Each ounce of a substance A supplies 2% of the nutrition a patient needs. Substance B supplies 10% of the required nutrition per ounce, and substance C supplies 16% of the required nutrition per ounce. If digestive restrictions require that substances A and C be given in equal amounts, and the amount of substance B be one-fifth of either of these other amounts, find the number of ounces of each substance that should be in the meal to provide 100% of the required nutrition.

Let's assume the number of ounces of substance A is x.

Since substance A supplies 2% of the required nutrition per ounce, the total nutrition provided by substance A alone is 0.02x.

According to the digestive restrictions, substance C should also be given in equal amounts as substance A, so the number of ounces of substance C is also x.

Since substance C supplies 16% of the required nutrition per ounce, the total nutrition provided by substance C alone is 0.16x.

Substance B is one-fifth the amount of substance A or C, so the number of ounces of substance B is (1/5)x.

Since substance B supplies 10% of the required nutrition per ounce, the total nutrition provided by substance B alone is 0.1(1/5)x = 0.02x.

To provide 100% of the required nutrition, the total nutrition provided by all three substances should be 1. We can set up the equation:

0.02x + 0.02x + 0.16x = 1

Combining like terms:

0.4x = 1

Dividing both sides by 0.4:

x = 2.5

Therefore, the number of ounces of substance A should be 2.5, substance B should be (1/5) * 2.5 = 0.5, and substance C should be 2.5.

To solve this problem, let's assign variables to the unknown quantities.

Let's say the number of ounces of substance A is x.
Since substances A and C should be in equal amounts, the number of ounces of substance C is also x.
The amount of substance B should be one-fifth of either of these other amounts, so the number of ounces of substance B is (x/5).

Now, let's calculate the nutrition contribution of each substance per ounce.

Substance A supplies 2% of the required nutrition per ounce. Therefore, the nutrition contribution per ounce from substance A is (2/100) * x.

Substance B supplies 10% of the required nutrition per ounce. Therefore, the nutrition contribution per ounce from substance B is (10/100) * (x/5).

Substance C supplies 16% of the required nutrition per ounce. Therefore, the nutrition contribution per ounce from substance C is (16/100) * x.

To find the number of ounces of each substance that should be in the meal to provide 100% of the required nutrition, we need the sum of the nutrition contribution from each substance to add up to 100%.

So, we have the equation:

(2/100) * x + (10/100) * (x/5) + (16/100) * x = 100%

We can simplify this equation:

(2/100) * x + (10/100) * (x/5) + (16/100) * x = 1

Simplifying the fractions:

(1/50) * x + (1/50) * (x/5) + (4/25) * x = 1

Multiplying through by 50:

x + (1/5) * x + 2 * x = 50

Simplifying:

x + (1/5) * x + 2 * x = 50
x + (1/5) * x + 2 * x = 50

Combining like terms:

(1 + 1/5 + 2) * x = 50
(6/5) * x = 50

Dividing both sides by (6/5):

x = 50 / (6/5)
x = 50 * (5/6)
x = 41.67

Since x represents the number of ounces of substance A, we can round up to the nearest whole number:

Number of ounces of substance A = 42

Since substances A and C are equal, the number of ounces of substance C is also 42.

And the amount of substance B is one-fifth of either of these amounts:

Number of ounces of substance B = (42/5) = 8.4

So, to provide 100% of the required nutrition, the meal should contain:
- 42 ounces of substance A
- 42 ounces of substance C
- 8.4 ounces of substance B

If A and C are x, then B = x/5. So

.02x + .10(x/5) + .16x = 1.00