) the t1 values go away, and I can solve a single variable equation easily.

 

.8t₂ = 1.5

t₂ = 1.5 / .8

t₂ = 1.875 hours

So if  t1 + t2 = 3.5, then

 

t1 = 3.5 - 1.875

t1 = 1.625

 

And distance is

d2 = 7.8 * 1.875 = 14.625 miles 

d1 - 7 * 1.625  =  11.375 miles

Proof :

d1 + d2 = 26

11.375 + 14.625 = 26 Check!

t1 + t2 = 3.5
1.625 + 1.875 = 3.5 Check!

 

(I did this with combining formulae for two reasons: 1 - good to remind you that the things you learned previously are not dead ends but doorways! And 2 - the math of combining the numbers was easier! :)

Answer 2

t = d / r

 

The distance in the first part is: d.

The time spent is: d / 7

 

The distance in the second part is: 26 - d

The time spent is: (26 - d) / 7.8

 

The total time (you add) is

 

d / 7 + (26 - d) / 7.8 = 3.5      multiply both sides by LCM 54.6

7.8d + 7(26 - d) = 3.5(54.6)   distribute, multiply

7.8d + 182 - 7d =  191.1        combine

0.8d + 183 = 191.1                subtract 183 from both sides

0.8d = 8.1                              divide both sides by 0.8

d = 10.125

 

He ran 10.125 miles in the first part.

 

Now you determine how far he ran in the second.