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.
Answer: (i) 48 books are arranged in each stack. (ii) 7 stacks are used to arrange all the Mathematics books. (iii) (a) The total number of stacks used is 14. (iii) (b) The height of each stack of History books is 86.4 cm, Science books is 105.6 cm, and Mathematics books is 120.0 cm.
Step-by-step solution
Step 1: Find the HCF of the number of books
To arrange the books in the minimum number of stacks, with each stack containing books of only one subject and the same number of books, we need to find the highest common factor (HCF) of the number of books for each subject. This HCF will represent the number of books in each stack. We start by finding the prime factorization of 96, 240, and 336.
Step 2: Calculate the HCF
The HCF is found by taking the lowest power of the common prime factors. In this case, the common prime factors are 2 and 3. The lowest power of 2 is and the lowest power of 3 is . Multiplying these gives us the HCF.
Step 3: Answer (i): Books per stack
The HCF we calculated, 48, represents the maximum number of books that can be placed in each stack while ensuring all stacks have the same number of books and contain only one subject.
Step 4: Answer (ii): Stacks for Mathematics books
To find the number of stacks required for Mathematics books, we divide the total number of Mathematics books by the number of books per stack.
Step 5: Answer (iii) (a): Total number of stacks
To find the total number of stacks, we calculate the number of stacks for each subject (History, Science, and Mathematics) by dividing the total books of that subject by the number of books per stack, and then sum these values.
Step 6: Answer (iii) (b): Height of each stack
If the thickness of each book is given, the height of each stack is found by multiplying the number of books in a stack (which is 48) by the thickness of a single book for that subject.
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