Question

​Verify the following properties:
(i) 3/7 × {5/6 + 12/13} = (3/7 × 5/6) + (3/7 × 12/13)
(ii) -15/4 × (3/7 + (-12)/5) = (-15/4 × 3/7) + (-15/4 × (-12)/5)
(iii) (-8/3 + -13/12) × 5/6 = (-8/3 × 5/6) + (-13/12 × 5/6)
(iv) -16/7 × {-8/9 + (-7)/6} = (-16/7 × (-8)/9) + ((-16)/7 × -7/6)

Answer 1

$\bigstar\:\mathfrak{(i)}$$\:\:\bf{\dfrac{3}{7}}\:\times\:{\Large{(}}\:{\dfrac{5}{6}}\:+\:{\dfrac{12}{13}}\:{\Large{)}}\:=\:{\Large{(}}\:{\dfrac{3}{7}}\:\times\:{\dfrac{5}{6}}\:{\Large{)}}\:+\:{\Large{(}}\:{\dfrac{3}{7}}\:\times\:{\dfrac{12}{13}}\:{\Large{)}}$

${ }$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{3}{7}}\:\times\:{\dfrac{137}{78}}\:=\:{\dfrac{1365\:+\:1512}{3822}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{137}{182}}\:=\:\cancel{\dfrac{2877}{3822}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{137}{182}}\:=\:\cancel{\dfrac{959}{1274}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{137}{182}}\:=\:{\dfrac{137}{182}}$

${ }$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{\underline{\underline{\textsf{\textbf{\pink{Hence,\:Verified\:!!}}}}}}$

${ }$

$\:\:\:\:\:\:\:\:━━━━━━━━━━━━━━━━━━━$

${ }$

$\bigstar\:\mathfrak{(ii)}$$\:\:\bf{\dfrac{-15}{4}}\:\times\:{\Large{(}}\:{\dfrac{3}{7}}\:+\:{\dfrac{(-12)}{5}}\:=\:{\Large{(}}\:{\dfrac{-15}{4}}\:\times\:{\dfrac{3}{7}}\:{\Large{)}}\:+\:{\Large{(}}\:{\dfrac{-15}{4}}\:\times\:{\dfrac{(-12)}{5}}\:{\Large{)}}$

${ }$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{-15}{4}}\:\times\:{\Large{(}}\:{\dfrac{15\:-\:84}{35}}\:{\Large{)}}\:=\:{\Large{(}}\:{\dfrac{-45}{28}}\:{\Large{)}}\:+\:{\Large{(}}\:{\dfrac{180}{20}}\:{\Large{)}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{-15}{4}}\:\times\:{\dfrac{-69}{35}}\:=\:{\dfrac{-900\:+\:5040}{560}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf\cancel{\dfrac{1035}{140}}\:=\:\cancel{\dfrac{4140}{560}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{207}{28}}\:=\:\cancel{\dfrac{828}{112}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{207}{28}}\:=\:\cancel{\dfrac{414}{56}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{207}{28}}\:=\:{\dfrac{207}{28}}$

${ }$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{\underline{\underline{\textsf{\textbf{\pink{Hence,\:Verified\:!!}}}}}}$

${ }$

$\:\:\:\:\:\:\:\:━━━━━━━━━━━━━━━━━━━$

${ }$

$\bigstar\:\mathfrak{(iii)}$$\:\:\bf{\Large{(}}\:{\dfrac{-8}{3}}\:+\:{\dfrac{-13}{12}}\:{\Large{)}}\:\times\:{\dfrac{5}{6}}\:=\:{\Large{(}}\:{\dfrac{-8}{3}}\:\times\:{\dfrac{5}{6}}\:{\Large{)}}+\:{\Large{(}}\:{\dfrac{-13}{12}}\:\times\:{\dfrac{5}{6}}\:{\Large{)}}$

${ }$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\Large{(}}\:{\dfrac{-96\:-\:39}{36}}\:{\Large{)}}\:\times\:{\dfrac{5}{6}}\:=\:{\Large{(}}\:{\dfrac{-40}{18}}\:{\Large{)}}\:+\:{\Large{(}}\:{\dfrac{-65}{72}}\:{\Large{)}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{135}{36}}\:\times\:{\dfrac{5}{6}}\:=\:{\dfrac{-40}{18}}\:-\:{\dfrac{65}{72}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{675}{216}}\:=\:{\dfrac{-2880\:-\:1170}{1296}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf\cancel{\dfrac{675}{216}}\:=\:\cancel{\dfrac{4050}{1296}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf\cancel{\dfrac{225}{72}}\:=\:\cancel{\dfrac{1350}{432}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf\cancel{\dfrac{75}{24}}\:=\:\cancel{\dfrac{450}{144}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf\cancel{\dfrac{25}{8}}\:=\:\cancel{\dfrac{150}{48}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{25}{8}}\:=\:\cancel{\dfrac{50}{16}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{25}{8}}\:=\:{\dfrac{25}{8}}$

${ }$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{\underline{\underline{\textsf{\textbf{\pink{Hence,\:Verified\:!!}}}}}}$

${ }$

$\:\:\:\:\:\:\:\:━━━━━━━━━━━━━━━━━━━$

${ }$

$\bigstar\:\mathfrak{(iv)}$$\:\:\bf{\dfrac{-16}{7}}\:\times\:{\Large{(}}\:{\dfrac{-8}{9}}\:+{\dfrac{(-7)}{6}}\:{\Large{)}}\:=\:{\Large{(}}\:{\dfrac{-16}{7}}\:\times\:{\dfrac{(-8)}{9}}\:{\Large{)}}\:+\:{\Large{(}}\:{\dfrac{(-16)}{7}}\:\times\:{\dfrac{(-7)}{6}}\:{\Large{)}}$

${ }$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{-16}{7}}\:\times\:{\Large{(}}\:{\dfrac{-48\:-\:63}{54}}\:{\Large{)}}\:=\:{\Large{(}}\:{\dfrac{128}{63}}\:{\Large{)}}\:+\:{\Large{(}}\:{\dfrac{112}{42}}\:{\Large{)}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{-16}{7}}\:\times\:{\dfrac{111}{54}}\:=\:{\dfrac{5376\:+\:7056}{2646}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf\cancel{\dfrac{1776}{378}}\:=\:\cancel{\dfrac{12432}{2646}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf\cancel{\dfrac{888}{189}}\:=\:\cancel{\dfrac{6216}{1323}}$

${ }$

$\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{296}{63}}\:=\:{\dfrac{296}{63}}$

${ }$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{\underline{\underline{\textsf{\textbf{\pink{Hence,\:Verified\:!!}}}}}}$

$\:\:\:\:\:\:\:\:━━━━━━━━━━━━━━━━━━━$