Question

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

1. A grocer mixes two grades of coffee which sell for 70 cents and 80 cents per pound, respectively. How much of each must he take to make a mixture of 80 pounds which he can sell for 76 cents per pound?

2. How many quarts of pure alcohol must be added to 40 quarts of a mixture that is 35% alcohol to make a mixture that will be 48% alcohol?

3. If a container contains a mixture of 5 gallons of white paint and 11 gallons of brown paint, how much white paint must be added to the container so that the new mixture will be two-thirds white paint?

4. Can the sum of three consecutive odd integers be (a) 25? (b) 45?

5. A tank can be filled in 9 hours by one pipe alone, in 12 hours by a second pipe alone, and can be drained when full, by a third pipe, in 15 hours. How long would it take to fill the tank if it is empty, and if all pipes are in operations?

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

`(35)/(100) * 40=14` quarts

Let x be the number of quarts of alcohol to be added, then
Volume of alcohol in the new mixture

`(48)/(100) * (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

`=(5)/(5+11)=(5)/(16)`

Let x be the number of gallons of white paint to be added, then

`(5+x)/(16+x)=(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

`(1)/(t) = (1)/(9) + (1)/(12) - (1)/(15) = (20+15-12)/(180) = (23)/(180) \\ \\ \Rightarrow t= (180)/(23) =7.83`
Therefore, it will take 7.83 hours to get the tank filled up.