Question

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 to a public library: History = 96, Science = 240, Biography = 336. These books have to be stacked in such a way that all books are stored topic wise and each stack contains equal number of books.

Q1) What is the maximum number of books in each stack?

a) 48
b)3360
c)1680
d)672

Q2) What will be the decimal expansion of a rational number formed by taking the number of Science books as numerator and number Biography books as denominator?

a) Terminating
b)Non-terminating non-repeating c)Non-termating repeating
d)None of these​

Answer 1

Answer:

acbd is correct order mate.

Answer 2

1) The maximum number of books in each stack = 48 [ The correct option is a) 48 ]

2) The decimal expansion of a rational number formed by taking the number of Science books as numerator and number Biography books as denominator is Non-termating repeating [ The correct option is c) Non-termating repeating ]

Given :

• 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 to a public library: History = 96, Science = 240, Biography = 336

• These books have to be stacked in such a way that all books are stored topic wise and each stack contains equal number of books.

To find :

1) The maximum number of books in each stack is

a) 48

b) 3360

c) 1680

d) 672

2) The decimal expansion of a rational number formed by taking the number of Science books as numerator and number Biography books as denominator is

a) Terminating

b) Non-terminating non-repeating

c) Non-termating repeating

d) None of these

Concept :

$\displaystyle\sf{Fraction = \frac{Numerator}{Denominator} }$

A fraction is said to be terminating if prime factorisation of the denominator contains only prime factors 2 and 5

If the denominator is of the form

$\sf{Denominator = {2}^{m} \times {5}^{n} }$

Then the fraction terminates after N decimal places

Where N = max { m , n }

Solution :

Step 1 of 2 :

Calculate maximum number of books in each stack

Some people in India also started celebrating this day and donated the following number of books to a public library

History = 96

Science = 240

Biography = 336

These books have to be stacked in such a way that all books are stored topic wise and each stack contains equal number of books

The maximum number of books in each stack will be HCF of 96 , 240 , 336

Now, prime factorisation of 96 , 240 , 336 We get

96 = 2 × 2 × 2 × 2 × 2 × 3

240 = 2 × 2 × 2 × 2 × 3 × 5

336 = 2 × 2 × 2 × 2 × 3 × 7

∴ The maximum number of books in each stack

= HCF of 96 , 240 , 336

= 2 × 2 × 2 × 2 × 3

= 48

Hence the correct option is a) 48

Step 2 of 2 :

Find nature of the rational number

The rational number formed by taking the number of Science books as numerator and number Biography books as denominator

Numerator = Number of Science books = 240

Denominator = Number Biography books = 336

∴ The Rational number formed

$\displaystyle\sf{ = \frac{Numerator}{Denominator} }$

$\displaystyle \sf = \frac{240}{336}$

$\displaystyle \sf = \frac{5}{7}$

We know that a rational number is said to be terminating if prime factorisation of the denominator contains only prime factors 2 and 5

The denominator of the given rational number is simplified to 7

Since the prime factorisation of the denominator does not contain prime factors as 2 and 5

So the decimal expansion of a rational number formed by taking the number of Science books as numerator and number Biography books as denominator is non-termating repeating

Hence the correct option is c) Non-termating repeating

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