Operations with Signed Numbers: Very Difficult Problems with Solutions

Solution:
m - (m - n) : (-2) ⋅ (-5) = (-4) - [-4 - (-6)] : (-2) ⋅ (-5) =
-4 - 2 : (-2) ⋅ (-5) = -4 - (-1) ⋅ (-5) = -4 - (+5) = -9

Solution:
Evaluate the expression
[x : (y - 3)] ⋅ (-4) - [xy + (-3)] : (-1) =
[(-5) : (-2 - 3)] ⋅ (-4) - [(-5) ⋅ (-2) + (-3)] : (-1) =
[(-5) : (-5)] ⋅ (-4) - [10 + (-3)] : (-1) =
(+1) ⋅ (-4) - (+7) : (-1) = -4 - (-7) = -4 + 7 = 3

Solution:
[tex][2.4 - (0.3 - 0.21) \cdot 2 + 0.44 : (-2)]:\frac{4}{5} =[/tex]
[tex][2.4 - 0.09 \cdot 2 + (-0.22)]:\frac{4}{5} =[/tex]
[tex](2.4 - 0.18 - 0.22):\frac{2}{5} = 2 : \frac{4}{5} = \frac{5}{2} = 2.5[/tex]

Solution:
[tex]3:\left(-\frac{3}{5}\right)-\left(-\frac{4}{5}:2\right)+5\cdot \left[-0.4 - \frac{4}{10}:(-2)\right] = [/tex]
[tex]-5+\frac{2}{5}+5\cdot \left(-0.4 + \frac{1}{5}\right) = [/tex]
[tex]-5+0,4+5\cdot \left(-0.4 + 0.2\right) = [/tex]
[tex]-4.6+5\cdot (- 0.2) = -4.6 - 1 = -5.6[/tex]

Solution:
Each fraction can be written as a difference: [tex]\frac{1}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1}[/tex].
All the middle terms cancel:
[tex]\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\ldots+\left(\frac{1}{14}-\frac{1}{15}\right)=1-\frac{1}{15}=\frac{14}{15}[/tex]
[tex]N=-3\times \frac{14}{15}=-\frac{14}{5}=-2\frac{4}{5}[/tex]

Solution:
Each of the four numbers divides [tex]10[/tex], so it is one of [tex]\pm 1, \pm 2, \pm 5, \pm 10[/tex].
The product of their absolute values is [tex]10[/tex]. Only two different integers have absolute value [tex]1[/tex] (namely [tex]1[/tex] and [tex]-1[/tex]), so the absolute values are [tex]1, 1, 2, 5[/tex] and two of the numbers are [tex]1[/tex] and [tex]-1[/tex]. The other two have product [tex]-10[/tex]:
[tex]1\times (-1)\times 2\times (-5)=10[/tex]: [tex](-5)\times 2=-10[/tex]
[tex]1\times (-1)\times (-2)\times 5=10[/tex]: [tex](-2)\times 5=-10[/tex]
In both cases the product of the smallest and the largest number is [tex]-10[/tex].

Solution:
[tex]7\frac{3}{7}\div (-4)=\frac{52}{7}\times \left(-\frac{1}{4}\right)=-\frac{13}{7}[/tex], [tex]9. 12\div (-5. 7)=-1. 6[/tex]
[tex]x\div \left(-\frac{13}{7}\right)-1. 6=-8. 6 \Rightarrow -\frac{7x}{13}=-7 \Rightarrow x=13[/tex]
[tex]13y=-1[/tex], [tex]\frac{x}{13}=1[/tex]
[tex]A=-|-1+1|+|-1|-|-(-1)|=0+1-1=0[/tex]