An agricultural research station recorded the annual grain harvest of a farm over a four-y… — GRE General Test

An agricultural research station recorded the annual grain harvest of a farm over a four-year period. In the second year, the harvest increased by $20\%$ compared to the first year. In the third year, the harvest increased by $25\%$ compared to the second year. In the fourth year, due to unfavorable weather, the harvest decreased by $20\%$ compared to the third year. If the grain harvest in the fourth year was $3,600$ metric tons, what was the grain harvest, in metric tons, in the first year?

A
2,400
B
2,880
3,000Answer
D
3,060
E
4,500

Answer

3,000 metric tons

To find the initial amount, represent the successive percentage changes as sequential decimal multipliers. A

20\%

increase corresponds to a multiplier of

1.20

, a

25\%

increase corresponds to

1.25

, and a

20\%

decrease corresponds to

0.80

. The net change factor is

1.20 \times 1.25 \times 0.80 = 1.20

. Since the fourth-year harvest equals

1.20

times the first-year harvest, dividing

3,600

1.20

yields

3,000

metric tons.

Step-by-Step Solution

Define variables and write multiplier expressions for each year's percent change

Let

x

be the harvest in the first year. Year 2 harvest

= 1.20x

; Year 3 harvest

= 1.25 \times (1.20x)

; Year 4 harvest

= 0.80 \times (1.25 \times 1.20x)

Percentage increases and decreases must be applied sequentially to the immediately preceding period's value.

Calculate the combined percentage multiplier for the four-year period

Combined multiplier

= 1.20 \times 1.25 \times 0.80 = 1.20

Multiplying the decimal factors yields the overall factor relating the first year's harvest to the fourth year's harvest.

Set up and solve the equation for the first year's harvest (

x

)

1.20x = 3,600 \implies x = \frac{3,600}{1.20} = 3,000

Dividing the fourth year's total harvest by the combined multiplier gives the initial first year harvest.

Key Concept

Successive Percentage Changes and Base Values

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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