Answers of all questions with full working correctly

1.	Answers of all questions with full working correctly
Answer» align="absmiddle" ALT="\large\sf\underline{QUESTION\:1\::}" class="latex-formula" id="TexFormula1" src="https://tex.z-dn.net/?f=%5Clarge%5Csf%5Cunderline%7BQuestion%5C%3A1%5C%3A%3A%7D" title="\large\sf\underline{Question\:1\::}"> Divide the sum of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ by their difference . ‎ ‎ $\large\sf\underline{Solution\::}$ $\small{\mathfrak{1^{st}\:step\:-}}$ Sum of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ . $\sf\longmapsto\:\frac{65}{12} + \frac{12}{7}$ LCM of 12 and 7 = 87 $\sf\longmapsto\:\frac{(65 \times 7)+(12 \times 12)}{84}$ Multiplying the terms in numerator $\sf\longmapsto\:\frac{455+144}{84}$ Adding the terms in numerator $\sf\longmapsto\:\frac{599}{84}$ ☆ ‎ $\small{\mathfrak{2^{nd}\:step\:-}}$ Difference of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ . $\sf\longmapsto\:\frac{65}{12} - \frac{12}{7}$ LCM of 12 and 7 = 87 $\sf\longmapsto\:\frac{(65 \times 7)-(12 \times 12)}{84}$ Multiplying the terms in numerator $\sf\longmapsto\:\frac{455-144}{84}$ Subtracting the terms in numerator $\sf\longmapsto\:\frac{311}{84}$ ☆ ‎ ‎ $\small{\mathfrak{3^{rd}\:step\:-}}$ Dividing the sum of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ by their difference . Sum of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ = $\sf\:\frac{599}{84}$ Difference of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ = $\sf\:\frac{311}{84}$ Dividing them : $\sf\longmapsto\:\frac{599}{84} \div \frac{311}{84}$ $\sf\longmapsto\:\frac{599}{\cancel{84}} \times \frac{\cancel{84}}{311}$ $\small{\underline{\boxed{\mathrm\pink{⟼\:\frac{599}{311}}}}}$ ★ _________________________ $\large\sf\underline{Question\:2\::}$ The sum of the ADDICTIVE inverse and multiplicative inverse of $\sf\:\frac{1}{5}$ is ? ‎ $\large\sf\underline{Solution\::}$ Addictive inverse of $\sf\:\frac{1}{5}$ = $\sf\:\frac{-1}{5}$ Multiplicative inverse of $\sf\:\frac{1}{5}$ = $\sf\:5$ Their sum : $\sf\longmapsto\:\frac{-1}{5}+5$ LCM of 5 and 1 = 5 $\sf\longmapsto\:\frac{-1+(5 \times 5)}{5}$ Multiplying the terms in numerator $\sf\longmapsto\:\frac{-1+25}{5}$ Subtracting the terms in numerator $\small{\underline{\boxed{\mathrm\pink{⟼\:\frac{24}{5}}}}}$ ★ ‎ _________________________ $\large\sf\underline{Question\:3\::}$ The PRODUCT of two RATIONAL numbers is $\sf\:\frac{-28}{81}$ . If obe of the number is $\sf\:\frac{14}{27}$ , then find the other number. ‎ $\large\sf\underline{Solution\::}$ Let the other number be x. So according to the question : $\sf\longmapsto\:\frac{14}{27} \times x =\frac{-28}{81}$ $\sf\longmapsto\:\frac{14x}{27} =\frac{-28}{81}$ Cross multiplying $\sf\longmapsto\:14x \times 81 =-28 \times 27$ Multiplying the terms $\sf\longmapsto\:1134x =-756$ Transposing 1134 to the other side $\sf\longmapsto\:x =\cancel{\frac{-756}{1134}}$ $\small{\underline{\boxed{\mathrm\pink{⟼\:x\:=\frac{-2}{3}}}}}$ ★ ‎ _________________________ $\large\sf\underline{Question\:4\::}$ If a = 7 , then the value of $\sf\:-[\frac{1-2a}{a-5}]$ is ______ ‎ $\large\sf\underline{Solution\::}$ $\sf\longmapsto\:-[\frac{1-2a}{a-5}]$ Substituiting the value of a as 7 $\sf\longmapsto\:-[\frac{1-2(7)}{7-5}]$ Simplifying the fraction $\sf\longmapsto\:-[\frac{1-14}{2}]$ $\sf\longmapsto\:-[\frac{-13}{2}]$ $\small{\underline{\boxed{\mathrm\pink{⟼\:\frac{13}{2}}}}}$ ★ ‎ _______________________ $\large\sf\underline{Question\:5\::}$ The multiplicative inverse of $\sf\:\frac{-a}{b}$ is ___ ‎ $\large\sf\underline{Solution\::}$ Multiplicative inverse of a number is a number which when multiplied by the original number given 1 . So multiplicative inverse of $\sf\:\frac{-a}{b}$ is : $\small{\underline{\boxed{\mathrm\pink{\frac{-b}{a}}}}}$ ★ Since : $\sf\longmapsto\:\frac{-a}{b} \times \frac{-b}{a}$ Multiplying the fraction $\sf\longmapsto\:\cancel{\frac{ab}{ab}}$ $\sf\longmapsto\:1$ ‎ ________________________ $\large\sf\underline{Question\:6\::}$ Which of the following properties of rational numbers is given below ? $\sf\:\frac{7}{4} \times (\frac{-8}{3} +\frac{-13}{12})=\frac{7}{4} \times \frac{-8}{3} +\frac{7}{4} \times \frac{-13}{12}$ ‎ $\large\sf\underline{Solution\::}$ So the given rational number indicates : $\small{\underline{\boxed{\mathrm\pink{Distribution\:of\:multiplication\:over\:addition.}}}}$ ★ ‎ _________________________ $\large\sf\underline{Question\:7\::}$ If $\sf\:x = \frac{2+3 \times 2}{-5}$ , then find the value of \| - x\| . ‎ $\large\sf\underline{Solution\::}$ Simplifying the value of x $\sf\longmapsto\:x = \frac{2+6}{-5}$ $\sf\longmapsto\:x = \frac{8}{-5}$ So \| - x\| = $\small{\underline{\boxed{\mathrm\pink{\frac{8}{5}}}}}$ ★

Answer»

align="absmiddle" ALT="\large\sf\underline{QUESTION\:1\::}" class="latex-formula" id="TexFormula1" src="https://tex.z-dn.net/?f=%5Clarge%5Csf%5Cunderline%7BQuestion%5C%3A1%5C%3A%3A%7D" title="\large\sf\underline{Question\:1\::}">

Divide the sum of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ by their difference .

‎

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

$\small{\mathfrak{1^{st}\:step\:-}}$

Sum of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ .

$\sf\longmapsto\:\frac{65}{12} + \frac{12}{7}$

LCM of 12 and 7 = 87

$\sf\longmapsto\:\frac{(65 \times 7)+(12 \times 12)}{84}$

Multiplying the terms in numerator

$\sf\longmapsto\:\frac{455+144}{84}$

Adding the terms in numerator

$\sf\longmapsto\:\frac{599}{84}$ ☆

‎

$\small{\mathfrak{2^{nd}\:step\:-}}$

Difference of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ .

$\sf\longmapsto\:\frac{65}{12} - \frac{12}{7}$

LCM of 12 and 7 = 87

$\sf\longmapsto\:\frac{(65 \times 7)-(12 \times 12)}{84}$

Multiplying the terms in numerator

$\sf\longmapsto\:\frac{455-144}{84}$

Subtracting the terms in numerator

$\sf\longmapsto\:\frac{311}{84}$ ☆

‎

‎ $\small{\mathfrak{3^{rd}\:step\:-}}$

Dividing the sum of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ by their difference .

Sum of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ = $\sf\:\frac{599}{84}$

Difference of $\sf\:\frac{65}{12}$ and $\sf\:\frac{12}{7}$ = $\sf\:\frac{311}{84}$

Dividing them :

$\sf\longmapsto\:\frac{599}{84} \div \frac{311}{84}$

$\sf\longmapsto\:\frac{599}{\cancel{84}} \times \frac{\cancel{84}}{311}$

$\small{\underline{\boxed{\mathrm\pink{⟼\:\frac{599}{311}}}}}$ ★

_________________________

$\large\sf\underline{Question\:2\::}$

The sum of the ADDICTIVE inverse and multiplicative inverse of $\sf\:\frac{1}{5}$ is ?

‎

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

Addictive inverse of $\sf\:\frac{1}{5}$ = $\sf\:\frac{-1}{5}$

Multiplicative inverse of $\sf\:\frac{1}{5}$ = $\sf\:5$

Their sum :

$\sf\longmapsto\:\frac{-1}{5}+5$

LCM of 5 and 1 = 5

$\sf\longmapsto\:\frac{-1+(5 \times 5)}{5}$

Multiplying the terms in numerator

$\sf\longmapsto\:\frac{-1+25}{5}$

Subtracting the terms in numerator

$\small{\underline{\boxed{\mathrm\pink{⟼\:\frac{24}{5}}}}}$ ★

‎ _________________________

$\large\sf\underline{Question\:3\::}$

The PRODUCT of two RATIONAL numbers is $\sf\:\frac{-28}{81}$ . If obe of the number is $\sf\:\frac{14}{27}$ , then find the other number.

‎

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

Let the other number be x.

So according to the question :

$\sf\longmapsto\:\frac{14}{27} \times x =\frac{-28}{81}$

$\sf\longmapsto\:\frac{14x}{27} =\frac{-28}{81}$

Cross multiplying

$\sf\longmapsto\:14x \times 81 =-28 \times 27$

Multiplying the terms

$\sf\longmapsto\:1134x =-756$

Transposing 1134 to the other side

$\sf\longmapsto\:x =\cancel{\frac{-756}{1134}}$

$\small{\underline{\boxed{\mathrm\pink{⟼\:x\:=\frac{-2}{3}}}}}$ ★

‎ _________________________

$\large\sf\underline{Question\:4\::}$

If a = 7 , then the value of $\sf\:-[\frac{1-2a}{a-5}]$ is ______

‎

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

$\sf\longmapsto\:-[\frac{1-2a}{a-5}]$

Substituiting the value of a as 7

$\sf\longmapsto\:-[\frac{1-2(7)}{7-5}]$

Simplifying the fraction

$\sf\longmapsto\:-[\frac{1-14}{2}]$

$\sf\longmapsto\:-[\frac{-13}{2}]$

$\small{\underline{\boxed{\mathrm\pink{⟼\:\frac{13}{2}}}}}$ ★

‎ _______________________

$\large\sf\underline{Question\:5\::}$

The multiplicative inverse of $\sf\:\frac{-a}{b}$ is ___

‎

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

Multiplicative inverse of a number is a number which when multiplied by the original number given 1 .

So multiplicative inverse of $\sf\:\frac{-a}{b}$ is :

$\small{\underline{\boxed{\mathrm\pink{\frac{-b}{a}}}}}$ ★

Since :

$\sf\longmapsto\:\frac{-a}{b} \times \frac{-b}{a}$

Multiplying the fraction

$\sf\longmapsto\:\cancel{\frac{ab}{ab}}$

$\sf\longmapsto\:1$

‎

________________________

$\large\sf\underline{Question\:6\::}$

Which of the following properties of rational numbers is given below ?

$\sf\:\frac{7}{4} \times (\frac{-8}{3} +\frac{-13}{12})=\frac{7}{4} \times \frac{-8}{3} +\frac{7}{4} \times \frac{-13}{12}$

‎

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

So the given rational number indicates :

$\small{\underline{\boxed{\mathrm\pink{Distribution\:of\:multiplication\:over\:addition.}}}}$ ★

‎

_________________________

$\large\sf\underline{Question\:7\::}$

If $\sf\:x = \frac{2+3 \times 2}{-5}$ , then find the value of | - x| .

‎

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

Simplifying the value of x

$\sf\longmapsto\:x = \frac{2+6}{-5}$

$\sf\longmapsto\:x = \frac{8}{-5}$

So | - x| = $\small{\underline{\boxed{\mathrm\pink{\frac{8}{5}}}}}$ ★