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21 Feb 2024, 06:41
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There is only one way to select an ingredient or combination of ingredients that results in more than 20% fat.: False
The percentage of carbohydrate will always be more than 40% in a mix containing three or more ingredients.: False
A blend cannot include four ingredients.: False
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
42% (03:05) correct
58%
(03:18)
wrong
based on 745
sessions
History
Date
Time
Result
Not Attempted Yet
The table below shows the composition of five ingredients as a percentage of four nutrients. These ingredients contain no other nutrients. When selecting ingredients to create a blend, they must either be used alone or combined in equal proportions. For example, if A, B, and C are chosen, they must be mixed in a 1:1:1 ratio. Regardless of the selection, the resulting blend must contain exactly 10% minerals.
For each of the following statements, select True if the statement can be verified to be true based on the information provided. Otherwise, select False..
| Composition | ||||
| Ingredients | Carbohydrate | Protein | Fat | Mineral |
|---|---|---|---|---|
| A | 90 | 10 | 0 | 0 |
| B | 50 | 30 | 10 | 10 |
| C | 20 | 30 | 40 | 10 |
| D | 10 | 50 | 35 | 5 |
| E | 40 | 35 | 0 | 25 |
For each of the following statements, select True if the statement can be verified to be true based on the information provided. Otherwise, select False..
| True | False | |
| There is only one way to select an ingredient or combination of ingredients that results in more than 20% fat. | ||
| The percentage of carbohydrate will always be more than 40% in a mix containing three or more ingredients. | ||
| A blend cannot include four ingredients. |
ShowHide Answer
Official Answer
There is only one way to select an ingredient or combination of ingredients that results in more than 20% fat.: False
The percentage of carbohydrate will always be more than 40% in a mix containing three or more ingredients.: False
A blend cannot include four ingredients.: False
30 Jan 2025, 04:24
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Bunuel
Official Solution:
Statement 1:
The requirement to be met is exactly 10% minerals.
Looking at the Minerals column (values: 0, 5, 10, 10, and 25), the possible ways to achieve this are:
1. Single ingredient: • B → Minerals = 10%, Fat = 10%. • C → Minerals = 10%, Fat = 40%.
2. Two ingredients: • B and C → Minerals = (10 + 10) / 2 = 10%, Fat = (10 + 40) / 2 = 25%.
3. Three ingredients: • A, D, and E → Minerals = (0 + 5 + 25) / 3 = 10%, Fat = (0 + 35 + 0) / 3 = 11.67%.
4. Four ingredients:
• A, B, D, and E → Minerals = (0 + 10 + 5 + 25) / 4 = 10%, Fat = (0 + 10 + 35 + 0) / 4 = 11.25%.
• A, C, D, and E → Minerals = (0 + 10 + 5 + 25) / 4 = 10%, Fat = (0 + 40 + 35 + 0) / 4 = 18.75%.
5. All five ingredients: • Minerals = (0 + 5 + 10 + 10 + 25) / 5 = 10%, Fat = (0 + 35 + 10 + 40 + 0) / 5 = 17%.
Since there are two ways to obtain more than 20% fat (C alone and B & C together), the first statement is False.
Statement 2:
The claim means that no valid combination of three or more ingredients should ever result in carbohydrates being 40% or less.
Checking carbohydrate content for valid selections:
Three ingredients:
• A, D, and E → Carbohydrates = (90 + 10 + 40) / 3 = 46.67%.
Since 46.67% is more than 40%, this combination satisfies the claim.
Four ingredients:
• A, B, D, and E → Carbohydrates = (90 + 50 + 10 + 40) / 4 = 47.5%.
Since 47.5% is more than 40%, this combination satisfies the claim.
• A, C, D, and E → Carbohydrates = (90 + 20 + 10 + 40) / 4 = 40%.
Since 40% is not more than 40%, this combination violates the claim.
Five ingredients:
• Carbohydrates = (90 + 10 + 40 + 20 + 50) / 5 = 42%.
Since 42% is more than 40%, this combination satisfies the claim.
Since we found one case (A, C, D, and E) where the carbohydrate percentage is exactly 40% and not more than 40%, the second statement is False.
Statement 3:
In the solution for the first statement, we found two cases where a combination of four ingredients resulted in exactly 10% minerals:
• A, B, D, and E → Minerals = (0 + 10 + 5 + 25) / 4 = 10%.
• A, C, D, and E → Minerals = (0 + 10 + 5 + 25) / 4 = 10%.
Since valid four-ingredient selections exist, the third statement is False.
Correct answer:
There is only one way to select an ingredient or combination of ingredients that results in more than 20% fat. "No"
The percentage of carbohydrate will always be more than 40% in a mix containing three or more ingredients. "No"
A blend cannot include four ingredients. "No"
06 Feb 2025, 21:14
Kudos
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1. The only restriction to making a mixture is the 10% minerals.
2. First, notice that including or discluding ingredients B and C doesn't change anything since they have 10% minerals and the ratio will always be 1:1:...:1.
3. Next, if we were to include A or D we would also need to include E to counter the negative weight and vice versa. However with E being 25% minerals, we must include all three. The math checks out: \(0 * \frac{1}{n} + 5 * \frac{1}{n} + 25 * \frac{1}{n} + 10 * \frac{n - 3}{n} = 30 * \frac{1}{n} + 10 * \frac{n - 3}{n} = 10 * \frac{n}{n} = 10\).
4. Now, let's go through all of the statements.
- There is only one way to select an ingredient or combination of ingredients that results in more than 20% fat. Picking C will give 40% fat and picking B and C will give \(\frac{10\% + 40\%}{2} = 25\%\). So, this statement is false.
- The percentage of carbohydrate will always be more than 40% in a mix containing three or more ingredients. Picking A, C, D, and E gives \(\frac{90\% + 20\% + 10\% + 40\%}{4} = 40\%\). So, this statement is false.
- A blend cannot include four ingredients. Just like in the statement above, we can pick A, C, D, and E. So, this statement is false.
2. First, notice that including or discluding ingredients B and C doesn't change anything since they have 10% minerals and the ratio will always be 1:1:...:1.
3. Next, if we were to include A or D we would also need to include E to counter the negative weight and vice versa. However with E being 25% minerals, we must include all three. The math checks out: \(0 * \frac{1}{n} + 5 * \frac{1}{n} + 25 * \frac{1}{n} + 10 * \frac{n - 3}{n} = 30 * \frac{1}{n} + 10 * \frac{n - 3}{n} = 10 * \frac{n}{n} = 10\).
4. Now, let's go through all of the statements.
- There is only one way to select an ingredient or combination of ingredients that results in more than 20% fat. Picking C will give 40% fat and picking B and C will give \(\frac{10\% + 40\%}{2} = 25\%\). So, this statement is false.
- The percentage of carbohydrate will always be more than 40% in a mix containing three or more ingredients. Picking A, C, D, and E gives \(\frac{90\% + 20\% + 10\% + 40\%}{4} = 40\%\). So, this statement is false.
- A blend cannot include four ingredients. Just like in the statement above, we can pick A, C, D, and E. So, this statement is false.
