The table given below shows the amount, in grams, of carbohydrate, protein, fat and all other nutrients, per 100 grams of nutrients in seven food grains. The first column shows the foodgrain category and the second column its codename. The table has some missing values.
The following additional facts are known.
- Both the pseudo-cereals had higher amounts of carbohydrate as well as higher amounts of protein than any millet.
- Both the cereals had higher amounts of carbohydrate than any pseudo-cereal.
- All the missing values of carbohydrate amounts (in grams) for all the foodgrains are non-zero multiples of 5.
- All the missing values of protein, fat and other nutrients amounts (in grams) for all the foodgrains are non-zero multiples of 4.
- P1 contained double the amount of protein that M3 contains.
Solution
| Category | Code | Composition per 100g of food grains | Total | |||
|---|---|---|---|---|---|---|
| Carbs | Protein | Fat | Others | |||
| Cereal | C1 | 0 | 12 | 100 | ||
| C2 | 3 | 10 | 100 | |||
| Millet | M1 | 62 | 10 | 100 | ||
| M2 | 7 | 16 | 100 | |||
| M3 | 56 | 12 | 100 | |||
| Pseudo-cereal | P1 | 66 | 10 | 100 | ||
| P2 | 14 | 8 | 100 | |||
Understanding the set:
In this question, the very first and most important step will be to understand that the sum of every row will be 100.
The table given below shows the amount, in grams, of carbohydrate, protein, fat and all other nutrients per 100 grams.
Therefore, in a food grain, the total amount of nutrients will sum up to 100 grams.
Step 1: Add Multiplication Indicators
| Category | Code | Composition per 100g of food grains | Total | |||
|---|---|---|---|---|---|---|
| Carbs (5x) | Protein (4x) | Fat (4x) | Others (4x) | |||
| Cereal | C1 | 0 | 12 | 100 | ||
| C2 | 3 | 10 | 100 | |||
| Millet | M1 | 62 | 10 | 100 | ||
| M2 | 7 | 16 | 100 | |||
| M3 | 56 | 12 | 100 | |||
| Pseudo-cereal | P1 | 66 | 10 | 100 | ||
| P2 | 14 | 8 | 100 | |||
Clue 3 and clue 4 say:
All the missing values of carbohydrate amounts (in grams) for all the food grains are non-zero multiples of 5.
All the missing values of protein, fat and other nutrients amounts (in grams) for all the food grains are non-zero multiples of 4.
Step 2: Solving for C1
| Category | Code | Composition per 100g of food grains | Total | |||
|---|---|---|---|---|---|---|
| Carbs (5x) | Protein (4x) | Fat (4x) | Others (4x) | |||
| Cereal | C1 | 80 | 8 | 0 | 12 | 100 |
| C2 | 3 | 10 | 100 | |||
| Millet | M1 | 62 | 10 | 100 | ||
| M2 | 7 | 16 | 100 | |||
| M3 | 56 | 12 | 100 | |||
| Pseudo-cereal | P1 | 66 | 10 | 100 | ||
| P2 | 14 | 8 | 100 | |||
Using clue 2:
Both the cereals had higher amounts of carbohydrate than any pseudo-cereal.
This implies that carbs in C1 & C2 > carbs in P1 & P2
We can see that P1 = 66.
Therefore, C1 & C2 will be a multiple of 5 greater than 66.
Calculating for C1 first:
We know that the sum of all nutrients in a food grain = 100.
carbs + protein + fats + other = 100
In C1:
carbs + protein + 0 + 12 = 100
carbs + protein = 88
Therefore, the possibilities are: 70, 75, 80, 85.
In C1, we know protein is a multiple of 4.
If we put carbs = 70:
protein = 18 (not a multiple of 4)
If we put carbs = 75:
protein = 13 (not a multiple of 4)
If we put carbs = 85:
protein = 3 (not a multiple of 4)
If we put carbs = 80:
protein = 8 (a multiple of 4)
Therefore, the only possibility left for C1 is carbs = 80 and protein = 8.
Step 3: Solving for C2
| Category | Code | Composition per 100g of food grains | Total (should be 100) | |||
|---|---|---|---|---|---|---|
| Carbs (5x) | Protein (4x) | Fat (4x) | Others (4x) | |||
| Cereal | C1 | 80 | 8 | 0 | 12 | 100 |
| C2 | 75 | 12 | 3 | 10 | 100 | |
| Millet | M1 | 62 | 10 | 72 | ||
| M2 | 7 | 16 | 23 | |||
| M3 | 56 | 12 | 68 | |||
| Pseudo-cereal | P1 | 66 | 10 | 76 | ||
| P2 | 14 | 8 | 22 | |||
Calculating the same for C2:
We know P1 = 66.
Therefore, C2 will be a multiple of 5 greater than 66.
We know that the sum of all nutrients in a food grain = 100.
carbs + protein + fats + other = 100
In C2:
carbs + protein + 3 + 10 = 100
carbs + protein = 87
Therefore, the possibilities are: 70, 75, 80, 85.
But in C2, we know protein is a multiple of 4.
If we put carbs = 70:
protein = 17 (not a multiple of 4)
If we put carbs = 75:
protein = 12 (a multiple of 4)
If we put carbs = 85:
protein = 2 (not a multiple of 4)
If we put carbs = 80:
protein = 7 (not a multiple of 4)
Therefore, the only possibility left for C2 is carbs = 75 and protein = 12.
Step 4: Solving for P2
| Category | Code | Composition per 100g of food grains | Total (should be 100) | |||
|---|---|---|---|---|---|---|
| Carbs (5x) | Protein (4x) | Fat (4x) | Others (4x) | |||
| Cereal | C1 | 80 | 8 | 0 | 12 | 100 |
| C2 | 75 | 12 | 3 | 10 | 100 | |
| Millet | M1 | 62 | 10 | 72 | ||
| M2 | 7 | 16 | 23 | |||
| M3 | 56 | 12 | 68 | |||
| Pseudo-cereal | P1 | 66 | 10 | 76 | ||
| P2 | 70 | 14 | 8 | 8 | 100 | |
Now continuing with the 2nd clue and using clue 1 alongside, which says:
Both the pseudo-cereals had higher amounts of carbohydrate as well as higher amounts of protein than any millet.
We know carbs in P1 & P2 < carbs in C1 & C2.
And carbs in P1 & P2 > carbs in M1, M2 & M3 (according to clue 2).
Therefore, for P2:
We know P2 is a multiple of 5 and will be less than 75 but greater than 62.
The only possibilities are: 65 & 70.
But we know that all missing values of fat are multiples of 4.
If carbs in P2 = 65, then fats = 13 (not a multiple of 4).
If carbs in P2 = 70, then fats = 8 (a multiple of 4).
Therefore, carbs in P2 = 70 and fats = 8.
Step 5: Solving for M2
| Category | Code | Composition per 100g of food grains | Total (should be 100) | |||
|---|---|---|---|---|---|---|
| Carbs (5x) | Protein (4x) | Fat (4x) | Others (4x) | |||
| Cereal | C1 | 80 | 8 | 0 | 12 | 100 |
| C2 | 75 | 12 | 3 | 10 | 100 | |
| Millet | M1 | 62 | 10 | 72 | ||
| M2 | 65 | 12 | 7 | 16 | 100 | |
| M3 | 56 | 12 | 68 | |||
| Pseudo-cereal | P1 | 66 | 10 | 76 | ||
| P2 | 70 | 14 | 8 | 8 | 100 | |
Continuing with clue 1:
We know protein in P1 & P2 > protein in M1, M2 & M3.
Therefore, protein in M2 & M3 will be less than 14 (protein in P2).
The possibilities are: 0, 4, 8, 12.
We know: carbs + protein + fats + other = 100.
For M2: carbs + protein + 7 + 16 = 100.
carbs + proteins = 77
Now, trying all possibilities from the previous slide, we get:
If we put protein = 0:
carbs = 77 (not a multiple of 5)
If we put protein = 4:
carbs = 73 (not a multiple of 5)
If we put protein = 8:
carbs = 69 (not a multiple of 5)
If we put protein = 12:
carbs = 65 (a multiple of 5)
Therefore, in M2, carbs = 65 and protein = 12.
Final Solution: Complete Table
| Category | Code | Composition per 100g of food grains | |||
|---|---|---|---|---|---|
| Carbs (5x) | Protein (4x) | Fat (4x) | Others (4x) | ||
| Cereal | C1 | 80 | 8 | 0 | 12 |
| C2 | 75 | 12 | 3 | 10 | |
| Millet | M1 | 62 | 10 | 20 | 8 |
| M2 | 65 | 12 | 7 | 16 | |
| M3 | 56 | 8 | 12 | 24 | |
| Pseudo-cereal | P1 | 66 | 16 | 8 | 10 |
| P2 | 70 | 14 | 8 | 8 | |
We can do the same for M3 as well.
The protein in M3 can be 0, 4, 8 or 12.
Clue 5 says:
P1 contained double the amount of protein that M3 contains.
The protein in P1 thus can be 0, 8, 16 or 24.
Since this P1 protein also has to be more than M1 and M2 protein, it cannot be 0 or 8. This leaves only 16 or 24 as the valid values.
The protein and fats in P1 must add up to 100-66-10 = 24.
If P1 had 24 grams of protein, then it would have 0 grams of fat.
But in clue 4, we are given that all missing fats are non-zero multiples of 4.
Hence, the only possible protein value in P1 is 16, with 8 grams of fats. This gives us 8 grams of protein in M3 and 24 grams of others in M3.
For M1: carbs + protein + fats + other = 100
62 + 10 + fats + other = 100
fats + other = 28
Since both must be multiples of 4, the combinations could be: (4,24), (8,20), (12,16), (16,12), (20,8), (24,4).
Given the constraints and checking consistency, fats = 20 and other = 8 works for M1.
| Food grain Category | Codename of the food grain | Composition per hundred grams of nutrients in the food grains | |||
|---|---|---|---|---|---|
| Carbohydrate (multiple of 5) | Protein (multiple of 4) | Fat (multiple of 4) | Other nutrients (multiple of 4) | ||
| Cereal | C1 | 80 | 8 | 0 | 12 |
| C2 | 75 | 12 | 3 | 10 | |
| Millet | M1 | 62 | 10 | ||
| M2 | 65 | 12 | 7 | 16 | |
| M3 | 56 | 8 | 12 | 24 | |
| Pseudo-cereal | P1 | 66 | 16 | 8 | 10 |
| P2 | 70 | 14 | 8 | 8 | |
The numbers of grams of proteins in 100 grams of nutrients among given food grains in increasing order are 8, 8, 10, 12, 12, 14 and 16. The median value 12