Comprehension
A health-drink company’s R&D department is trying to make various diet formulations, which can be used for certain specific purposes. It is considering a choice of 5 alternative ingredients \( (O, P, Q, R, S) \), which can be used in different proportions in the formulations.
The cost per unit of each ingredient is: \[ O: \text{Rs. } 150,\quad P: \text{Rs. } 50,\quad Q: \text{Rs. } 200,\quad R: \text{Rs. } 500,\quad S: \text{Rs. } 100 \]
The table below gives the composition of these ingredients:
| Ingredient | Carbohydrate % | Protein % | Fat % | Minerals % |
|---|---|---|---|---|
| O | 50 | 30 | 10 | 10 |
| P | 80 | 20 | 0 | 0 |
| Q | 10 | 30 | 50 | 10 |
| R | 5 | 50 | 40 | 5 |
| S | 45 | 10 | 0 | 5 |
Question: 1
For a recuperating patient, the doctor recommended a diet containing 10% minerals and at least 30% protein. In how many different ways can we prepare this diet by mixing at least two ingredients?
For a recuperating patient, the doctor recommended a diet containing 10% minerals and at least 30% protein. In how many different ways can we prepare this diet by mixing at least two ingredients?
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When a nutrient percentage must be exact, pick ingredients whose weighted average can match that value.
Updated On: Jul 31, 2025
- One
- Two
- Three
- Four
- None
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The Correct Option is C
Solution and Explanation
Minerals exactly 10%: from the table, O (10%), Q (10%) meet directly. R has 5%, S has 5%, P has 0%.
To get exactly 10%, mixtures possible:
- O + Q (both have 10%, average = 10%). Protein = avg(30, 30) = 30% .
- O + R: minerals avg(10, 5) can be 10% by proper proportion; protein avg(30, 50) ≥ 30% .
- Q + R: minerals avg(10, 5) adjustable to 10%; protein avg(30, 50) ≥ 30% .
Thus 3 possible ways.
To get exactly 10%, mixtures possible:
- O + Q (both have 10%, average = 10%). Protein = avg(30, 30) = 30% .
- O + R: minerals avg(10, 5) can be 10% by proper proportion; protein avg(30, 50) ≥ 30% .
- Q + R: minerals avg(10, 5) adjustable to 10%; protein avg(30, 50) ≥ 30% .
Thus 3 possible ways.
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Question: 2
Which among the following is the formulation having the lowest cost per unit for a diet having 10% fat and at least 30% protein? (The diet has to be formed by mixing two ingredients.)
Which among the following is the formulation having the lowest cost per unit for a diet having 10% fat and at least 30% protein? (The diet has to be formed by mixing two ingredients.)
Show Hint
For cost minimisation, select cheapest pair that can satisfy nutrient constraints via appropriate mixing ratio.
Updated On: Jul 31, 2025
- P and Q
- P and S
- P and R
- Q and S
- R and S
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The Correct Option is D
Solution and Explanation
Fat exactly 10%: From the table, O (10%), Q (10%), R (40%), S (0%), P (0%).
For at least 30% protein: Q (30%), S (50%) qualify. O (30%) also qualifies.
Lowest cost: O (150), Q (200), S (100). Among combinations giving fat 10%: Q+S (avg fat = (10+0)/2 adjustable to 10%) has lowest combined cost average compared to O+Q or O+S.
For at least 30% protein: Q (30%), S (50%) qualify. O (30%) also qualifies.
Lowest cost: O (150), Q (200), S (100). Among combinations giving fat 10%: Q+S (avg fat = (10+0)/2 adjustable to 10%) has lowest combined cost average compared to O+Q or O+S.
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Question: 3
In what proportion P, Q and S should be mixed to make a diet having at least 60% carbohydrate at the lowest cost per unit?
In what proportion P, Q and S should be mixed to make a diet having at least 60% carbohydrate at the lowest cost per unit?
Show Hint
Test candidate ratios for both nutrient target and cost minimisation simultaneously.
Updated On: Jul 31, 2025
- 2:1:3
- 4:1:2
- 2:1:4
- 3:1:2
- 4:1:1
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The Correct Option is B
Solution and Explanation
Carbohydrates: P(80%), Q(10%), S(45%).
We want min cost while achieving ≥60%. Let ratio be \( 4:1:2 \):
Weighted carb% = \( (4\times80 + 1\times10 + 2\times45) / 7 = (320+10+90)/7 \approx 60 %\).
Cost = \( (4\times500 + 1\times200 + 2\times100) / 7 = 2400/7 \approx 342.86 \).
Checking other ratios gives higher cost; thus optimal = 4:1:2.
We want min cost while achieving ≥60%. Let ratio be \( 4:1:2 \):
Weighted carb% = \( (4\times80 + 1\times10 + 2\times45) / 7 = (320+10+90)/7 \approx 60 %\).
Cost = \( (4\times500 + 1\times200 + 2\times100) / 7 = 2400/7 \approx 342.86 \).
Checking other ratios gives higher cost; thus optimal = 4:1:2.
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Question: 4
The company is planning to launch a balanced diet required for adolescent children.
The diet must contain at least 30% each of carbohydrate and protein, no more than 25% fat and at least 5% minerals. Which one of the following combinations of equally mixed ingredients is feasible?
The company is planning to launch a balanced diet required for adolescent children.
The diet must contain at least 30% each of carbohydrate and protein, no more than 25% fat and at least 5% minerals. Which one of the following combinations of equally mixed ingredients is feasible?
Show Hint
Feasibility checks require testing all conditions simultaneously; one fail invalidates the pair.
Updated On: Jul 31, 2025
- O and P
- R and S
- P and S
- Q and R
- O and S
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The Correct Option is
Solution and Explanation
Check each pair’s average composition:
- O(50,30,10,10) + S(45,50,0,5): Avg carb = 47.5% ≥ 30%, protein = 40% ≥ 30%, fat = 5% ≤ 25%, minerals = 7.5% ≥ 5% ✓ feasible.
Others fail fat ≤ 25% or protein ≥ 30% condition.
- O(50,30,10,10) + S(45,50,0,5): Avg carb = 47.5% ≥ 30%, protein = 40% ≥ 30%, fat = 5% ≤ 25%, minerals = 7.5% ≥ 5% ✓ feasible.
Others fail fat ≤ 25% or protein ≥ 30% condition.
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