Three ships take 10, 12, and 21 days, respectively, to leave and return to Newar

Question

Three ships take 10, 12, and 21 days, respectively, to leave and return to Newark. They leave at the same time and take continuous trips to and from Newark. In how many days will all three ships be back in Newark on the same day?

Step I. Read the problem. What are you trying to find?

Step 2. Think and Estimate.

When you have a number of days that is a multiple of 10, where will you find the ship taking the 10-day trip? Why?

What if the number of days is a multiple of 12? Where is the ship with the 12-day trip?

Where are those two ships after 120 days? Why?

What is the name, and correct mathematical notation for the number of days we need to find that sees all three ships in the port of Newark on the same day?

Explanation / Answer

We are trying to find the time period in which all three ships return to Newark on the same day.

When the number of days is a multiple of 10 the ship taking the 10-day trip will return to Newark from where it started it's trip.

When the number of days is a multiple of 12 the ship taking the 12-day trip will return to Newark from where it started the trip.

Since 120 is a multiple of 12 and 10 both so both the ships return to Newark from where it started the trip on the same day.

The name of the quantity that depicts the number of days in which all three ships come on the port of Newark on the same day is least common multiple commonly denoted by LCM.

Therefore LCM(10,12,21) is required to be found.

10=2*5

12=2*2*3

21=3*7

Hence least common multiple of 10,12,21=(10,12,21)=2*3*2*5*7=420

Hence in 420 days all three ships will return to port of Newark on the same day after taking continuous trips to and fro Newark.

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