A 15% concentrate is to be mixed with a mixture having a concentration of 48% to obtain 25 gallons of a mixture with concentration of 30%.  How much of the 15% concentrate will be needed?

Let the amount of the 15% concentrate needed be \( x \) gallons. The amount of the 48% concentrate will be \( 25 - x \) gallons since the total mixture needs to be 25 gallons.

We can set up an equation based on the amount of pure concentrate in the two mixtures:

- The 15% concentrate contributes \( 0.15x \) gallons of pure concentrate.
- The 48% concentrate contributes \( 0.48(25 - x) \) gallons of pure concentrate.
- The final mixture has 30% concentration, so it contains \( 0.30 \times 25 = 7.5 \) gallons of pure concentrate.

Now, we can set up the equation:

\[
0.15x + 0.48(25 - x) = 7.5
\]

### Solving the equation:

\[
0.15x + 0.48 \times 25 - 0.48x = 7.5
\]

\[
0.15x + 12 - 0.48x = 7.5
\]

\[
-0.33x + 12 = 7.5
\]

\[
-0.33x = 7.5 - 12
\]

\[
-0.33x = -4.5
\]

\[
x = \frac{-4.5}{-0.33} = \frac{45}{3.3} \approx 13.64
\]

So, approximately **13.64 gallons** of the 15% concentrate will be needed.

Would you like further details or have any questions?

Here are 5 related questions:
1. What would be the formula for mixing two other concentrations?
2. How would the answer change if the target concentration were 40% instead of 30%?
3. How can you find the required amount of 48% concentrate?
4. Can you generalize the equation for any target concentration?
5. What happens if you increase the total volume of the mixture to 50 gallons?

**Tip**: Always verify your equation with units or labels (like gallons of concentrate) to ensure clarity in solving mixture problems.

A 15% concentrate is to be mixed with a mixture having a concentration of 48% to obtain 25 gallons of a mixture with concentration of 30%. How much of the 15% concentrate will be needed?

Solve the mixture problem of combining a 15% concentrate with a 48% concentrate to produce 25 gallons of a 30% mixture. Learn how to set up and solve the linear equation for the exact amounts needed. Suitable for students in Grades 8-10.

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