Question

PLEASE HELP ME ON THESE PROBLEMS, ITS URGENT! (Help for any of these would be greatly appreciated!)

Answer 1

1.) We can solve the problem by developing a linear model.
Let x represent the quantity to be used of the grade of coffee that sells for $70 per pound, and y represent the quantity to be used of the grade of coffee that sells for $80 per pound, then
$\left\begin{array}{c}x+y=80 . . . (1)\\70x+80y=76(80) . . . (2)\end{array}\right \\ \\ \Rightarrow \left\begin{array}{c}80x+80y=6400 . . . (3)\\70x+80y=6080 . . . (4)\end{array}\right \\ \\ \Rightarrow10x=320 \\ \\ \Rightarrow x=32 \\ \\ \Rightarrow y=80-32=48$
Therefore, 32 pounds of the grade of coffee that sells for $70 per pound, and 48 pounds of the grade of coffee that sells for $80 per pound should be used.


2.) Volume of alcohol in the original mixture
$\frac{35}{100} \times 40=14$ quarts

Let x be the number of quarts of alcohol to be added, then
Volume of alcohol in the new mixture
$\frac{48}{100} \times (40+x)=14+x \\ \\ \Rightarrow19.2+0.48x=14+x \\ \\ \Rightarrow0.52x=5.2 \\ \\ \Rightarrow x=10$

Therefore, 10 quarts of pure alcohol should be added


3.) Original proportion of white paint in the mixture
$=\frac{5}{5+11}=\frac{5}{16}$

Let x be the number of gallons of white paint to be added, then
$\frac{5+x}{16+x}=\frac{2}{3} \\ \\ \Rightarrow3(5+x)=2(16+x) \\ \\ \Rightarrow15+3x=32+2x \\ \\ \Rightarrow x=32-15=17$
Therefore, 17 gallons of white paint should be added.


4a.) Let the three consecutive odd integers be x - 2, x and x + 2, then
$x - 2 + x + x + 2 = 25 \\ \\ \Rightarrow3x=25\Rightarrow x=8.33$
which is not an integer.
Hence, the sum of three consecutive odd integers cannot be 25.

4b.) Let the three consecutive odd integers be x - 2, x and x + 2, then
$x - 2 + x + x + 2 = 45 \\ \\ \Rightarrow3x=45\Rightarrow x=15$
Thus, the three consecutive odd intergers whose sum is 45 are 13, 15 and 17.


5.) If all pipes are open and the tank was initially empty, let t be the time it will take to fill the tank, then
$\frac{1}{t} = \frac{1}{9} + \frac{1}{12} - \frac{1}{15} = \frac{20+15-12}{180} = \frac{23}{180} \\ \\ \Rightarrow t= \frac{180}{23} =7.83$
Therefore, it will take 7.83 hours to get the tank filled up.