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# halp

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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?

Guest May 16, 2018
#1
+88897
+2

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

CPhill  May 16, 2018
#2
+3035
+2

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

$$27-24=\boxed{3}$$

tertre  May 17, 2018
#3
+20009
+2

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?

$$\large{1.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & & x & \color{red} 10 \\ \hline \overline{\text{black fur}} & & & \small{24-10} \\ \hline & \small{24-17}& \color{red}17 & \color{red}24 \\ \end{array}$$

$$\large{2.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{10-x } & x & 10 \\ \hline \overline{\text{black fur}} & \small{7-(10-x) =} & \small{17-x} & 14 \\ & \small{x-3 } & & \\ \hline & 7 & 17 & 24 \\ \end{array}$$

$$\large{3.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{10-x } & x & 10 \\ & \small{\text{min: } 10-x=0 } & & \\ & \Rightarrow x_{min} = \color{red}10 & & \\ \hline \overline{\text{black fur}} & \small{x-3 } & \small{17-x} & 14 \\ & \small{\text{min: } x-3=0} & \small{\text{min: } 17-x=0 } & \\ & \Rightarrow x_{min} = \color{red}3 & \Rightarrow x_{min} = \color{red}17 \\ \hline & 7 & 17 & 24 \\ \end{array}$$

$$\large{4.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{10-x } & x & 10 \\ & \Rightarrow x_{min} = \color{red}10 & = min({\color{red}10},{\color{red}3},{\color{red}17}) & \\ & & = 3 & \\ \hline \overline{\text{black fur}} & \small{x-3 } & \small{17-x} & 14 \\ & \Rightarrow x_{min} = \color{red}3 & \Rightarrow x_{min} = \color{red}17 \\ \hline & 7 & 17 & 24 \\ \end{array}$$

$$\large{5.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{7 } & \color{red}3 & 10 \\ \hline \overline{\text{black fur}} & \small{0 } & \small{14} & 14 \\ \hline & 7 & 17 & 24 \\ \end{array}$$

heureka  May 17, 2018