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