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Difficulty: MediumRatios, Rates, and Proportions

An urban hydroponic farm uses a liquid nutrient mixture containing nitrogen, potassium, and calcium in a mass ratio of 3:5:83 : 5 : 8, respectively. A technician prepares a batch of solution where the amount of calcium exceeds the amount of nitrogen by exactly 15 grams15\text{ grams}. How many total grams of nutrient solution will be prepared?

  1. A
    24 grams24\text{ grams}
  2. B
    30 grams30\text{ grams}
  3. C
    45 grams45\text{ grams}
  4. 48 grams48\text{ grams}Answer
  5. E
    80 grams80\text{ grams}

Answer

The total mass of the nutrient solution prepared is 48 grams48\text{ grams}.
The given ratio of nitrogen to potassium to calcium is 3:5:83 : 5 : 8. The difference between calcium and nitrogen in the ratio is 8−3=58 - 3 = 5 parts. Since calcium exceeds nitrogen by 15 grams15\text{ grams}, each ratio part corresponds to 15÷5=3 grams15 \div 5 = 3\text{ grams}. The total mixture consists of 3+5+8=163 + 5 + 8 = 16 parts, so the total mass is 16×3 grams=48 grams16 \times 3\text{ grams} = 48\text{ grams}.

Step-by-Step Solution

1
Determine the difference in ratio parts between calcium and nitrogen
Calcium has 88 parts and nitrogen has 33 parts, so calcium exceeds nitrogen by 8−3=58 - 3 = 5 ratio parts.
The problem states that calcium exceeds nitrogen by 15 grams15\text{ grams}, which corresponds to the difference between their ratio values.
2
Calculate the mass represented by a single ratio part
15 grams5 parts=3 grams per part\frac{15\text{ grams}}{5\text{ parts}} = 3\text{ grams per part}.
Dividing the total difference in grams by the difference in ratio parts gives the value of one part.
3
Find the total number of ratio parts and compute the total mass
Total ratio parts = 3+5+8=163 + 5 + 8 = 16 parts. Total mass = 16 parts×3 grams per part=48 grams16 \text{ parts} \times 3\text{ grams per part} = 48\text{ grams}.
Multiplying the single part mass by the sum of all components yields the total mass of the solution.

Key Concept

Solving multi-part ratio problems by finding the value of one ratio unit using given part-to-part differences.
Estimated Time:1m 15s
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Topics

Ratios, Rates, and Proportions

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