Question

Solve :-

$\sf 1. \: \: \frac{2}{3} \: - ( \: \frac{1}{5} \: - \: \frac{3}{2} \: )$
$\sf 2. \: \: (\: \frac{2}{3} \: - \: \frac{1}{5} \: )- \: \frac{3}{2} \:$
$\sf 3. \: \: \frac{2}{3} \: \div ( \: \frac{1}{5} \: \div \: \frac{3}{2} \: )$
$\sf 4. \: \: ( \: \frac{2}{3} \: \div \: \frac{1}{5} \: ) \div \: \frac{3}{2} \:$


Note :-

Please explain step by step.​

Answer 1

Step-by-step explanation:

Solutions:-

1) Given that

2/3 -[ (1/5)-(3/2)]

LCM of 5 and 2 is 10

=>2/3 - [(2-15)/10]

=>2/3-(-13/10)

=>(2/3)+(13/10)

LCM of 3 and 10 is 30

=>(20+39)/30

=>59/30

2/3 -[ (1/5)-(3/2)] = 59/30

____________________________

2) Given that

(2/3 - 1/5)- 3/2

LCM of 3 and 5 is 15

=>[10-3]/15 -(3/2)

=>(7/15)-(3/2)

LCM of 15 and 2 is 30

= (14-45)/30

=>-31/30

(2/3 - 1/5)- 3/2 = -31/30

___________________________

3) Given that

(2/3)÷[1/5 ÷ 3/2]

Multiplicative inverse of 3/2 is 2/3

=>(2/3)÷[(1/5)×(2/3)]

=>(2/3)÷[(1×2)/(5×3)]

=>(2/3)÷[2/15]

Multiplicative inverse of 2/15 is 15/2

=>(2/3)×(15/2)

=>(2×15)/(3×2)

=>30/6

=>5

(2/3)÷[1/5 ÷ 3/2]=5

____________________________

4) Given that

(2/3 ÷ 1/5) ÷ (3/2)

Multiplicative inverse of 1/5 = 5

=>[(2/3)×(5)]÷(3/2)

=>[(2×5)/(3)]÷(3/2)

=>(10/3)÷(3/2)

Multiplicative inverse of 3/2 = 2/3

=>(10/3)×(2/3)

=>(10×2)/(3×3)

=>20/9

(2/3 ÷ 1/5) ÷ (3/2) = 20/9

_____________________________

Used formulae:-

The multiplicative inverse of a is 1/a

The product of two numbers is equal to 1 then they are called multiplicative inverses to each other

Answer 2

$\large\underline{\bold{ANSWER-1}}$

$\rm :\longmapsto\:\dfrac{2}{3} \: - \bigg( \: \dfrac{1}{5} \: - \: \dfrac{3}{2} \bigg)$

• Step 1 : We first solve the bracket. Take the LCM of 2 and 5 = 10

So,

$\rm :\longmapsto\:\dfrac{2}{3} - \bigg(\dfrac{2 - 15}{10} \bigg)$

$\rm :\longmapsto\:\dfrac{2}{3} - \bigg( \dfrac{ - 13}{10} \bigg)$

$\rm :\longmapsto\:\dfrac{2}{3} + \dfrac{13}{10}$

• Step 2 : Take out the LCM of 3 and 10 = 30

$\rm :\longmapsto\:\dfrac{20 + 39}{30}$

$\rm :\longmapsto\:\dfrac{59}{30}$

$\rm :\implies\: \boxed{ \bf{\dfrac{2}{3} \: - \bigg( \: \dfrac{1}{5} \: - \: \dfrac{3}{2} \bigg) = \dfrac{59}{30} }}$

$\large\underline{\bold{ANSWER-2}}$

$\rm :\longmapsto\: \bigg(\: \dfrac{2}{3} \: - \: \dfrac{1}{5} \: \bigg)- \: \dfrac{3}{2}$

• Step 1 : We first solve the bracket. Take the LCM of 3 and 5 = 15

$\rm :\longmapsto\:\bigg( \dfrac{10 - 3}{15} \bigg) - \dfrac{3}{2}$

$\rm :\longmapsto\:\dfrac{7}{15} - \dfrac{3}{2}$

• Step 2 : Take out the LCM of 15 and 2 = 30

$\rm :\longmapsto\:\dfrac{14 - 45}{30}$

$\rm :\longmapsto\:\dfrac{ - 31}{30}$

$\rm :\implies\: \boxed{ \bf{\bigg(\: \dfrac{2}{3} \: - \: \dfrac{1}{5} \: \bigg)- \: \dfrac{3}{2} = \: - \: \dfrac{31}{30} }}$

$\large\underline{\bold{ANSWER-3}}$

$\rm :\longmapsto\:\dfrac{2}{3} \: \div \bigg( \: \dfrac{1}{5} \: \div \: \dfrac{3}{2} \: \bigg)$

$\rm :\longmapsto\:\dfrac{2}{3} \div \bigg(\dfrac{1}{5} \times \dfrac{2}{3} \bigg)$

$\rm :\longmapsto\:\dfrac{2}{3} \div \dfrac{2}{15}$

$\rm :\longmapsto\:\dfrac{2}{3} \times \dfrac{15}{2}$

$\rm :\longmapsto\:5$

$\rm :\implies\: \boxed{ \bf{\dfrac{2}{3} \: \div \bigg( \: \dfrac{1}{5} \: \div \: \dfrac{3}{2} \: \bigg) \: = \: 5}}$

$\large\underline{\bold{ANSWER-4}}$

$\rm :\longmapsto\: \bigg( \: \dfrac{2}{3} \: \div \: \dfrac{1}{5} \: \bigg) \div \: \dfrac{3}{2}$

$\rm :\longmapsto\:\bigg( \: \dfrac{2}{3} \: \times \: \dfrac{5}{1} \: \bigg) \div \: \dfrac{3}{2}$

$\rm :\longmapsto\:\bigg( \: \dfrac{10}{3} \: \: \bigg) \div \: \dfrac{3}{2}$

$\rm :\longmapsto\:\dfrac{10}{3} \times \dfrac{2}{3}$

$\rm :\longmapsto\:\dfrac{20}{9}$

$\rm :\implies\: \boxed{ \bf{\bigg( \: \dfrac{2}{3} \: \div \: \dfrac{1}{5} \: \bigg) \div \: \dfrac{3}{2} \: = \: \dfrac{20}{9} }}$

Additional Information :-

Fraction :-

• Fractions represent equal parts of a whole or a collection. 
• Fraction of a whole: When we divide a whole into equal parts, each part is a fraction of the whole. 

For example, 

• There are total of 5 children.

• 3 out of 5 are girls. So, the fraction of girls is three-fifths ( 3⁄5 ).

• 2 out of 5 are boys. So, the fraction of boys is two-fifths ( 2⁄5 ).

• A fraction has two parts. The number on the top of the line is called the numerator. It tells how many equal parts of the whole or collection are taken.  The number below the line is called the denominator.  It shows the total divisible number of equal parts the whole into or the total number of equal parts which are there in a collection. 

Unit fractions

• Fractions with numerator 1 are called unit fractions.

Proper fractions

• Fractions in which the numerator is less than the denominator are called proper fractions

Improper fractions

• Fractions in which the numerator is more than or equal to the denominator are called improper fractions.

Mixed fractions

• Mixed fractions consist of a whole number along with a proper fraction.