WORD PROBLEMS ON LCM

LCM :

The smallest among the common multiples of two or more numbers is called their least common multiple ( LCM)

To see different methods to find LCM, please visit the page

"Methods of finding LCM"

Question 1 :

Henry walks 60 cm in one step and Rahim walks 75 cm in one step. Determine the shortest that either of them can measure by an exact number of steps.

How many steps will each take to cover that distance?

Solution :

Distance covered by Hendry in each steps

60, 120, 180, 240, ........

Distance covered by Rahim in each steps

75, 150, 225, ...........

Instead of writing multiples like this, we can find least common multiple by ladder method. 

Least common multiple  =  300

So the required shortest distance covered by Henry and Rahim is 300 cm.

Number of steps taken by Henry :

=  300/60

=  5

Number of steps taken by Rahim :

=  300/75

=  4

So, Hendry and Rahim will take 5 and 4 steps respectively.

Problem 2 :

Three boy start running round a circular track in a sports stadium. They start at the same time and run in the same direction. The first boy takes 10 minutes, the second boy takes 12 minutes and the third boy takes 15 minutes to go round once.

After how many minutes will they be together again ?

How many rounds would each run before they are together again ?

Solution :

Time taken by each boy to run around the circular park

1^st boy  =  10, 20, 30, .....

2^nd boy  =  12, 24, 36, .....

3^rd boy  =  15, 30, 45......

L.C.M  =  2 x 2 x 5 x 3

= 60  

After 60 minutes those three boys will meet together.

Number of rounds by 1st boy  =  60/10

=  6 rounds

Number of rounds by 2nd boy  =  60/12

=  5 rounds

Number of rounds by 2nd boy  =  60/15

=  4 rounds

Problem 3 :

On a highway, banyan trees are planted on one side and neem trees on the other side, starting from the same place. Banyan trees are planted at intervals of 32 meter and neem trees a intervals of 40 meter.

At what distance from the beginning would a banyan tree and neem tree stand together on either side?

Solution :

Distance of Banyan trees planted on one side of the road.

32, 64, 96, .......

Distance of Neem trees planted on other side of the road.

40, 80, 120, .......

LCM  =  2 x 2 x 2 x 4 x 5

  =  160

The banyan tree and neem tree stand together on 160 m apart.

Problem 4 :

What is the smallest number which when divided by 36 and 48 leaves a remainder of 6 ?

Solution :

For this, let us find the lcm of 36 and 48.

LCM  =  2 x 2 x 3 x 3 x 4

  =  144

When we divide 144 by 36 and 48, we will get the remainder 0.

Since we should get the remainder 6, let us add 144 by 6.

Problem 5 :

In a school, the students are arranged in rows for mass drill. If 28, 35 or 42 students are kept in a row and no row is kept incomplete, every time 10 students are left over. Find the least number of students assembled for the parade.

Solution :

28  =  7 x 2 x 2

35  =  7 x 5

42  =  2 x 3 x 7

LCM  =  2 x 2 x 5 x 3 x 7

=  420

Here 420 is exactly divisible by 28, 35 and 42.

But, according to the question every time 10 students are left over. 

=  430 (420 + 10)

So,  the least number of students assembled for the parade is 430.

Problem 6 :

Garlands having 48, 30 or 72 rose flowers in each are to be prepared. Find the minimum number of flowers that would be required to make the garlands. How many garlands of each kind can be prepared?

Solution :

48  =  2 x 2 x 2 x 2 x 3

30  =  2 x 3 x 5

72  =  2 x 2 x 2 x 3 x 3

L.C.M  =  2 x 2 x 2 x 2 x 3 x 3 x 5

  =  720  

Number of flowers to make 1^st type garland :

= 720/48

=  15

Number of flowers to make 2^nd type garland :

=  720/30

=  24

Number of flowers to make 3^rd type garland :

=  720/72

=  10

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