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GMAT Club Olympics 2025 (Day 3): Alice and Bob are on an escalator

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harshnaicker

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Total = 80 steps
speed of escalator = s steps/second

ALICE
Speed = s1 steps per second (upward)
s1-s=2
s1 = s+2

BOB
Speed = s2 steps per second (downward)
s2 - (-s) = 3
s2 = 3 - s

Given relation:
4*80/(s+2) = 80/(3-s)
s = 2
s1 = 4
s2 = 1

Time in seconds after which Alice and Bob meet: 80/(s1+s2) = 16 seconds
Number of steps Alice would have walked before reaching the top:
Time taken = 80/4 = 20 seconds
Number of steps = Speed relative to escalator * time
= 2*20 = 40

Answers are:
t=16
n=40

Bunuel

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Alice and Bob are on an escalator with 80 steps from bottom to top. The escalator moves upward at a constant rate of s steps per second.

Alice starts from the bottom and walks upward at a constant rate of 2 steps per second relative to the escalator. Bob starts from the top and walks downward at a constant rate of 3 steps per second relative to the escalator.

Bob takes four times as long to reach the bottom as Alice takes to reach the top.

Select for t the time in seconds after which Alice and Bob meet, and select for n the number of steps Alice would have walked on the escalator by the time she reaches the top. Make only two selections, one in each column.

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03 Jul 2025, 00:58

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D=80 Steps
Speed of A=Sa=2+s , Ta=80/2+s
Speed of B=Sb=3-s , Tb=80/3-s
Tb=4Ta
80/3-S=320/2+S
S=2 seconds
Ta=80/4=20
Total steps for A :2*20=40
A and B to meet : 80/2+s+3-s =80/5=16s

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speed of escalator = $s$ steps/sec
speed of Alice going up =$ s + 2 $ steps/sec, since she is moving in same direction as escalator
speed of Bob going down =$ 3 - s $ steps/sec, since he is moving in the opposite direction of the escalator

Time taken by B = 4 * Time taken by A
$\frac {80}{3-s} = 4 * \frac{80} {s+2}$
$\\
s+2 = 12 - 4s\\
5s = 10 \\
$
s= 2 steps/sec

Effective speed of Alice = 4 steps/sec
Effective speed of Bob = 1 step/sec

Time when Alice and Bob will meet is t
$4t + t = 80$
t = 16 seconds

Time taken by alice to reach the top = 80/4 = 20 seconds.
Alice covers 2 steps per second, in 20 seconds, she will cover 40 steps.
n = 40
Alice is travelling at 2 steps/sec and escalator is also at 2 steps/sec, so each Alice and escalator cover 40 steps each - in total covering 80 steps at their effective 4 steps/sec.

Select t =16, n = 40

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First we need to find the speed of the escalator.

Let speed of Escalator =x
Time taken by Alice = y

2 steps per second relative to the escalator. Bob starts from the top and walks downward at a constant rate of 3 steps per second relative to the escalator.

Bob takes four times as long to reach the bottom as Alice takes to reach the top.

(2+x) * y=80
(3-x) * 4y= 80

Solve to get
Y=20

plug in y in the equation to get x=2

now we know the speed of the escalator is 2 steps per second.

Alice climbed 2*20 steps= 40=n

In 16 seconds Alice climbs 64 steps and bob climbs 16 steps. And thats where they meet and high-five.
T=16

Bunuel

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Alice :
Distance = 80 steps
Speed = 2 + s steps/ second (Since she is moving in the direction of escalator so both speeds of escalator and Alice would be added)
or we can say for every 2 steps Alice take, escalator additionally moves her upwards "s" steps ahead
Time = 80 / (2 + s) ---------- Using formula T = D/ S

Bob :
Distance = 80 steps
Speed = 3 - s steps/ second (Since he is moving in the opposite direction of escalator so both speeds of escalator and Bob would be subtracted)
or we can say for every 3 steps Bob take downwards, escalator moves him upwards in opposite direction by "s" steps
Time = 80 / (3 - s) ---------- Using formula T = D/ S

Now as per question: "Bob takes four times as long to reach the bottom as Alice takes to reach the top"
So, 4 * (80/ 2 + s) = (80/ 3 - s)
320/ (2+s) = 80/ (3-s)
s = 2

Now, we know the speed of escalator as 2 steps/ second

That means for every second Alice moves 4 steps upwards and Bob moves only 1 step downwards

We can form an equation here to find out after how many seconds both will meet and that would be, let t = seconds after both meet
So, 4t = 80-t (Meaning for every 1 second Alice travels 4 steps towards Bob and Bob travels 1 step less than 80 towards Alice)
So solving the equation : 4t = 80-t
5t = 80
t = 16 seconds

So, after 16 seconds both of them will meet

Now let's find the number of steps Alice would have travelled till she reaches the top
Distance = 80 steps
Speed = 2 + 2 = 4 steps
Time = 80/ 4 = 20 seconds

So in 20 seconds Alice alone would have travelled 20 * 2 steps = 40 steps and rest of the 40 steps were covered by escalator.

Bunuel

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Alice starts from the bottom and walks upward at a constant rate of 2 steps per second relative to the escalator
=> Rate of steps of Alice: 2 + s (steps/second)
Bob starts from the top and walks downward at a constant rate of 3 steps per second relative to the escalator.
=> Rate of steps of Bob: 3 - s (steps/second)

To finish the escalator, Time of Bob = 4 Time Alice
=> 80/(3-s) = 4*80/(2+s) => s = 2
=> Rate of steps of Alice = 2+2 = 4
Rate of steps of Bob : 3 - 2 = 1

The time in seconds after which Alice and Bob meet (t): 4*t + 1*t = 80 => 5*t = 80 => t=16
The number of steps Alice would have walked on the escalator by the time she reaches the top (n): n = 80/4 = 20

Answer: t = 16, n = 20

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Total = 80 Steps
Speed of escalator = c = s (UP)
Let walking speeds Alice and Bob be A & B respectively
Let relative speeds of Alice and Bob be A' & B' respectively
Moving Up, A = 2 sps.; A' = (2 + s) sps. (Same direction as the escalator)
Moving down, B = 3 sps.; B' = (3 - s) sps., s<2 (Opposite direction as the escalator)
From the details,
80/(3-s) = 4 * 80/(2 + s)
2 + s = 12 - 4s
s = 2 sps. ~ (1)

Now,
The question asks for the time taken for both Alice and Bob to meet; That is, it would be at a particular point and the time taken to reach this point will be the same for both.
Let it x be the distance between the point at which they meet and the bottom of the elevator and t the time taken.
t = x/(3 - s) = (80 - x)/(2 + s)
5x = 240 - 80 x 2; (s = 2) from (1)
x = 16 steps ~ (1)

Substituting in either L.H.S. or R.H.S. we get, x/(3 - s) = 16 seconds

Now to find time taken by Alice to reach to the top, we have Time = 80/(2 + s) = 80/4 = 20 seconds.
To find the number of steps, n = Time * speed = 20 * 2 = 40 steps

Therefore, t the time in seconds after which Alice and Bob meet = 16 seconds
n the number of steps Alice would have walked on the escalator by the time she reaches the top = 40 steps

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03 Jul 2025, 03:12

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Escalator: 80 steps, moves up at s s s steps/sec.
Alice walks up at 2 steps/sec (relative to escalator).
Bob walks down at 3 steps/sec (relative to escalator).

Alice's speed (relative to ground): s+2 steps/sec (up)
Bob's speed (relative to ground): 3−s steps/sec (down)

Time for Alice to reach the top: TA=80/(s+2)
Time for Bob to reach the bottom: TB=80/(3−s)
Given: TB=4TA
Set up the equation:
80/(3−s)=4*80/(s+2)
12-4*s=s+2
5*s = 10
s=2 m/sec (Elevator speed)

Alice’s Time to the Top
TA=80/(2+2)=80/4=20 seconds

Number of Steps Alice Walks
She walks at 2 steps/sec for 20 seconds:
n=2×20=40 steps

Step 5: When Do Alice and Bob Meet?

Alice goes up at 2+2=4 steps/sec.
Bob goes down at 3−2=1 step/sec.

Let t = time when they meet.

Alice covers 4t steps from the bottom.
Bob covers 1t steps from the top, so from the bottom he is at 80−t.

Set equal:
4t=80−t ⟹ 5t=80 ⟹ t=16 seconds

Final Answers

Time after which Alice and Bob meet (t): 16 seconds
Number of steps Alice walks to the top (n) : 40 steps

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Bunuel

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Total Steps =80
Speed of Alice = 2+s
Speed of Bob = 3-s

time to reach top and bottom by Alice and bob respectively => (2+s)/80, (3-s)/80
80/(2+s)*4= 80/(3-s) => 12-4s=2+s ===> s=2:

the time in seconds after which Alice and Bob Meet = 80/(4+1)=16 secs, time taken by alice to reach to= 80/4 = 20 secs, Number of steps by Alice in 20 secs = 20*2= 40 steps

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Given:

Alice walks up at 2 steps/second relative to escalator
Bob walks down at 3 steps/second relative to escalator
Escalator moves up at s steps/second
Bob takes 4 times as long as Alice

Finding escalator speed:

Since both Alice and the escalator are moving in the same direction (upward), their speeds ADD together
Alice's speed upward = (2 + s) steps/second
Time for Alice to reach top = 80/(2 + s). -- time=distance/speed

Since Bob walks down while the escalator moves up, they're moving in OPPOSITE directions so subtract
Bob's actual speed downward = (3 - s) steps/second
Time for Bob to reach bottom = 80/(3 - s)

Since Bob takes 4 times as long: 80/(3 - s) = 4 × 80/(2 + s)

Simplifying:

80/(3 - s) = 320/(2 + s)
80(2 + s) = 320(3 - s)
160 + 80s = 960 - 320s
400s = 800
s = 2 steps/second

Finding when they meet:
With s = 2:

Alice moves up at 2 + 2 = 4 steps/second
Bob moves down at 3 - 2 = 1 step/second

At time t:

Alice's position from bottom: 4t
Bob's position from bottom: 80 - t

They meet when:
4t = 80 - t
5t = 80
t = 16 seconds

Finding Alice's walking distance:
Alice reaches the top in: 80/4 = 20 seconds
In 20 seconds, she walks: 2 × 20 = 40 steps relative to escalator n = 40 steps

Answers:

t = 16 seconds
n = 40 steps

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Let escalator speed = s
Bob’s time : 80 / (3 − s)
Alice’s time : 80 / (s + 2)

Since Bob takes 4x time as Alice: 80 / (3 − s) = 4 * (80 / (s + 2)) -> s = 2.
Alice’s actual speed upward : s + 2 = 4 step/sec.
Bob’s actual speed downward : 3 − s = 1 step/sec.

Together they complete the 80-step gap at 4 + 1 = 5 step/sec or in (80/5) = 16 seconds.
Alice reaches the top in 80/4 = 20 secs and she walks (2*20) = 40 steps.

Answer:
t = 16
n = 40

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You have to draw the diagram properly.

First, write down the speed of Alice as 2+s considering escalator speed. (Moving in same direction, hence addition of speeds.)
Similarly, write down the Bob's speed as 3-s considering escalator speed. (Moving in opposite direction.)

To get the time by they meet, let's use meeting/catch-up formula : 2+s - 3+s = 80/t =>t =16secs

To get how much Alice would have climbed up, we need d = s*T.

Let's find s first, 2+s = d/T & 3-s = d/4T, using this we will get s =2.
Using s value, T = 80/(2+2) =20secs.

=> d = 2*20 = 40secs

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Speeds are Alice s+2 and Bob 3-s
Finding speed of elevator with info about times:

80/(3-s) = 4*(80/(s+2))

s+2=12-4s
5s=10
s=2

speed of Alice is 4 and speed of Bob is 1

They meet in the time t=80/(4+1)=16

Alice takes 80/4=20 seconds to reach the top.
Taking relative speed of Alice, which is 2, in these 20 seconds, she walks 2*20=40 steps

The right answers are: t=16 and n=40

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Escalator's speed $=$ $s$ steps/second
Alice's speed relative to the escalator $=$ $2$ steps/second
Alice's actual speed $=$ $2+s$ steps/second (adding $s$ because Alice is moving in the same direction as the escalator. Upwards)
Bob's speed relative to the escalator $=$ $3$ steps/second
Bob's actual speed $=$ $3-s$ steps/second (subtracting $s$ because Bob is moving in the opposite direction as the escalator. Downwards)

Making an equation for the gap -> The gap between them was $80$ steps. The gap closed at a speed of $(2+s)+(3-s) = 5$ steps/second
The time in which the gap closes is also the time when Alice and Bob meet.
$80 = 5(t)$
$t = 16$ seconds

Bob takes four times as long to reach the bottom as Alice takes to reach the top -> $T_B = 4T_A$
We know both of them travelled $80$ steps in total and we know their respective speeds. We can find $T_A$ and $T_B$

$T_A = \frac{80}{2+s}$ and $T_B = \frac{80}{3-s}$

$\frac{80}{3-s} = 4(\frac{80}{2+s})$

$s = 2$

Alice's actual speed is $2+s = 2+2 = 4$ steps/second
Total time taken by Alice $= \frac{80}{4} = 20$ seconds
Now, the number of steps Alice walked is different from the number of steps she travelled. The escalator was helping her at a rate of $2$ steps/second. Her speed relative to the escalator is $2$ steps/second. That means in $20$ seconds, she took $2*20 = 40$ steps.
$n = 40$

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03 Jul 2025, 04:40

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D= 80 (steps from bottom to top)

Given, Alice's speed = s+2 (as Alice and the escalator are moving in the same direction)
Bob's speed = 3-s (as Bob and the escalator are moving in opposite direction)

Now, 80/(3-s) = 4[80/(s+2)]

solve this to get s=2

Alice's speed - s+2 = 4 steps/sec
Bob's speed - 3-s = 1 step/sec

1. Time in seconds after which Alice and Bob meet = length of the escalator/relative speed = 80/(4+1) (As Bob and Alice walk in opposite direction, we need to sum their speed)

=80/5 = 16sec

2. Alice's speed relative to the escalator = 2 steps/sec
Time Alice was on the escalator to reach the top = 80/4 = 20 seconds.
Therefore, n = 2*20 = 40

16 and 40

Bunuel

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t=16
n=40

Let's assume escalator's speed is s.
so,
Alice's speed = 2 + s
Bob's speed = 3 - s

According to the question

80/(3-s) = 4*80/(2+s)
s = 2 steps per second

t is the time in seconds after which Alice and Bob meet so,
Alice's dist. + Bob's dist. = 80 steps
4*t + 1*t = 80
t = 16 sec

Since Alice's speed = escalator's speed, they would both cover have the total distance
so n = 40 steps

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The answer should be t= 16
and n= 40
As per the question,
4 * (80 / (s+2) ) = 80 / (3-s)
s= 2 m/s

Bunuel

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03 Jul 2025, 05:54

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escalator speed $- s$

speed of Alice $= s+2$ & speed of Bob $=3-s$

Time taken by Bob is 4 times Alice to cover 80 steps

$(s+2)x=80$ & $(3-s)4x=80$

$(s+2)x= (3-s)4x$

Speed of escalator, $s=2$

Time taken to meet, t:

$t$ is same for both of them when they meet.

Lets take the steps travelled by Alice at $t$ seconds as $x$

$\frac{x}{(s+2)}=\frac{(80-x)}{(3-s)}$

$\frac{x}{(4)}=(80-x)$

$x=64$

$t=16$

Steps $n$ Alice walked:

Since $s=$ Alice's relative speed=2 steps per second, she covered half the steps by walking relative to escalator

$n=40$

t: 16
n: 40

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03 Jul 2025, 05:57

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Escalator = 80 steps

Let A = Alice and B = Bob

Velocity A: 2 steps / second + escalator velocity (Up)

Velocity B: 3 steps/ seconds - escalator velocity (down)

Let s = escalator velocity.

When they meet, we will have: distA + distB = totalDist

They will meet in t seconds, so:

t(3-s) + t(2+s) = 80
5t = 80
t = 16

We know that “ Bob takes four times as long to reach the bottom as Alice takes to reach the top.”, then:

(2+s) = 4(3-s)
2+s = 12 - 4s
5s = 10
s = 2

The escalator velocity is 2. A “real velocity” will be 2+2 =4 steps/second, where she only really walks 2, the half. So, from the 80 steps, she would walk 40.

Final answers:
t = 16
n = 40

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745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards