Percents: Very Difficult Problems with Solutions

Solution:
Let [tex]m\%[/tex] vote for the Mathematicians' Party. The people who love mathematics are all of these [tex]m\%[/tex] plus [tex]20\%[/tex] of the other [tex](100-m)\%[/tex]:
[tex]m+0.2(100-m)=52 \Rightarrow 0.8m=32 \Rightarrow m=40[/tex]

Solution:
Let there be [tex]n[/tex] students, [tex]k[/tex] of whom did not get an excellent grade. Then [tex]\frac{k}{n}[/tex] is between [tex]2.4\%[/tex] and [tex]3.2\%[/tex]. Since [tex]k\neq 0[/tex], we have [tex]k\geq 1[/tex] and
[tex]\dfrac{1}{n}\leq \dfrac{k}{n}\leq 0.032 \Rightarrow n\geq \dfrac{1}{0.032}=31.25[/tex]
So [tex]n\geq 32[/tex]. For [tex]n=32[/tex] and [tex]k=1[/tex]: [tex]\frac{31}{32}=96.875\%[/tex], which is in the interval.

Solution:
Take [tex]100[/tex] problems. [tex]80[/tex] are interesting and [tex]10\%[/tex] of them, i.e. [tex]8[/tex], contain errors. In total [tex]10[/tex] contain errors, so [tex]10-8=2[/tex] of the [tex]20[/tex] uninteresting problems contain errors. The correct uninteresting problems are [tex]18[/tex]:
[tex]\dfrac{18}{20}=90\%[/tex]

Solution:
Let there be [tex]G[/tex] gentlemen and [tex]L[/tex] ladies. The dancing gentlemen are as many as the dancing ladies:
[tex]0.5G=0.75L \Rightarrow G=1.5L[/tex]
[tex]\dfrac{0.5G+0.75L}{G+L}=\dfrac{0.75L+0.75L}{1.5L+L}=\dfrac{1.5L}{2.5L}=60\%[/tex]

Solution:
When we add the two areas, the common part is counted twice:
[tex]85\%+60\%-100\%=45\%[/tex]
So the part of the square outside the circle is [tex]60\%-45\%=15\%[/tex] of the figure. Compared with the square:
[tex]\dfrac{15}{60}=25\%[/tex]