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The rational number that does not belong with the other three is -7/3 as it is smaller than –1 whereas rest of the numbers are greater than –1.

(3/4) by (5/7)

The product of two rational numbers = (product of their numerator)/ (product of their denominator)

The above question can be written as (3/4) × (5/7)

We have,

= (3×5)/ (4×7)

= (15/28)

= (7/8) + (1/6) – (1/12)

= ((14 + 1)/16) – (1/12)

= (15/16) – (1/12)

= (45-4)/48

= 41/48

(-12/5) by (10/-3)

The product of two rational numbers = (product of their numerator)/ (product of their denominator)

First we write each of the given numbers with a positive denominator.

(10/-3) = [(10× (-1))/ (-3×-1)]

= (-10/3)

The above question can be written as (-12/5) × (-10/3)

We have,

= (-12×-10)/ (5×3)

On simplifying,

= (-4×-2)/ (1×1)

= 8

(9/8) by (32/3)

The product of two rational numbers = (product of their numerator)/ (product of their denominator)

The above question can be written as (9/8) × (32/3)

We have,

= (9×32)/ (8×3)

On simplifying,

= (3×4)/ (1×1)

= 12

(-2/3) by (6/7)

The product of two rational numbers = (product of their numerator)/ (product of their denominator)

The above question can be written as (-2/3) × (6/7)

We have,

= (-2×6)/ (3×7)

On simplifying,

= (-2×2)/ (1×7)

= (-4/7)

= (3/7) – (2/21) × (-5/6)

= (3/7) – (1/21) × (-5/3)

= (3/7) – (-5/63)

= (3/7) + (5/63)

= (27 + 5)/63

= 32/63

Given that product of two rational numbers = 15

One of the number = -10

∴ other number 

 = 15/-10

= -3/2

It is given that the product of two rational numbers is \(\frac{-8}{9}\).

If one of the number is \(-\frac{4}{15}\) we have to find the other number.

So, the other number is obtained by dividing the product by the given number.

Therefore,

Other number = \(\frac{\frac{-8}{9}}{\frac{-4}{15}}\) = \(\frac{8\times 15}{9\times 4}\) = \(\frac{2\times 5}{3}\) = \(\frac{10}{3}\)

For finding rational numbers between two numbers:

Add the numbers are divide by 2, this will give a number between the numbers. Now take the new number and add with any of the numbers and repeat the process. You can keep on repeating the process and new numbers will be obtained. A rational number lying between \(\frac{1}{5}\) and \(\frac{1}{2}\)

= \(\frac{\frac{2}{5}+\frac{1}{2}}{2}\)

= \((\frac{\frac{2+5}{10}}{2})\)

= \(\frac{7}{20}\)

Now,

A rational number between \(\frac{1}{5}\) and \(\frac{1}{2}\)

 = \(\frac{-\frac{1}{5}+\frac{7}{20}}{2}\)

= \((\frac{\frac{4+7}{20}}{2})\)

= \(\frac{11}{40}\)

Hence,

The two rational numbers lying between \(\frac{1}{5}\) and \(\frac{1}{2}\) are \(\frac{7}{20}\) and \(\frac{11}{40}\)

Given product of two numbers = (-8/9)

One of them is (-4/15)

Let the required number be x

x × (-4/15) = (-8/9)

x = (-8/9) ÷ (-4/15)

x = (-8/9) × (15/-4)

x = (-120/-36)

x = (10/3)

Given product of two rational numbers = -8/9

One of the number = -4/15

∴ other number 

 = (-8/9) /(-4/15)

= (-8/9) × (15/-4)

= (-2/3) × (5/-1)

= (-2×5) /(3×-1)

= -10/-3

= 10/3

= (3/7) × (28/15) ÷ (14/5)

= (1/1) × (4/5) ÷ (14/5)

= (4/5) ÷ (14/5)

= (4/5) × (5/14)

= (2/1) × (1/7)

= 2/7

Given product = (-23/9)

One number is (-1/6)

Let the required number be x

x × (-1/6) = (-23/9)

x = (-23/9) ÷ (-1/6)

x = (-23/9) × (-6/1)

x = (-138/9)

x = (46/3)

Let the required number be ‘x’

Now,

According to the question,

x \(\times \frac{-1}{6}\) = \(\frac{-23}{9}\)

x = \(\frac{\frac{23}{9}}{\frac{-1}{6}}\)

x = \(\frac{23}{9}\times \frac{6}{1}\)

x = \(\frac{46}{3}\)

(-7/30) × (5/14)

The product of two rational numbers = (product of their numerator)/ (product of their denominator)

We have,

= (-7×5)/ (30×14)

On simplifying,

= (-1×1)/ (6×2)

= -1/12

Let us assume the other number be y.

Given, product of two rational number = -7

One number = -5

Then,

= y × (-5) = -7

= y = -7/ (-5)

= y = 7/5

So, the other number is 7/5

Given product = (-5/7)

One number is (-15/28)

Let the required number be x

x × (-15/28) = (-5/7)

x = (-5/7) ÷ (-15/28)

x = (-5/7) × (28/-15)

x = (-4/-3)

x = (4/3)

Let us assume the other number be y.

Given, product of two rational number = -5/7

One number = -15/20

Then,

= y × (-15/20) = -5/7

= y = (-5/7)/ (-15/20)

= y = (-5/7) × (-20/15)

= y = (-1/7) × (-20/3)

= y = -20/21

So, the other number is -20/21

Let the required number be ‘x’

Now,

According to the question,

\(x\times \frac{-15}{28}\) = \(\frac{-5}{7}\)

x = \(\frac{\frac{-5}{7}}{\frac{-15}{28}}\)

x = \(\frac{-5}{7}\) x \(\frac{28}{-15}\)

x =   \(\frac{1}{1}\) x \(\frac{4}{3}\)

x = \(\frac{4}{3}\)

(3/20) × (4/5)

The product of two rational numbers = (product of their numerator)/ (product of their denominator)

We have,

= (3×4)/ (20×5)

On simplifying,

= (3×1)/ (5×5)

= 3/25

Let a number be  = x

So,
\(x\times\frac{-1}{6}=\frac{-23}{9}\)

x = (-23/9)/(-1/6)

x = (-23/9) × (6/-1)

= (-23/3) × (2×-1)

= (-23×-2)/(3×1)

= 46/3

Let the required number be ‘x’

Now,

According to the question,

\(x\times \frac{-8}{13}\) = 24

x = \(\frac{\frac{24}{1}}{\frac{-8}{13}}\)

x =   \(\frac{24}{1}\) x \(\frac{13}{8}\)

x = \(\frac{3}{1}\) x \(\frac{13}{1}\)

x = -39

Let us assume the other number be y.

Given, product of two rational number = 24

One number = -8/13

Then,

= y × (-8/13) = 24

= y = 24/ (-8/13)

= y = (24/1) × (-13/8)

= y = (3/1) × (-13/1)

= y = -39

So, the other number is -39

(7/6) by (24)

The product of two rational numbers = (product of their numerator)/ (product of their denominator)

The above question can be written as (7/6) × (24)

We have,

= (7×24)/ (6×1)

On simplifying,

= (7×4)/ (1×1)

= 28

(-13/15) by (-25/26)

The product of two rational numbers = (product of their numerator)/ (product of their denominator)

The above question can be written as (-13/15) × (-25/26)

We have,

= (-13×-25)/ (15×26)

On simplifying,

= (-1×-5)/ (3×2)

= (5/6)

Let a number be  = x

So,
\(x\times\frac{-15}{28}=\frac{-5}{7}\)

x = (-5/7) / (-15/28)

x = (-5/7) × (28/-15)

= (-1/1) × (4×-3)

= 4/3

Given product = 24

One of the number is = (-8/13)

Let the required number be x

x × (-8/13) = 24

x = 24 ÷ (-8/13)

x = 24 × (13/-8)

x = -39

Let a number = x

So,

\(x\times\frac{-8}{13}=24\)

x = (24) / (-8/13)

x = (24) × (13/-8)

= (3) × (13×-1)

= -39

Let a number be = x

So,
\(x\times\frac{-3}{4}=\frac{2}{3}\)

x = (2/3) / (-3/4)

x = (2/3) × (4/-3)

= -8/9

Given product = (-2/3)

One of the number is = (-3/4)

Let the required number be x

x × (-3/4) = (-2/3)

x = (-2/3) ÷ (-3/4)

x = (-2/3) × (4/-3)

x = (-8/-9)

x = (8/9)

Given x = (2/3), y = (3/2)

(x + y) ÷ (x – y) = ((2/3) + (3/2)) ÷ ((2/3) – (3/2))

= (4 + 9)/6 ÷ (4 – 9)/6

= (4 + 9)/6 × (6/ (4 – 9)

= (4 + 9)/ (4 -9)

= (13/-5)

Given x = (5/4), y = (-1/3)

(x + y) ÷ (x – y) = ((5/4) + (-1/3)) ÷ ((5/4) – (-1/3))

= (15 – 4)/12 ÷ (15 + 4)/12

= (15 – 4)/12 × (12/ (15 + 4)

= (15 – 4)/ (15 + 4)

= (11/19)

0.9 = \(\frac{9}{10}\)and 0.8 = \(\frac{8}{10} = \frac{4}{5}\)

(i) 0.57 = \(\frac{57}{100}\)

(∵ two digits are there after the decimal point) 

(ii) 0.176 = \(\frac{176}{1000}\)

(iii) 1.00001 = \(\frac{100001}{100000}\)

(iv) 25.125 = \(\frac{25125}{1000}\)

Given x = (2/5), y = (1/2)

(x + y) ÷ (x – y) = ((2/5) + (1/2)) ÷ ((2/5) – (1/2))

= (4 + 5)/10 ÷ (4 -5)/10

= (4 + 5)/10 × (10/ (4 – 5)

= (4 + 5)/ (4 -5)

= (9/-1)

Correct option is (B) 5

\(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\) \(+.....+\frac12\)  (10 times)

= \(10\times\frac12\) = 5.

Correct option is  B) 5

(i) True

\(\frac{-3}{5}\)is a negative number.

All negative numbers are less than 0 and therefore, 

lie to the left of 0 on the number line.

Hence, \(\frac{-3}{5}\)lies to the left of 0 on the number line.

(ii) False

 \(\frac{-12}{7}\) is a negative number.

All negative numbers are less than 0 and therefore, 

lie to the left of 0 on the number line.

Hence, \(\frac{-12}{7}\)lies to the left of 0 on the number line.

(iii) True

\(\frac{1}{3}\)is a positive number

All positive numbers are greater than 0 and therefore, 

lie to the right of 0 on the number line.

Hence,\(\frac{1}{3}\)lies to the right of 0 on the number line.

\(\frac{-5}{2}\)is a negative number.

All negative numbers are less than 0 and therefore, 

lie to the left of 0 on the number line.

Hence, \(\frac{-5}{2}\)lies to the left of 0 on the number line.

Therefore, 

the rational numbers, \(\frac{1}{3}\) and \(\frac{-5}{2}\) are on opposite sides of 0 on the number line.

(iv) False

\(\frac{-18}{-13} = \frac{-18\times-1}{-13\times-1} = \frac{18}{13}\)

\(\frac{18}{13}\)is a positive number.

All positive numbers are greater than 0 and therefore, 

lie to the right of 0 on the number line.

Hence, \(\frac{18}{13}\)lies to the right of 0 on the number line.

7/9 = (7 + 5/9 + 5) is-

False

(7/9 = (7-5/9-5) is-

False

7/9 = (7 - 5/9 - 5) is-

False

7/9 = (7 x 5/9 x 5) is-

True

No, the reverse is not true. The sum of any two consecutive natural numbers need not be a perfect square of a number. Example: 35 + 36 = 71, not a perfect square.

(i) 8120

= (8 x 1000) + (1 x 100) + (2 x 10) + 0 x 1

= (8 x 10³) + (1 x 10²) + (2 x 10¹)

(ii) 20305 

= (2 x 10000) + (0 x 1000) + (3 x 100) + (0 x 10) + (5 x 1)

= (2 x 10⁴) + (3 x 10²) + 5

(iii) 3652.01 

= 3000 + 600 + 50 + 2 + 0/10 + 1/100

= (3 x 1000) +(6 x 100) + (5 x 10) + (2 x 1) + (1 x 1100)

= (3 x 10³) + (6 x 10²) + (5 x 10¹) + 2 + (1 x 10^-2)

(iv) 9426.521

= (9 x 1000) + (4 x 100) + (2 x 10) + (6 x 1) + (5/10) + (2/100) + (1/1000)

= (9 x 10³) + (4 x 10²) + (2 x 10¹) + 6 + (5 x 10^-1) + (2 x 10^-2) + (1 x 10^-3)

3/5 + 13/5

= (3 + 13)/5

= 16/5

= 3 1/5

Area of room \(=65\frac{1}{4}\,m^2\)

Breadth of room \(=5\frac{7}{16}\,m\)

Length of room = Area of room \(\div\) Breadth of room

\(=65\frac{1}{4}\,m^2\div5\frac{7}{16}\,m\)

\(=\frac{261}{4}\,m^2\div\frac{87}{16}\,m\)

\(=\frac{261}{4}\,m^2\times\frac{16}{87}\,m\)

\(=\frac{4176}{348}\,m\)

\(=12\,m\)

Product of two fractions \(=9\frac{3}{5}\)

First fraction \(=9\frac{3}{7}\)

Second fraction = Product of two fractions \(\div\) First fraction

\(=9\frac{3}{5}\div9\frac{3}{7}\)

\(=\frac{48}{5}\div\frac{66}{7}\)

\(=\frac{48}{5}\times\frac{7}{66}\)

\(=\frac{336}{330}\)

\(=\frac{56}{55}\)

\(=1\frac{1}{55}\)

Second fraction \(=1\frac{1}{55}\)

Fraction of boys \(=\frac{5}{8}\)

Fraction of girls \(=1-\frac{5}{8}=\frac{3}{8}\)

Number of girl \(=240\)

Number of girls = Total students \(\times\frac{3}{8}\)

\(\Rightarrow\) 240 = Total students \(\times\frac{3}{8}\)

\(\Rightarrow\) Total students = 240 \(\div\) \(\frac{3}{8}\)

\(=240\times\frac{8}{3}\)

\(=\frac{240\times8}{3}\)

\(=\frac{1920}{3}=640\)

Total students \(=640\)

Number of boys = Total students - Number of girls

\(=640-240=400\)

Number of boys \(=\) 400

Correct option is (B) 43/9

Let x = \(4.\bar{7}\)

\(\Rightarrow\) x = 4.777...           ______(1)

Multiply equation (1) by 10, we obtain

10x = 47.777...          ______(2)

Subtract equation (1) from (2), we get

10x - x = 47.777... - 4.777....

\(\Rightarrow\) 9x = 43

\(\Rightarrow\) x = \(\frac{43}{9}\).

Correct option is   B) \(\frac{43}{9}\)

Correct option is (D) 13

Let x = \(0.\bar{4}\)

\(\Rightarrow\) x = 0.444...           ______(1)

Multiply equation (1) by 10, we obtain

10x = 4.444...          ______(2)

Subtract equation (1) from (2), we get

10x - x = 4.444... - 0.444...

\(\Rightarrow\) 9x = 4

\(\Rightarrow\) x = \(\frac49\) = \(\frac{p}{q}\)

Then p = 4 & q = 9

\(\Rightarrow\) p+q = 4+9 = 13.

Correct option is  D) 13