GMAT Club Olympics 2024 (Day 10): A car is traveling on a straight

|-----------100 miles------------10hours--------------|
|----------x mph----------|------------y mph----------|
T1= 50/x                         T2= 50/y

50/x+50/y= 10
y= 5x/x-5

We see all answer choices are integers, which means x and y both will be integers. Plugging in values for x to solve for y:

x=3
y= 15/-2 NA

x=5
y= 25/0 NA

x=7
y= 35/2 NA

x=10
y= 50/5 = 10 (Given x>=y)
Hence, x=y=10
 

The car travels for a total of 10 hours to cover 100 miles.
The car travels at a constant rate of x mph for exactly half of the travel time.
The car travels at a constant rate of y mph for the remaining half of the travel time.
It is known that x is greater than or equal to y.
Now, let's break it down:

In the first 5 hours, the car goes at speed x, so it travels 5×x miles.
In the next 5 hours, the car goes at speed y, so it travels 5×y miles.
Altogether, the distance is 5x+5y miles, and we know this is 100 miles.
Simplifying this, we get x+y=20.
From the table, the values of x and y must add up to 20, with x being at least as large as y.
Looking at the table: If x=10, and y=10, then x+y=20. This works!If x=15, and y=5, then
x+y=20. This also works!
So, the values that fit are:x=10 and y=10
These values add up to 20, matching the conditions

This is quite a straightforward question and easy to solve once we understand the wording. 

It is stated that a car travels for half of the time at a constant rate of X mph and the other half at a constant rate of Y mph. Since the total time for the travel was 10 hours, we can conclude that the car drove for 5 hours at X mph and then another 5 hours at Y mph. Moreover, the overall distance for the journey is 100 miles. 

This is means that the average speed for the journey was 100 miles/10 hours = 10 mph. 

We can also express the distance as 

100 = 5*X + 5*Y
or 
X + Y = 20

It states in the question that X > Y OR X = Y   *sorry I do not know how to do the greater than or equal symbol*

So X must be equal to or greater than 10. There is only one answer choice that is equal to or greater than 10 whilst being lower than 20 and that is 10. 

So X and Y are both 10 mph. 
­

 
­Total distance = 100 miles
Total time = 10 hours
Constant speed for first 5 hours = \(x\) mph
Constant speed for remaining 5 hours = \(y\) mph
\(x >= y\)

Distance = Speed * Time
Average speed for entire journey = 100/10 = 10 mph
(Total distance / Total time = Average speed)

\(=> (x*5 + y*5)/10 = 10\)
\(=> x + y = 20\)

As we know \(x >= y \)

1st column: 10
2nd column: 10

­According to the given information,
5*x + 5*y=100
x+y=20
Given , x>=y
so the values of x and y  among the given options that satisfy the given condition are x=10 and y=10

Time:     5 hours                      5 hours
­|------------------------------|-------------------------|
Speed:      x                                 y


D=SxT
100 = 5x + 5y
20 = x + y

Using each option:
A. Let's take x=3.
Using above equation, y = 20-3 = 17
17 is not available in options.

B.  Let's take x=5.
Using above equation, y = 20-5 = 15
15 is not available in options.

C. Let's take x=7.
Using above equation, y = 20-7 = 13
13 is not available in options.

D. Let's take x=10.
Using above equation, y = 20-10 = 10
10 is available in options.

E. Let's take x=20.
Using above equation, y = 20-20 = 0
0 is not available in options.

F.  Let's take x=50.
As 50>20, we can not take x=50.

Only D fits the bill and hence is correct.

­A car is traveling on a straight stretch of a road at a constant rate of x mph, for exactly one half of the travel time. The remaining time, the car travels at a constant rate of y mph. The car does not stop and takes exactly 10 hours to complete the journey of 100 miles. It is also known that x is greater than or equal to y. Select values of x and y that are jointly consistent with the information provided. Make only two selections, one in each column.

Now, we know that Speed=D/t

Total time=10 hrs, half time=5hrs.

For first half, Speed=x mph; t=5hrs
For second half, Speed=y mph, t=5hrs

Total D=100
Therefore 5x+5y=100
or x+y=20.

From the options, only x=10, y=10 fulfill this criteria. Hence 10,10.

 
­We know that the total travel time is 10 hours (so half of that is 5 hours) and that the total distance is 100 miles. We also know that x is greater than or equal to y.

That means that:
\(D_x=5*x\) and \(D_y=5*y\)

\(D_x+D_y=100\)
\(5*x+5*y=100\)
\(x+y=20\)

Looking at the possible options of 3, 5, 7, 10, 20, 50, the only possible way to get 20 is with 10 and 10, which fits the inequality.

Therefore,
x = 10
y = 10
 ­

­We can find the average speed of the trip as total distance / total time = 100/10 = 10 miles/hr

Also, as per formula for average speed, when x mph and y mph is the speed for each half of travel, the average speed is the harmonic mean of the two.

2/ (1/x +1/y) = 10

=> 1/x + 1/y = 1/5

The only two values of x and y from the options that satisfy the above equation are if x and y both equal 10.

Therefore, x=10 and y=10

 

total time=10 hrs
took 5, 5 hrs for each with speed x mph , y mph

5x+5y=100
=> x+y=20, x>=y

(10,10) holds true

D, D

A car is traveling on a straight stretch of a road at a constant rate of x mph, for exactly one half of the travel time. The remaining time, the car travels at a constant rate of y mph. The car does not stop and takes exactly 10 hours to complete the journey of 100 miles. It is also known that x is greater than or equal to y.

Answer choices for values of x and y: 3, 5, 7, 10, 20, 50

Ans.­ Given: 

x≥y.................(i)

Total time t=10 hours

Total distance = 100 miles
x mph * (t/2) + y mph * (t/2) = 100 miles
5x+5y=100
x+y=20.....................(ii)


From the given answer choices,
only x=10 and y=10 satisfy both (i) and (ii)


 

Let T= Total time = 10 Hours
Let D = Total Distance = 100

Initial Speed = x
Speed later = y

We know x >= y

and both speed are maintained for same time

Hence 5 hours for x and 5 hours for y

Hence Total distance = Speed (x) * Time for (x) + Speed (y) * Time for (y)
100 = 5x + 5y
Therefore 20 = x + y

When we plug in the given options while maintaining x>= y

We get x=10 = y

Total distance = 100 miles
Total time = 10 hours

Car takes T/2 or 5 hours with speed x miles/hr
Distance travelled = 5x

Similalarly, for rest 5 hours, speed is y miles/hr
Distance travelled = 5y

5x + 5y = 100
x + y = 20
x>=y

x=10
y=10

We can analyze the information provided to find jointly consistent values for x and y (speeds during the two halves) using the given list:

Given

1. Total Distance:100 miles
2. Total Time:10 hours
3. Speed Relationship:** x ≥ y (x is greater than or equal to y)
4. Available Speeds (list):** 3, 5, 7, 10, 20, 50 mph

Key Observation: The car spends half the time at speed x and the other half at speed y. Therefore, the average speed must be the total distance divided by the total time:

5x + 5y = 100

x + y = 20.

The only pair of number in the list which meet this equation is x =10 =y­

­A car is traveling on a straight stretch of a road at a constant rate of x mph, for exactly one-half of the travel time. For the remaining time, the car travels at a constant rate of y mph. The car does not stop and takes exactly 10 hours to complete the journey of 100 miles. It is also known that x is greater than or equal to y. Select values of x and y that are jointly consistent with the information provided. Make only two selections, one in each column.



Solution: Given that,

• The car doesn't stop at any point during the 10-hour trip
• x ≥ y

Total Journey distance = 100 miles
Total time taken to cover 100 miles = 10 hours
Speed during the first 5 hours of the journey = x
Speed during the second 5 hours of the journey = y

We need to find x and y

Since, Distance = Speed * Time
d1 = x * 5
and d2 = y * 5
Also, d1 + d2 = 100
So, 5x + 5y = 100
x + y = 20
Since x ≥ y
x + y ≥ 2y
2y ≤ 20
y ≤ 10

Let's analyze given options:
y cannot be = 20 or greater as x + y = 20 and x ≥ y
If y = 10, then x = 20 - 10 = 10 (Option given)
If y = 7, then x = 20 - 7 = 13 (Option not given)
If y = 5, then x = 20 - 5 = 15 (Option not given)
If y = 3, then x = 20 - 3 = 17 (Option not given)

Thus x = 10 and y = 10
 ­

­We are given, D = 100 miles and T = 10 hours.

Now, the car travels for x mph for 5 hours (T/2) and for y mph for the next 5 hours.
We know, Speed = Distance/ Time
Hence, distance travelled in each 5 hour lap when:
Speed is x mph = 5x
Speed is y mph = 5y

5x + 5y = 100
x + y = 20 ......... (1)

Now, as per the question, either x>y or x=y. ......... (2)
Also, the car doesn't stop. .......... (3)

And as per options, only 10 is such value which is fulfilling the conditions (1), (2) and (3) above for both x and y.

Hence, x=y=10.

time of journey =10 hour
distance journey= 100 miles
The car does not stop

Constant rate of x mph, for exactly one half of the travel time
x*(10/2)=5x= distance travelled during 5 hours

The remaining time, the car travels at a constant rate of y mph.
Remaining time 10-5= 5
y*5= 5y=distance travelled during the last 5 hours

5x+5y=100
5(x+y)=100
x+y=20

as x is greater than or equal to y
the only option which we can add up 20 is 10 for x and 10 for y

10 +10 = 20

x=10
y=10