Question

Melissa drove to the mountains last weekend. There was heavy traffic on the way there, and the trip took 10 hours. When Melissa drove home, there was no traffic and the trip only took 7 hours. If her average rate was 18 miles per hour faster on the trip home, how far away does Melissa live from the mountains?
Do not do any rounding.

Answer 1

Okay let's say that the mountains are at a distance say x away from home...

Answer 2

In other words she had to travel x units distance... is that fine?

Answer 3

yes

Answer 4

Now, she took 10 hours for going from home to mountains.. So speed = distance/time

That means.. speed = x/10

Answer 5

Now, when she returned, she had to travel x miles again... this time she took 7 hours.. So, speed will be x/7

Answer 6

Any doubt till this step?

Answer 7

nope

Answer 8

Great!

Answer 9

Now, the question says... "If her average rate was 18 miles per hour faster on the trip home"

That means, \(\frac{x}{7} = \frac{x}{10} + 18\)

Answer 10

Now, can you solve this equation?

Answer 11

hmmm how do I solve that

Answer 12

First let's simplify this a bit..

Answer 13

\(\frac{x}{7} = \frac{x}{10} + 18\)

First let's multiply 10 on both sides, so that we can get rid of that 10 in denominator

Answer 14

\(10\frac{x}{7} = x + 18*10\)

Now, we multiply on both sides by 7 to get rid of that 7 in denominator..

Answer 15

We get, \(7*10*x = 7*x + 7*18*10\)

Answer 16

I will simplify this:

\(70x = 7x + 1260\)

Answer 17

Any problem till this step?

Answer 18

nope

Answer 19

Great!

Now, let's bring 7x to the left hand side, we get:

\(70x - 7x = 1260\)
\(\implies 63x = 1260\)
\(\implies x = \frac{1260}{63}\)

Answer 20

so 20?

Answer 21

Perfect!!

Answer 22

So, she lives 20 miles away from mountains...

Answer 23

It was wrong :(

Answer 24

Wait... let me check ..

Answer 25

Damn! I did a calculation mistake... The equation was:

\(\frac{x}{7} = \frac{x}{10} + 18\)

\(\implies 10x = 7x + 1260\)

\(\implies 3x = 1260\)

\(\implies x = \frac{1260}{3} = 420\)

Answer 26

So, the answer should have been 420 miles...

Answer 27

that was right. lol

Answer 28

Can you help me with 2-3 more? :o

Answer 29

Yeah sure

Answer 30

Ok it's
Two trains leave the station at the same time, one heading east and the other west. The eastbound train travels 14 miles per hour slower than the westbound train. If the two trains are 750 miles apart after 5 hours, what is the rate of the eastbound train?
Do not do any rounding.

Answer 31

Now, both trains are travelling in opposite directions so their distance will increase with time...

Answer 32

mhm

Answer 33

Let's assume the speed of eastward train = v1

And the speed of westward train = v2

Answer 34

Now, it is given that: "The eastbound train travels 14 miles per hour slower than the westbound train."

Thus, v1 = v2 - 14

Answer 35

We also know that:

Distance = Speed * Time

Thus, distance traveled by Eastward train:

Distance1 = v1 * 5

Similarly, distance traveled by Westward train:

Distance2 = v2*5

Answer 36

mhhmmm

Answer 37

Now, total distance between trains after 5 hours will be:

Distance1 + Distance2 = 750

=> v1*5 + v2*5 = 750

We also know v1 = v2 - 14

Put this value of v1 in first equation

Answer 38

ok

Answer 39

So, we get:

(v2 - 14)*5 + v2 * 5 = 750

=> 5* v2 - 5 * 14 + v2 * 5 = 750

=> 10*v2 - 70 = 750

=> 10 * v2 = 750 + 70

So, what will be the value of v2 ?

Answer 40

82?

Answer 41

Good :)

Answer 42

Now, v1 = v2 - 14

What will be v1?

Answer 43

ummm 14?

Answer 44

No no ... see you know v2 = 82.. just subtract 14 from it

Answer 45

OOHH

Answer 46

68

Answer 47

Great!

Answer 48

so its 68? :o

Answer 49

Yeps! So the speed of eastward train is 68 miles/hour

Answer 50

I believe I have 2 more problems left. :D

Answer 51

next one is,

A car passes a landmark on a highway traveling at a constant rate of 45 kilometers per hour. Two hours later, a second car passes the same landmark traveling in the same direction at 70 kilometers per hour. How much time after the second car passes the landmark will it overtake the first car?

Answer 52

See... this time both cars are travelling in same direction so distance between them will decrease

Answer 53

i see

Answer 54

Now, second car will overtake first car when distance between them becomes zero

Answer 55

mhm

Answer 56

Now, distance traveled by first car = d1 = speed * time = 45 * t

Answer 57

I see

Answer 58

Now, second car traveled the same landmark 2 hours after the first car

Answer 59

Thus, during this time, first car traveled an extra distance of 45*2 = 90 km

Answer 60

Any doubt?

Answer 61

nopp

Answer 62

Great!

Answer 63

Now, in same time (t hours), car 2 will travel a distance = 70*t

Answer 64

To overtake the first car, this distance must be equal to 90 + 45*t .. so we get:

70*t = 90 + 45*t

Answer 65

Any doubt?

Answer 66

no :p

Answer 67

OKay ... So we can solve this equation to find out t...

Answer 68

so um 75t = 135? right?

Answer 69

No no ... see...

70t = 90 + 45t

=> 70t - 45t = 90

=> 25t = 90

Calculate t from here

Answer 70

um 3.6?

Answer 71

Very good :D

Answer 72

hopefully this is the last one..
Ivanna drove to the mountains last weekend. There was heavy traffic on the way there, and the trip took 8 hours. When Ivanna drove home, there was no traffic and the trip only took
6 hours. If her average rate was 16 miles per hour faster on the trip home, how far away does Ivanna live from the mountains?

Answer 73

It's exactly like the one we did before the last one...

Answer 74

First, assume the distance between home and mountains = x

Then, you know that speed = distance/ time

So, on her way to mountains, her speed will be x/8 = v1

And on her way back to home, her speed will be x/6 = v2

Answer 75

"her average rate was 16 miles per hour faster on the trip home"

This means that v2 = 16 + v2

=> x/6 = 16 + x/8

Answer 76

mhmm

Answer 77

Now can you solve this equation?

Answer 78

I multiply the 16 and 8 right?

Answer 79

Exactly!

Answer 80

No no not by 16 and 8 but by 6 and 8

Answer 81

Notice that denominators are 6 and 8...

Answer 82

ohh yes

Answer 83

Can you tell me what will you obtain?

Answer 84

would it be 3?