Mixture Problems: Problems with Solutions

Solution:
The alloy contains [tex]200-112=88[/tex] ounces of zinc.
[tex]112:88=14:11[/tex]

Solution:
Take one part as the unit. The pure alcohol is [tex]4\times 0.075=0.3[/tex] parts, and the whole solution is [tex]1+4=5[/tex] parts:
[tex]\dfrac{0.3}{5}=0.06=6\%[/tex]

Solution:
The amount of salt does not change: [tex]0.1\times 9=0.9[/tex]
[tex]0.9=0.06(9+x) \Rightarrow 9+x=15 \Rightarrow x=6[/tex]

Solution:
Veal in the first mixture: [tex]0.75\times 12=9[/tex] lb. The second mixture is [tex]100\%-49\%=51\%[/tex] veal: [tex]0.51\times 20=10.2[/tex] lb.
[tex]\dfrac{9+10.2}{12+20}=\dfrac{19.2}{32}=0.6=60\%[/tex]

Solution:
The sugar does not change: [tex]0.5\times 2=1[/tex] lb. The new solution weighs [tex]1\div 0.3=\frac{10}{3}[/tex] lb:
[tex]\dfrac{10}{3}:2=5:3[/tex]
Shortcut: with the same amount of sugar, the amounts are in the inverse ratio of the concentrations, [tex]50:30=5:3[/tex].

Solution:
The solution is [tex]80[/tex] fl oz, and [tex]0.35\times 80=28[/tex] fl oz of it is the substance, all of which came from the concentrate:
[tex]\dfrac{28}{50}=0.56=56\%[/tex]

Solution:
Take [tex]x[/tex] of the first alloy and [tex]y[/tex] of the second:
[tex]0.21x+0.28y=0.24(x+y) \Rightarrow 0.04y=0.03x \Rightarrow x:y=4:3[/tex]
Check: [tex]\dfrac{4\times 21+3\times 28}{7}=\dfrac{168}{7}=24[/tex]

Solution:
[tex]0.26x+0.68\times 40=0.32(x+40)[/tex]
[tex]0.26x+27.2=0.32x+12.8 \Rightarrow 0.06x=14.4 \Rightarrow x=240[/tex]

Solution:
Fineness [tex]800[/tex] means [tex]80\%[/tex] gold. Let [tex]x[/tex] lb of the [tex]700[/tex] alloy be used:
[tex]0.7x+0.85(1.5-x)=0.8\times 1.5[/tex]
[tex]1.275-0.15x=1.2 \Rightarrow x=0.5[/tex]
So [tex]0.5[/tex] lb of the [tex]700[/tex] alloy and [tex]1[/tex] lb of the [tex]850[/tex] alloy were used.

Solution:
Let [tex]x[/tex] lb of the cheaper flour be used. Counting the cost in cents:
[tex]90x+95(100-x)=93\times 100 \Rightarrow 9500-5x=9300 \Rightarrow x=40[/tex]
[tex]40[/tex] lb of the cheaper flour and [tex]60[/tex] lb of the other were used.

Solution:
The dry matter does not change. The dried mushrooms contain [tex]0.88\times 2=1.76[/tex] lb of it, and fresh mushrooms are [tex]22\%[/tex] dry matter:
[tex]0.22x=1.76 \Rightarrow x=8[/tex]