WORD PROBLEMS INVOLVING HCF AND LCM

Problem 1 :

Kamal has 6 cans of regular soda and 15 cans of diet soda. He wants to create some identical refreshment tables that will operate during the American football game. He also doesn't want to have any sodas left over. What is the greatest number of refreshment tables that Kamal can stock?

Solution:

Number of cans Kamal has of regular soda = 6

Number of cans Kamal has of diet soda = 15

Each table should contain same number of soda. So, we find greatest common factor of (6, 15).

= 3

So, he can create identical tables with 3 tables in each.

Problem 2 :

At a family reunion, each of Sana's aunts and uncles is getting photographed once. The aunts are taking pictures in groups of 5 and the uncles are taking pictures in groups of 10. If Sana has the same total number of aunts and uncles, what is the minimum number of aunts that Sana must have?

Solution:

Uncles are taking picture in group of 10

Aunts are taking pictures in group of 5

LCM of 5 and 10 is 10.

Hence, minimum number of aunts that Sana must have is 10.

Problem 3 :

Sapphire and Abe are shelving books at a public library. Sapphire shelves 5 books at a time, whereas Abe shelves 6 at a time. If they end up shelving the same number of books, what is the smallest number of books each could have shelved.

Solution:

Sapphire will shelve the number of books as multiple of 5.

Number of books she is shelving are :

5, 10, 15, 20, ...........

Abe will shelve the number of books as multiple of 6.

6, 12, 18, ...............

LCM of 5 and 6 is 30.

So, 30 is the smallest number of books each could have shelved.

Problem 4 :

For a dinner party, Abraham is creating individual servings of starters. He has 9 carrot sticks and 18 celery sticks. If he wants each serving to be identical, with no food left over, what is the greatest number of servings Abraham can create?

Solution:

Abraham has 9 carrot sticks and 18 Celery sticks.

In order to calculate the greatest number of servings Abraham can create, to find the Greatest Common Factor.

9 = 3 × 3 = 3²

18 = 2 × 3 × 3 = 2 × 3²

GCF = 3² = 9

Therefore, the greatest number of servings Abraham can create is 9 servings.

Problem 5 :

To encourage public transportation, Russom wants to give some friends envelopes with bus tickets and subway tickets in them. If he has 18 bus tickets and 12 subway tickets to split equally among the envelopes, and wants no tickets left over, what is the greatest number of envelopes Russom can make?

Solution:

Greatest Common Factor of 18 and 12 

12 = 1, 2, 3, 4, 6, 12

18 = 1, 2, 3, 6, 9, 18

GCF(18, 12) = 6

So, Russom can make 6 envelopes.

Problem 6 :

Miley and Cole ended up making the same number of biscuits for a bake sale at school, even though Miley made them in batches of 7 biscuits and Cole made them in batches of 11 biscuits. What is the smallest number of biscuits each must have baked?

Solution:

LCM of 7 and 11 is 77.

So, the smallest number of biscuits each has baked is 77.

Problem 7 :

Veronica is making emergency-preparedness kits to share with friends. She has 20 bottles of water and 12 cans of food, which she would like to distribute equally among the kits, with nothing left over. What is the greatest number of kits Veronica can make?

Solution:

20 bottles of water

20 = 2 × 2 × 5

12 cans of food

12 = 2 × 2 × 3

HCF = 2 × 2 = 4

Greatest number of Kits = 4

Each kit have 5 bottles of water and 3 cans of food.

Problem 8 :

Colton has 16 blue marbles and 8 white ones. If he wants to place them in identical groups without any marbles left over, what is the greatest number of groups Colton can make?

Solution:

For blue marbles,

16 = 8 × 2 (8 groups with 2 marbles in everyone of them)

For white marbles,

8 = 4 × 2 (4 groups with 2)

8 + 4 = 12

So, the greatest number of groups Colton can make is 12.

Problem 9 :

This afternoon, Sara noticed that the number of the page assigned for homework is divisible by both 12 and 2. What is the smallest possible page number that could have been assigned?

Solution:

Number of pages assigned for homework = x

The x is the number which divisible by both 2 and 12.

2 = 1 × 2 (prime number)

12 = 1 × 2 × 2 × 3 (composite number)

The smallest possible page number that could have been assigned will be equal to lowest common multiple of 1, 1, 2, 2 and 3.

x = 1 × 1 × 2 × 2 × 3

x = 12

Problem 10 :

Egbert is making trail mix out of 18 bags of nuts and 9 bags of dried fruit. He wants each new portion of trail mix to be identical, containing the same combination of bags of nuts and bags of dried fruit, with no bags left over. What is the greatest number of portions of trail mix Egbert can make?

Solution:

Egbert is making trail mix out of 18 bags of nuts and 9 bags of dried fruit.

HCF of (18, 9) = 9

Egbert can make 9 trail mix out.

And the number of bags of nuts in 1 trail mix is,

= 18/9

= 2

Number of a bag of dried fruits in 1 trail mix,

= 9/9

= 1

Hence, the greatest number of portions of trail mix Egbert can make is 18.