A nutrient ratio fixes a serving ratio
A Sumo wrestler would like to have fish fillet, salad and fries for breakfast. As he is on a special diet, he must make sure that his intake (in grams) of protein, carbohydrates and fats per meal is in the ratio of 4:8:3. The table below shows the nutritional breakdown for one serving of each item.
| Item | Protein (g) | Carbohydrates (g) | Fats (g) |
|---|---|---|---|
| Fish Fillet | 150 | 60 | 25 |
| Salad with dressing | 15 | 30 | 5 |
| Fries | 5 | 250 | 110 |
Calculate the ratio of the servings of fish fillet, salad and fries that the wrestler should take.
[2009/NJC/Prelim/II/2]
Solution
Ratios in linear systems
A Sumo wrestler wants fish fillet, salad and fries for breakfast. His intake of protein, carbohydrates and fats per meal must be in the ratio . One serving contains: fish fillet 150 g protein, 60 g carbohydrate, 25 g fat; salad 15, 30, 5; fries 5, 250, 110. Calculate the ratio of the servings of fish fillet, salad and fries.
Hint
Let be the numbers of servings of fish, salad and fries.
Total protein : carbohydrate : fat , so they are , , for some constant .
Divide every unknown by so the right-hand sides become known numbers.
You may choose to calculate the ratio. The resulting servings describe one normalized meal, not a required meal size: multiplying all servings by the same positive factor preserves the ratio.
simplifies to by dividing by .
Answer · Q
Fish:salad:fries .
Write the three nutrient totals
Protein from each food times its servings, and so on. The right-hand sides are in the required ratio, with unknown scale .
Scale away the unknown
For a nonempty meal, . Dividing each equation by gives a system whose right-hand sides are the plain numbers 4, 8, 3.
Solve
By calculator, or by elimination.
Form the ratio
The common factor cancels in a ratio.
Explore the idea