Q) February 14 is celebrated as International Book Giving Day and many countries in the world celebrate this day. Some people in India also started celebrating this day and donated the following number of books of various subjects to a public library : History = 96, Science = 240, Mathematics = 336.

These books have to be arranged in minimum number of stacks such that each stack contains books of only one subject and the number of books on each stack is the same.

Based on the above information, answer the following questions:
(i) How many books are arranged in each stack?
(ii) How many stacks are used to arrange all the Mathematics books?
(iii) (a) Determine the total number of stacks that will be used for arranging all the books.

OR

(iii)(b) If the thickness of each book of History, Science and Mathematics is 1.8 cm, 2.2 cm and 2.5 cm respectively, then find the height of each stack of History, Science and Mathematics books.

Ans:

### STEP BY STEP SOLUTION

(i) Books in each stack:

Let’s first understand the question.

In this question, some hint words are given. If you read it carefully, you would find how to attempt.

Read this line: “These books have to be arranged in minimum number of stacks such that each stack contains books of only one subject and the number of books on each stack is the same.”

Here,

Number of books on each stack is same.. it means, it is a factor and it is common across all books of 3 subjects.

Each stack contains books of only one subject .. means books are not mixed and the factor will be taken out separately from each subject.

Books to be arranged in minimum number of stacks means books in each stack has to be maximum … it means the factor which is common to all the books is highest.

It means we have to take highest common factor or HCF of all the three subjects books.

Let’s find HCF of the 3 numbers:

History: 96 = 2 x 2 × 2 × 2 x 2 x 3

Science: 240 = 2 x 2 x 2 x 2 x 3 × 5

Mathematics : 336 = 2 × 2 x 2 x 2 x 3 × 7

HCF of 96, 240, 336 = 2 x 2 x 2 x 2 x 3 = 48

Hence, there will be 48 books in each stack across each of the 3 subjects.

(ii) Number of stacks for Maths:

Mathematics contains 336 books and we are distributing these books in stacks of 48 books each

Therefore number stacks made of Mathematics books = = 7

Therefore, Mathematics books will be arranged in 6 stacks,

(iii) (a) Total number of stacks for all books:

To calculate total number of stacks, we need to first calculate number of stacks for each subject and then add them all.

Number of stacks for History books: = 2

Number of stacks for Science books: = 5

Number of stacks for Mathematics books: = 7

Total number of stacks across all 3 subjects = Stacks for History + Stacks for Science + Stacks for Mathematics

= 2 + 5 + 7 = 14

Therefore, total 14 stacks will be made across all the books

OR

(iii) (b) height of each stack:

Here we have thickness of each subject’s book (given) and we already know number of books in each stack (=HCF value). Hence, if we multiply number of books with thickness of subject’s book, we will get the height of stack for that subject.

Height of History books stack = 48 x 1.8 = 86.4 cm

Height of Science books stack = 48 x 2.2 = 105.6 cm

Height of Mathematics books stack = 48 x 2.5 = 120 cm

Therefore the height of each stack if History, Science and Mathematics books is 86.4 cm, 105.6 cm and 120 cm respectively.

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