Bracket placement

Add to expression 1 + 2x3 - 4x5:6
a / one pair of brackets so that the result is as large as possible
b / one pair of brackets so that the result is as small as possible

Final Answer:

a =  5.6667
b =  -4.6667

Step-by-step explanation:

$$\begin{gathered} q_1=1+2\cdot 3-4\cdot 5/6=\frac{11}{3}=3\frac{2}{3}\doteq 3.6667 \\ {q}_2=(1+2)\cdot 3-4\cdot 5/6=\frac{17}{3}=5\frac{2}{3}\doteq 5.6667 \\ {q}_3=(1+2\cdot 3-4)\cdot 5/6=\frac{5}{2}=2\frac{1}{2}=2.5 \\ {q}_4=1+2\cdot (3-4)\cdot 5/6=-\frac{2}{3}\doteq -0.6667 \\ {w}_4=1+(2\cdot 3-4)\cdot 5/6=\frac{8}{3}=2\frac{2}{3}\doteq 2.6667 \\ {q}_5=(1+2\cdot 3-4)\cdot 5/6=\frac{5}{2}=2\frac{1}{2}=2.5 \\ {q}_6=(1+2\cdot 3-4\cdot 5)/6=-\frac{13}{6}=-2\frac{1}{6}\doteq -2.1667 \\ {q}_7=1+(2\cdot 3-4\cdot 5)/6=-\frac{4}{3}=-1\frac{1}{3}\doteq -1.3333 \\ {q}_8=1+2\cdot (3-4\cdot 5)/6=-\frac{14}{3}=-4\frac{2}{3}\doteq -4.6667 \\ {q}_9=1+2\cdot 3-(4\cdot 5)/6=\frac{11}{3}=3\frac{2}{3}\doteq 3.6667 \\ a=q_2=5.6667 \end{gathered}$$

$b=q_8=(-4.6667)=-4.6667$

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