Question

Ravish runs a book shop at school of Math, Gurgaon. He received 480 chemistry books, 192 physics
books and 672 Mathematics books of class XI. He wishes to average these books in minimum numbers of
stacks such that each stack consists of the books on only one subject and the number of books in each
stack is the same.

Answer 1

Appropriate Question

Ravish runs a book shop at school of Math, Gurgaon. He received 480 chemistry books, 192 physics books and 672 Mathematics books of class XI. He wishes to average these books in minimum numbers of stacks such that each stack consists of the books on only one subject and the number of books in each stack is the same. Find the total number of stacks.

$\large\underline{\sf{Solution-}}$

Given that,

• Ravish runs a book shop at school of Math, Gurgaon. He received 480 chemistry books, 192 physics books and 672 Mathematics books of class XI.

• He wishes to average these books in minimum numbers of stacks such that each stack consists of the books on only one subject and the number of books in each stack is the same.

So, Number of books in each stack = HCF of 480, 192, 672

To find the HCF of 480, 192, 672, we use the concept of Prime factorization.

So,

$\red{\rm :\longmapsto\:Prime \: factors \: of \: 480 = {2}^{5} \times 3 \times 5}$

$\green{\rm :\longmapsto\:Prime \: factors \: of \: 192= {2}^{6} \times 3}$

$\blue{\rm :\longmapsto\:Prime \: factors \: of \: 672= {2}^{5} \times 3 \times 7}$

So,

$\purple{\rm\implies \:HCF(480,192,672) = {2}^{5} = 32}$

So, it means number of books in each stack is 32.

$\red{\rm :\longmapsto\:Number \: of \: stacks \: for \: Chemistry =\dfrac{480}{32} = 15}$

$\green{\rm :\longmapsto\:Number \: of \: stacks \: for \: Physics =\dfrac{192}{32} = 6}$

$\pink{\rm :\longmapsto\:Number \: of \: stacks \: for \: Mathematics =\dfrac{672}{32} = 21}$

So,

$\purple{\rm\implies \:Total \: Number \: of \: stacks = 15 + 6 + 21 = 42}$

Answer 2

Given: Ravish runs a book shop at school of Math, Gurgaon. He received 480 chemistry books, 192 physics books and 672 Mathematics books of class XI.

To find: Find the total number of stacks.

Solution: Know that, Ravish wants to average the books in minimum numbers of stacks such that each stack consists of the books on only one subject and the number of books in each stack is the same.

Therefore,

Find the number of books in each stack.

Number of books in each stack $=$ HCF of $480, 192, 672$.

$\[\begin{array}{l}480 = {2^5} \times 3 \times 5\\192 = {2^6} \times 3\\672 = {2^5} \times 3 \times 7\\ \Rightarrow HCF\left( {480,192,672} \right) = {2^5}\\ \Rightarrow HCF\left( {480,192,672} \right) = 32\end{array}\]$

Understand that, the number of books in each stack is 32.

Find the number of stacks for each of the subjects.

Number of stacks for Chemistry is,

$\[\begin{array}{l} = \frac{{480}}{{32}}\\ = 15\end{array}\]$

Number of stacks for Physics is,

$\[\begin{array}{l} = \frac{{192}}{{32}}\\ = 6\end{array}\]$

Number of stacks for Mathematics is,

$\[\begin{array}{l} = \frac{{672}}{{32}}\\ = 21\end{array}\]$

Find the total number of stacks.

Total number of stacks$=15+6+21$

                                     $=42$

Hence, the total number of stacks is 42.