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Difficulty: HardRatios, Rates, and Proportions

A municipal authority allocates water from a central reservoir to three sectors: Agriculture, Industry, and Residential. Initially, the ratio of the volume of water allocated to Agriculture to that of Industry is 5:35 : 3, and the ratio of the volume allocated to Industry to that of Residential is 4:54 : 5. During a drought, the total supply is reallocated such that Agriculture's allocation is decreased by 20%20\%, Industry's allocation is decreased by 10%10\%, and Residential's allocation is increased by 12%12\%. After these adjustments, what is the ratio of Agriculture's new allocation to Residential's new allocation?

  1. 20:2120 : 21Answer
  2. B
    5:75 : 7
  3. C
    4:34 : 3
  4. D
    21:2021 : 20
  5. E
    15:1415 : 14

Answer

The ratio of Agriculture's new allocation to Residential's new allocation is 20:2120 : 21.
To find the new ratio, first unify the given ratios Agriculture : Industry (5:35 : 3) and Industry : Residential (4:54 : 5) by finding a common multiplier for Industry (12). This yields an overall initial ratio of 20:12:1520 : 12 : 15. Applying a 20%20\% decrease to Agriculture gives 20×0.80=1620 \times 0.80 = 16, and applying a 12%12\% increase to Residential gives 15×1.12=16.815 \times 1.12 = 16.8. The ratio of the new allocations is 16:16.816 : 16.8, which simplifies to 160:168=20:21160 : 168 = 20 : 21.

Step-by-Step Solution

1
Express the three initial allocations in a unified three-part ratio.
Agriculture : Industry = 5:3=20:125 : 3 = 20 : 12, Industry : Residential = 4:5=12:154 : 5 = 12 : 15. Therefore, Agriculture : Industry : Residential = 20:12:1520 : 12 : 15.
Industry is the common element linking both ratios, so its ratio component must be equalized (LCM of 3 and 4 is 12).
2
Assign algebraic representations to the initial allocations based on the unified ratio.
Let Agriculture's initial allocation be 20x20x, Industry's initial allocation be 12x12x, and Residential's initial allocation be 15x15x.
This allows for exact percentage calculations on consistent base values.
3
Calculate the updated allocations after applying the specified percentage adjustments.
Agriculture's new allocation = 20x×(1−0.20)=16x20x \times (1 - 0.20) = 16x. Residential's new allocation = 15x×(1+0.12)=16.8x15x \times (1 + 0.12) = 16.8x.
A 20%20\% decrease reduces a quantity to 80%80\% of its original value, and a 12%12\% increase expands it to 112%112\% of its original value.
4
Compute and simplify the ratio of Agriculture's new allocation to Residential's new allocation.
\frac{\text{Agriculture}_{\text{new}}}{\text{Residential}_{\text{new}}} = \frac{16x}{16.8x} = \frac{160}{168} = \frac{20}{21}.
Multiplying both terms by 10 eliminates decimals, and dividing both by their greatest common divisor (8) yields the simplified integer ratio 20:2120 : 21.

Key Concept

Combining relative ratios through a common quantity and calculating proportional adjustments.

Alternative Method

Instead of setting a variable xx, assume a concrete initial volume for Industry equal to 1212 units. Consequently, Agriculture is 2020 units and Residential is 1515 units. Agriculture's new volume is 20−4=1620 - 4 = 16 units, and Residential's new volume is 15+1.8=16.815 + 1.8 = 16.8 units. The ratio 1616.8=2021\frac{16}{16.8} = \frac{20}{21} is obtained immediately.
Estimated Time:2m 0s
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Topics

Ratios, Rates, and Proportions

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