Let's solve the problem step by step, filling in the values into the table as we go.
1. Amount of juice in the original solution:
- The original solution is 10 gallons and it contains 15% juice.
- Amount of juice in the original solution is calculated as:
[tex]\[
a = 10 \times \frac{15}{100} = 1.5 \text{ gallons}
\][/tex]
2. Original amount of solution in gallons:
- Given that the original solution is 10 gallons.
[tex]\[
b = 10 \text{ gallons}
\][/tex]
3. Amount of juice added:
- No additional juice is added, we are only adding pure water.
[tex]\[
c = 0 \text{ gallons}
\][/tex]
4. Total solution added:
- Let's denote the amount of pure water added as [tex]\( d \)[/tex].
- We can use the resulting percentage to find [tex]\( d \)[/tex].
- The new solution should have a 5\% juice concentration.
- The amount of juice in the final solution will be the same as the amount of juice in the original solution since we are adding only water.
- To find [tex]\( d \)[/tex], we use the relationship that the final concentration should be 5% when the juice amount remains 1.5 gallons.
- Let [tex]\( x \)[/tex] be the amount of pure water added.
- The new total amount of solution is [tex]\( 10 + x \)[/tex] gallons.
- The concentration equation is:
[tex]\[
\frac{1.5}{10 + x} \times 100 = 5
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
1.5 = 0.05 \times (10 + x)
\][/tex]
[tex]\[
1.5 = 0.5 + 0.05x
\][/tex]
[tex]\[
1 = 0.05x
\][/tex]
[tex]\[
x = 20
\][/tex]
- Therefore, the amount of pure water added is:
[tex]\[
d = 20 \text{ gallons}
\][/tex]
Using the above calculations, we can now fill in the table:
| | Original | Added | New |
|---------------------------|----------|-------|-------------------|
| Amount of juice | [tex]\( a = 1.5 \)[/tex] | [tex]\( c = 0 \)[/tex] | [tex]\( 1.5 \)[/tex] |
| Total solution | [tex]\( b = 10 \)[/tex] | [tex]\( d = 20 \)[/tex] | [tex]\( 30 \)[/tex] |
So, the gallons of pure water to be added to the solution are 20 gallons.
