a The amount of time spent traveling is equal to the ratio of the distance traveled to the corresponding rate.
B
b Expand the expression that you found in Part A by 10.
C
c Substitute the given rate into the expression that you found in Part B.
A
a 1/r+1/0.7r=1.7/0.7r
B
b 17/7r
C
c ≈ 0.81 hours, or 48.6 minutes
Practice makes perfect
a We are told that the distance between Jane's house and her grandparents' house is 1 mile. Her rate walking alone to her grandparents' home is represented by r. However, while she is returning home her rate is 70 % of her rate walking when she is alone. Let's write her rate for each situation.
Alone:& r
With Grandfather:& 70 r/100 ⇒ 0.7r
We want to find the total amount of time that Jane spends walking. To do so let's recall that the amount of time t spent traveling is the ratio of the distance d traveled to the corresponding rate r.
Time=Distance/Speed ⇔ t=d/r
Since we know the corresponding rates in terms of r and the distance is 1 mile, we can write the time expressions in terms of distance and rate for each case.
Distance (mi)
Rate (mi/h)
Time (h)
Walking Alone
1
r
t_1=1/r
Walking With Grandfather
1
0.7r
t_2=1/0.7r
Now we will add the expressions t_1 and t_2 to find the total amount of time.
Finally, we can write the following expression for the amount of time Jane spends walking.
1/r+1/0.7r=1.7/0.7r
b We are asked to simplify the expression that we found in Part A. To do so we will get rid of the decimals by expanding the fraction by 10. Let's do it!
1.7 * 10/0.7r * 10 ⇒ 17/7r
Since there are no common factors to cancel out, we can conclude the simplification.
c Jane's rate when she walks alone is given as 3 miles per hour. To find the total time that she spends walking, both to her grandparents' house and back, we will use the expression that we simplified in Part B.
17/7r
Let's substitute r= 3 to calculate the time.
Now let's convert 0.81 hours to minutes by using a conversion factor.
60min/1h
Multiplying 0.81 hours by this conversion factor will convert it to minutes.
