Answer :
Sure, let’s solve each part step-by-step:
1. How much of an 8% solution should we use to make 100 g of a 3% solution?
- We want to create 100 grams of a solution that has 3% solute.
- Let [tex]\( x \)[/tex] be the amount (in grams) of the 8% solution we need.
- The amount of solute in the 8% solution is [tex]\( 0.08 \times x \)[/tex].
- The total solute needed in the final solution is [tex]\( 0.03 \times 100 = 3 \)[/tex] grams.
Setting up the equation:
[tex]\[
0.08 \times x = 3
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{3}{0.08} = 37.5
\][/tex]
So, we need 37.5 grams of the 8% solution.
2. How many pounds of chocolate that is 45% cocoa should be mixed with cocoa mass to obtain a chocolate that is 78% cocoa?
- Let [tex]\( z \)[/tex] be the pounds of 45% cocoa chocolate.
- Let [tex]\( y \)[/tex] be the pounds of cocoa mass (which is 100% cocoa).
- Our goal is to have a mixture that is 78% cocoa.
The equation for the cocoa content is:
[tex]\[
0.45z + 1.00y = 0.78(z + y)
\][/tex]
Unfortunately, without specific values for either [tex]\( z \)[/tex] or [tex]\( y \)[/tex], this equation requires additional information to find a unique solution. More information, like one of the amounts or ratio to mix, would be necessary to solve this equation completely.
This step-by-step explanation shows the amount needed to create the desired percentage solutions.
1. How much of an 8% solution should we use to make 100 g of a 3% solution?
- We want to create 100 grams of a solution that has 3% solute.
- Let [tex]\( x \)[/tex] be the amount (in grams) of the 8% solution we need.
- The amount of solute in the 8% solution is [tex]\( 0.08 \times x \)[/tex].
- The total solute needed in the final solution is [tex]\( 0.03 \times 100 = 3 \)[/tex] grams.
Setting up the equation:
[tex]\[
0.08 \times x = 3
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{3}{0.08} = 37.5
\][/tex]
So, we need 37.5 grams of the 8% solution.
2. How many pounds of chocolate that is 45% cocoa should be mixed with cocoa mass to obtain a chocolate that is 78% cocoa?
- Let [tex]\( z \)[/tex] be the pounds of 45% cocoa chocolate.
- Let [tex]\( y \)[/tex] be the pounds of cocoa mass (which is 100% cocoa).
- Our goal is to have a mixture that is 78% cocoa.
The equation for the cocoa content is:
[tex]\[
0.45z + 1.00y = 0.78(z + y)
\][/tex]
Unfortunately, without specific values for either [tex]\( z \)[/tex] or [tex]\( y \)[/tex], this equation requires additional information to find a unique solution. More information, like one of the amounts or ratio to mix, would be necessary to solve this equation completely.
This step-by-step explanation shows the amount needed to create the desired percentage solutions.
