Tough PS question

rishianand7 wrote:On Main Street, a number of popular franchise restaurants are located at specific intervals. The Maghreb Sands coffee company maintains one store every 7/3 blocks, starting with the first store at the intersection of Main Street and Broadway. Two other restaurants at the intersection of Main and Broadway have installed locations in the same fashion. The Shock and Raw cafe has a franchise every 9/2 blocks, and Bronto’s Cupcakes appears every 4/3 blocks. If a pedestrian walks down Main Street from the Broadway intersection, how many full blocks would she travel before encountering the next
intersection with all three restaurants?

252
256
504
1512
3024

Let each block = 6 units.

Since (7/3) * 6 = 14, a Maghreb restaurant is located every 14 units.
Since (9/2) * 6 = 27, a Shock and Raw restaurant is located every 27 units.
Since (4/3) * 6 = 8, a Bronto's restaurant is located every 8 units.

Each time the pedestrian travels a multiple of 14 units, 27 units and 8 units, she will pass all 3 restaurants at once.
Thus, the FIRST TIME she will pass all 3 restaurants at once occurs when she has traveled the LEAST COMMON MULTIPLE (LCM) of 14, 27 and 8.

14 = 2*7.
27 = 3*3*3.
8 = 2*2*2.

The LCM must include each prime factor the GREATEST NUMBER OF TIMES it occurs within any single value.
2 occurs THREE times within 8.
3 occurs THREE times within 27.
7 occurs ONE time within 14.
Thus, the LCM must include three 2's, three 3's, and one 7:
2*2*2*3*3*3*7 = 1512.

Thus, 1512 units must be traveled for the pedestrian to pass all 3 restaurants at once.
Since each block = 6 units, we get:
1512/6 = 252 blocks.

The correct answer is A

rishianand7 wrote:On Main Street, a number of popular franchise restaurants are located at specific intervals. The Maghreb Sands coffee company maintains one store every 7/3 blocks, starting with the first store at the intersection of Main Street and Broadway. Two other restaurants at the intersection of Main and Broadway have installed locations in the same fashion. The Shock and Raw cafe has a franchise every 9/2 blocks, and Bronto’s Cupcakes appears every 4/3 blocks. If a pedestrian walks down Main Street from the Broadway intersection, how many full blocks would she travel before encountering the next
intersection with all three restaurants?

252
256
504
1512
3024

An alternate approach is to plug in the answers, which represent the number of blocks that the pedestrian must travel to pass all 3 restaurants at once.
Since the restaurants are located every 7/3 blocks, every 9/2 blocks, and every 4/3 blocks, the correct answer choice must yield an integer value when divided by 7/3, and 9/2, and 4/3.
We must start with the SMALLEST answer choice, since we need to determine the FIRST TIME the pedestrian will pass all 3 restaurants at once,

Answer choice A: 252
Maghreb:
Since a Maghreb restaurant is located every 7/3 blocks, we get:
252/(7/3) = (252*3)/7= 36*3 = 108.
Implication: there are exactly 108 Maghreb restaurants over the course of the first 252 blocks, with the 108th Maghreb restaurant located at the 252nd block.

Shock and Raw:
Since a Shock and Raw is located every 9/2 blocks, we get:
252/(9/2) = (252*2)/9 = 28*3 = 84.
Implication: there are exactly 84 Shock and Raw restaurants over the course of the first 252 blocks, with the 84th Shock and Raw restaurant located at the 252nd block.

Bronto's:
Since a Bronto's restaurant is located every 4/3 blocks, we get:
252/(4/3) = (252*3)/4 = 63*3 = 189.
Implication: there are exactly 189 Bronto's restaurants over the course of the first 252 blocks, with the 189th Bronto's restaurant located at the 252nd block.

Success!
Each restaurant is located at the 252nd block.

The correct answer is A.

Is there a typo in the question?

On Main Street, a number of popular franchise restaurants are located at specific intervals. The Maghreb Sands coffee company maintains one store every 7/3 blocks, starting with the first store at the intersection of Main Street and Boardwalk. Two other restaurants at the intersection of Main and Broadway have installed locations in the same fashion.

Are boardwalk and broadway 2 different streets?

Yes. I've corrected the typo in my posts above in order to avoid confusion.