Examkin

Question

Difficulty: HardPercentages, Percent Change, and Interest

An agricultural cooperative harvested crops from three distinct farming zones: Zone A, Zone B, and Zone C. In Season 1, Zone A accounted for 40%40\% of the cooperative's total crop yield, Zone B accounted for 35%35\%, and Zone C accounted for the remaining 25%25\%.

In Season 2:
- The crop yield in Zone A increased by 15%15\% compared to Season 1.
- The crop yield in Zone B decreased by 20%20\% compared to Season 1.
- The crop yield in Zone C increased by p%p\% compared to Season 1.

If the cooperative's total crop yield in Season 2 was 6%6\% greater than its total crop yield in Season 1, what is the value of pp?

  1. A
    7
  2. B
    11
  3. C
    24
  4. 28Answer
  5. E
    32

Answer

The value of pp is 2828.
The correct value of pp is 2828. Calculating the weighted percentage contributions of each zone shows that Zone A contributes a yield equal to 46%46\% of the original total yield and Zone B contributes 28%28\%, making their combined Season 2 contribution 74%74\% of the original total yield. To reach the required total Season 2 yield of 106%106\% of the original total, Zone C must contribute 106%−74%=32%106\% - 74\% = 32\% of the original total yield. Since Zone C initially produced 25%25\% of the total yield, the percentage increase for Zone C is 32−2525×100%=725×100%=28%\frac{32 - 25}{25} \times 100\% = \frac{7}{25} \times 100\% = 28\%.

Step-by-Step Solution

1
Define total initial yield and express Season 1 yields for each zone in terms of total yield YY.
Zone A yield = 0.40Y0.40Y, Zone B yield = 0.35Y0.35Y, Zone C yield = 0.25Y0.25Y.
Establishing baseline values based on given initial percentage shares.
2
Calculate the Season 2 yields for Zones A and B after their respective percentage changes.
Zone A yield in Season 2 = 0.40Y×1.15=0.46Y0.40Y \times 1.15 = 0.46Y. Zone B yield in Season 2 = 0.35Y×0.80=0.28Y0.35Y \times 0.80 = 0.28Y.
Applying a 15% increase to Zone A and a 20% decrease to Zone B.
3
Set up the total Season 2 yield equation using the given 6% overall increase.
0.46Y+0.28Y+0.25Y(1+p100)=1.06Y0.46Y + 0.28Y + 0.25Y\left(1 + \frac{p}{100}\right) = 1.06Y.
The sum of all three zone yields in Season 2 equals 106% of the total Season 1 yield.
4
Solve for pp.
0.74+0.25(1+p100)=1.06  ⟹  0.25(1+p100)=0.32  ⟹  1+p100=1.28  ⟹  p=280.74 + 0.25\left(1 + \frac{p}{100}\right) = 1.06 \implies 0.25\left(1 + \frac{p}{100}\right) = 0.32 \implies 1 + \frac{p}{100} = 1.28 \implies p = 28.
Subtracting 0.74 from 1.06 gives 0.32, and dividing 0.32 by 0.25 yields 1.28, representing a 28% increase.

Key Concept

Weighted Percent Change and Base Values
Rate this question

Topics

Percentages, Percent Change, and Interest

More questions from this topic