Let the number of ounces of substance A be denoted by $x$, substance B by $y$, and substance C by $z$.

Step 1: Set up the equations

We know from the problem:

1. Substance A supplies 5% of the required nutrition per ounce, so the contribution of substance A is $0.05x$.
2. Substance B supplies 15% of the required nutrition per ounce, so the contribution of substance B is $0.15y$.
3. Substance C supplies 12% of the required nutrition per ounce, so the contribution of substance C is $0.12z$.

The total contribution of all three substances must sum to 100%, so:

0.05x + 0.15y + 0.12z = 1 \quad \text{(Equation 1: for 100% nutrition)}

Step 2: Use the given restrictions

• First restriction: Substances A and C must be given in equal amounts: $x = z \quad \text{(Equation 2)}$

• Second restriction: The amount of substance B must be one-fifth of either of the amounts of A or C: $y = \frac{1}{5}x \quad \text{(Equation 3)}$

Step 3: Substitute the restrictions into Equation 1

Using $x = z$ and $y = \frac{1}{5}x$, substitute into Equation 1:

$0.05x + 0.15\left(\frac{1}{5}x\right) + 0.12x = 1$

Step 4: Simplify the equation

Simplify the terms:

$0.05x + 0.15 \times \frac{1}{5}x + 0.12x = 1$ $0.05x + 0.03x + 0.12x = 1$ $0.20x = 1$

Step 5: Solve for $x$

$x = \frac{1}{0.20} = 5$

Step 6: Find $y$ and $z$

From $x = z$, we know that: $z = 5$

From $y = \frac{1}{5}x$, we know that: $y = \frac{1}{5} \times 5 = 1$

Final Answer

The number of ounces of each substance is:

• Substance A: $5$ ounces
• Substance B: $1$ ounce
• Substance C: $5$ ounces

Would you like further details or have any questions? Here are some related questions:

1. What happens if the nutrition requirements change for one of the substances?
2. How would the system change if substances A and C didn’t have to be equal?
3. How can you express the problem using matrices?
4. What if substance B had a different percentage contribution?
5. How can you verify the solution using substitution?

Tip: Always check that all constraints are satisfied when solving a system of linear equations!