Math, asked by thepokester123, 6 months ago

here were three types of fruit in a box. The ratio of the number of mangoes to the total number of apples and oranges was 2:5. The ratio of the number of apples to the number of oranges was 9:1. There were 30 more apples than mangoes. After some mangoes were added into the box, 70% of the fruits in the box were mangoes.
a) How many mangoes were there in the end?
b) How many mangoes were added in the box?

Answers

Answered by Anonymous
2

Step-by-step explanation:

\color{red} {{{\Large {\bf{To\:\:Simplify\::\frac{\sin ^4(x)-\cos ^4(x)}{\sin ^2(x)-\cos ^2(x)}}}}}}

\color{green}{{{\large {\bf{Your\:\:Answer\::\frac{\sin ^4(x)-\cos ^4(x)}{\sin ^2(x)-\cos ^2(x)}=1}}}}}

\color{yellow} {\Huge {\sf{Solution:}}}

\color{blue} {\large {\bf{Factor\:\sin ^4(x)-\cos ^4(x)}}}

\tt \color{blue} {\mathrm{Rewrite\:}\sin ^4(x)-\cos ^4(x)\mathrm{\:as\:}(\sin ^2(x))^2-(\cos ^2(x))^2=(\sin ^2(x))^2-(\cos ^2(x))^2}

\color{fuchsia} {\normalsize {\mathrm{Apply\:exponent\:rule}:\quad \:a^{bc}=(a^b)^c}}

\color{fuchsia} {\normalsize \sin ^4(x)=(\sin ^2(x))^2}

\color{fuchsia} {\normalsize =(\sin ^2(x))^2-\cos ^4(x)}=

\color{fuchsia} {\normalsize \mathrm{Apply\:exponent\:rule}:\quad \:a^{bc}=(a^b)^c}

\color{fuchsia} {\normalsize \cos ^4(x)=(\cos ^2(x))^2}

\color{fuchsia} {\normalsize =(\sin ^2(x))^2-(\cos ^2(x))^2}=

\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}x^2-y^2=(x+y)(x-y)

(\sin ^2(x))^2-(\cos ^2(x))^2=(\sin ^2(x)+\cos ^2(x))(\sin ^2(x)-\cos ^2(x))

=(\sin ^2(x)+\cos ^2(x))(\sin ^2(x)-\cos ^2(x))=

\color{blue} {\large {\bf{Factor\:\sin ^2(x)-\cos ^2(x)}}}

\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}x^2-y^2=(x+y)(x-y)

\sin ^2(x)-\cos ^2(x)=(\sin (x)+\cos (x))(\sin (x)-\cos (x))

(x)=(sin(x)+cos(x))(sin(x)−cos(x))

=(\sin (x)+\cos (x))(\sin (x)-\cos (x))=(sin(x)+cos(x))(sin(x)−cos(x))

\large=(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))\ \textless \ br /\ \textgreater \ (x))(sin(x)+cos(x))(sin(x)−cos(x))\ \textless \ br /\ \textgreater \ \large =\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{\sin ^2(x)-\cos ^2(x)}=

\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}x^2-y^2=(x+y)(x-y)

\sin ^2(x)-\cos ^2(x)=(\sin (x)+\cos (x))(\sin (x)

 (x)=(sin(x)+cos(x))(sin(x)−cos(x))\ \textless \ br /\ \textgreater \ =\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{(\sin (x)+\cos (x))(\sin (x)-\cos (x))}=

\mathrm{Cancel\:}\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{(\sin (x)+\cos (x))(\sin (x)-\cos (x))}:\quad \sin ^2(x)+\cos ^2(x)Cancel \ \textless \ br /\ \textgreater \ (sin(x)+cos(x))(sin(x)−cos(x))

\mathrm{Cancel\:the\:common\:factor:}\:\sin (x)+\cos(x)Cancelthecommonfactor:sin(x)+cos(x)

=\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)-\cos (x))}{\sin (x)-\cos (x)}=

\mathrm{Cancel\:the\:common\:factor:}\:\sin (x)-\cos

\mathrm{Use\:the\:following\:identity}:\quad \cos ^2(x)+\sin

\huge \boxed{\color{red} {\ \huge =1}}

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