Ann and Bea leave X-ville at the same time and travel towards Y-ville,

Target question: When they meet, how far has Bea traveled?

Statement 1: Ann’s speed is 30 kilometers per hour greater than Bea’s speed
We can see that is not sufficient if we examine some EXTREME CASES:
Case a: Ann's speed = 30.00000001 kilometers per hour, and Bea's speed = 0.00000001 kilometers per hour. In this case, Bea travels almost 0 kilometers
Case b: Ann's speed = 40 kilometers per hour, and Bea's speed = 10 kilometers per hour. In this case, Bea travels more than 0 kilometers
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Ann’s speed is twice Bea’s speed
One option here is to test a bunch of cases to see what happens. If we do this, we'll find that we keep getting the same answer to the target question

Alternatively, we can use some algebra:
Let B = the distance Bea traveled
Let R = Bea's speed.

NOTE: the total distance from Townville to Villageton and then BACK TO Townville = 140 kilometers.

So, 140 - B = the distance Ann traveled
And 2R = Ann's speed (since her speed is TWICE Bea's speed)

From here, let's create a WORD EQUATION that uses distance and speed.
How about: Ann's travel time = Bea's travel time

Time = distance/rate, so we get:
(140 - B)/2R = B/R
Cross multiply to get: (B)(2R) = (R)(140 - B)
Expand: 2BR = 140R - BR
Add BR to both sides: 3BR = 140R
Divide both sides by R to get: 3B = 140
Divide both sides by 3 to get: B = 140/3
In other words, Bea traveled 140/3 kilometers
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer:

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Ann and Bea leave X-ville at the same time and travel towards Y-ville, which is 70 kilometers away. Their individual speeds are constant, but Ann’s speed is greater than Bea’s speed. Upon reaching Y-ville, Ann immediately turns around and drives toward X-ville until she meets Bea. When they meet, how far has Bea traveled?

1) Ann’s speed is 30 kilometers per hour greater than Bea’s speed

Common sense: B's speed 20 kmph A s speed 50 kmph

B's speed 5kmph A's speed 35kmph

these will yield diff. result.
NS.

2) Ann’s speed is twice Bea’s speed

A-2x, A travelled 2xt.
B- x, B travelled xt.

2xt + xt = 70 + 70

xt = 140/3.

Suff.

B.

Hi
Distance travelled by Ann on her journey beck from Y to X – c.
Total distance travelled by Ann d_1 = 70 + c
Total distance travelled by Bea d_2 = 70 – c
1) Ann’s speed is 30 kilometers per hour greater than Bea’s speed
Bea’s speed = v
Ann’s speed = v + 30
(70 + c )/ (v + 30) = (70 – c) / v
70c + cv = (70 – c)(v + 30)
c(v + 15) = 1050
We can derive many values for v and c. Insufficient.
2) Ann’s speed is twice Bea’s speed
(70 + c) / 2v = (70 – c) / v
70v + cv = 140v – 2cv
3cv = 70v
c = 70/3
Distance travelled by Bea: 70 – 70/3 = 140/3 Sufficient.
Answer B.

Let \(a\) be speed of Ann & \(b\) be the speed of Bea.

Let the distance traveled by Bea be \(D\). Hence distance traveled by Ann is \(70 + D\)

Since the time required is same, we get \(\frac{(70+D)}{a}\) = \(\frac{D}{b}\)

Statement 1: \(a = b + 30\)

Substituting this in above equation we get an equation with 2 unknowns.

Hence Statement 1 alone is Not Sufficient.

Statement 2: \(a = 2b\)

Substituting this, we can find the value of \(D\).

Hence Statement 2 is sufficient.


Answer B.


Thanks,
GyM
_________________

Let Speeds of Ann and Bea be\(S_{A}\) and \(S_{B}\) respectively ----- (1)

Let Time of Ann and Bea be \(T_{A}\) and \(T_{B}\) respectively ------- (2)

Let distance covered by Ann after completing 70Km is D.

Therefore,
Total distance covered by Ann = 70 + D --------- (3)
Total distance covered by Bea = 70 - D ---------- (4) (As Bea is D kms far from completing 70 kms)

As we know both meet at same time there we can equate their times, i.e.,

\(T_{A}\) = \(T_{B}\) -------- (5)

Now we know \(S = \frac{D}{T}\) ------------ (6)

Using (1), (2), (3), (4), (5), (6), we get ->

\(\frac{70 + D}{S_{A}} = \frac{70 - D}{S_{B}}\) ---------- (7)

We need to determine (70 - D)



Here we determine,

\(S_{A}\) = 30 + \(S_{B}\)

Substituting same in equation (7), we get

\(\frac{70 + D}{30 + S_{B}} = \frac{70 - D}{S_{B}}\)

This equation cannot be used to determine value of D or (70 - D).

Hence Insufficient.



Here we determine,

\(S_{A}\) = 2 * \(S_{B}\)

Substituting same in equation (7), we get

\(\frac{70 + D}{2 * S_{B}} = \frac{70 - D}{S_{B}}\)

Now here we can cancel \(S_{B}\) and determine the value of D, henceforth (70 - D).

Hence Sufficient.

Hence Option B
_________________

Ann and Bea leave X-ville at the same time and travel towards Y-ville, which is 70 kilometers away. Their individual speeds are constant, but Ann’s speed is greater than Bea’s speed. Upon reaching Y-ville, Ann immediately turns around and drives toward X-ville until she meets Bea. When they meet, how far has Bea traveled?

1) Ann’s speed is 30 kilometers per hour greater than Bea’s speed

distance = rate * time

Clearly insufficient. Unable to solve for distance with 2 variables.

2) Ann’s speed is twice Bea’s speed

let 2x = Ann's speed
let x = Bea's speed

2x * time = 70
x * time = 70

Add the two equations together: 3x * time = 140
distance = 140/3

Statement 2 is sufficient.

Answer is B.
_________________

When A and B finally meet, they will have together combined to cover the 70 km distance TWICE

Combined distance of A and B = 140 km

A and B both depart from the same location at the same instant of time. Therefore, A and B will have traveled for the SAME TIME when A turns around at Y-ville to come meet B.

Concept: when the Time traveled is constant, the Speed at which each travels will be DIRECTLY PROPORTIONAL to the DISTANCE each covers

Or

Ratio of Speeds = Ratio of Distance Covered

To determine how much of the 140 km Bea covers, if we can determine the relative speed of the two then we can determine how much of the total distance Bea covered with respect to Anne

S1: speed of A = 30 + speed of B

Case 1: B = 10 mph
A = 40 mph

Ratio of speeds = A : B = 40 : 10 = 4 : 1

Bea will cover (1/5) of the total combined distance of 140 while Anne will cover (4 / 5)

Distance Bea covers = 140 * (1/5) = 28 miles

Case 2: B = 20
A = 50

Ratio of Speeds = A : B = 5 : 2

Bea covers (2 /7) of the 140 combined distance = (140) (2 / 7) = 40 miles

S1 not sufficient

S2: Anne speed = 2 (Bea Speed)

Ratio of speeds = A : B = 2 : 1

Bea covers (1 / 3) of the 140 distance

S2 is sufficient alone
Bea distance = (140 / 3)

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