Jenny's grandmother has 24 cats. Seventeen of the cats do not catch mice. Ten of the cats have black fur. What is the smallest possible number of cats that do not catch mice that have black fur?
+128407 Note that 24 - 10 = 14 don't have black fur
And suppose that these 14 are part of the 17 that don't catch mice
Then.....3 that catch mice must also have black fur....and this is the smallest number that do not catch mice but have black fur
+4609 We can draw a Venn diagram(sorry, don't have one).
So, 17 of the cats don't catch mice, and 10 have black fur.
Thus, 17+10=27
\(27-24=\boxed{3}\)
+26367 Jenny's grandmother has 24 cats.
Seventeen of the cats do not catch mice.
Ten of the cats have black fur.
What is the smallest possible number of cats that do not catch mice that have black fur?
\(\large{1.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & & x & \color{red} 10 \\ \hline \overline{\text{black fur}} & & & \small{24-10} \\ \hline & \small{24-17}& \color{red}17 & \color{red}24 \\ \end{array}\)
\(\large{2.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{10-x } & x & 10 \\ \hline \overline{\text{black fur}} & \small{7-(10-x) =} & \small{17-x} & 14 \\ & \small{x-3 } & & \\ \hline & 7 & 17 & 24 \\ \end{array} \)
\(\large{3.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{10-x } & x & 10 \\ & \small{\text{min: } 10-x=0 } & & \\ & \Rightarrow x_{min} = \color{red}10 & & \\ \hline \overline{\text{black fur}} & \small{x-3 } & \small{17-x} & 14 \\ & \small{\text{min: } x-3=0} & \small{\text{min: } 17-x=0 } & \\ & \Rightarrow x_{min} = \color{red}3 & \Rightarrow x_{min} = \color{red}17 \\ \hline & 7 & 17 & 24 \\ \end{array}\)
\(\large{4.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{10-x } & x & 10 \\ & \Rightarrow x_{min} = \color{red}10 & = min({\color{red}10},{\color{red}3},{\color{red}17}) & \\ & & = 3 & \\ \hline \overline{\text{black fur}} & \small{x-3 } & \small{17-x} & 14 \\ & \Rightarrow x_{min} = \color{red}3 & \Rightarrow x_{min} = \color{red}17 \\ \hline & 7 & 17 & 24 \\ \end{array} \)
\(\large{5.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{7 } & \color{red}3 & 10 \\ \hline \overline{\text{black fur}} & \small{0 } & \small{14} & 14 \\ \hline & 7 & 17 & 24 \\ \end{array}\)
