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Sagot :
To determine how many gallons of pure water should be added to 10 gallons of a solution containing 15% juice so that the resulting solution contains 5% juice, let's consider the following steps:
### Step-by-Step Solution
1. Find the amount of juice in the original solution (a):
- We start with 10 gallons of solution.
- 15% of 10 gallons is juice.
- [tex]\( a = \frac{15}{100} \times 10 = 1.5 \)[/tex] gallons of juice.
2. Identify the original amount of solution in gallons (b):
- The original solution is 10 gallons.
- [tex]\( b = 10 \)[/tex] gallons.
3. Determine the amount of juice added (c):
- We are adding pure water, which contains 0% juice.
- [tex]\( c = 0 \)[/tex].
4. Calculate the amount of total solution added (d):
- Let [tex]\( x \)[/tex] be the number of gallons of pure water added.
- Adding pure water will increase the total volume of the solution.
We know the total solution after adding [tex]\( x \)[/tex] gallons of pure water must have 5% juice.
5. Relate the new total amount of solution and the new percentage of juice:
- Let the new total solution be [tex]\( 10 + x \)[/tex] gallons.
- The amount of juice remains the same initially (1.5 gallons).
- For 5% juice, [tex]\( \frac{1.5}{10 + x} = 0.05 \)[/tex].
6. Solve 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]\( x = \frac{1}{0.05} = 20 \)[/tex].
Hence, the amount of pure water to be added is 20 gallons.
### Summary in a Table
| | Original | Added | New |
|-------------------------------|----------|-------|--------------|
| Amount of juice [tex]\(a\)[/tex] | 1.5 | 0 | 1.5 |
| Total solution [tex]\(b\)[/tex] and [tex]\(d\)[/tex]| 10 | 20 | 30 (10+20) |
### Completed Table
What is the amount of juice in the original solution? [tex]\(a = 1.5\)[/tex] \\
What is the original amount of solution in gallons? [tex]\(b = 10\)[/tex] \\
How much juice is added? [tex]\(c = 0\)[/tex] \\
How much total solution is added? [tex]\(d = 20\)[/tex]
### Step-by-Step Solution
1. Find the amount of juice in the original solution (a):
- We start with 10 gallons of solution.
- 15% of 10 gallons is juice.
- [tex]\( a = \frac{15}{100} \times 10 = 1.5 \)[/tex] gallons of juice.
2. Identify the original amount of solution in gallons (b):
- The original solution is 10 gallons.
- [tex]\( b = 10 \)[/tex] gallons.
3. Determine the amount of juice added (c):
- We are adding pure water, which contains 0% juice.
- [tex]\( c = 0 \)[/tex].
4. Calculate the amount of total solution added (d):
- Let [tex]\( x \)[/tex] be the number of gallons of pure water added.
- Adding pure water will increase the total volume of the solution.
We know the total solution after adding [tex]\( x \)[/tex] gallons of pure water must have 5% juice.
5. Relate the new total amount of solution and the new percentage of juice:
- Let the new total solution be [tex]\( 10 + x \)[/tex] gallons.
- The amount of juice remains the same initially (1.5 gallons).
- For 5% juice, [tex]\( \frac{1.5}{10 + x} = 0.05 \)[/tex].
6. Solve 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]\( x = \frac{1}{0.05} = 20 \)[/tex].
Hence, the amount of pure water to be added is 20 gallons.
### Summary in a Table
| | Original | Added | New |
|-------------------------------|----------|-------|--------------|
| Amount of juice [tex]\(a\)[/tex] | 1.5 | 0 | 1.5 |
| Total solution [tex]\(b\)[/tex] and [tex]\(d\)[/tex]| 10 | 20 | 30 (10+20) |
### Completed Table
What is the amount of juice in the original solution? [tex]\(a = 1.5\)[/tex] \\
What is the original amount of solution in gallons? [tex]\(b = 10\)[/tex] \\
How much juice is added? [tex]\(c = 0\)[/tex] \\
How much total solution is added? [tex]\(d = 20\)[/tex]
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