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Difficulty: MediumPercentages, Percent Change, and Interest

At the beginning of the year, a cloud storage service had a certain number of active paid subscriptions. During the first six months of the year, the number of active subscriptions increased by 20%20\%. During the second six months of the year, the number of active subscriptions decreased by 15%15\% relative to the total at the middle of the year. If the service had 12,24012,240 active subscriptions at the end of the year, what was the number of active subscriptions at the beginning of the year?

Answer: 12000 subscriptions

Answer

12000
To find the initial subscription count, model the successive percentage changes sequentially. Increasing the initial amount NN by 20%20\% yields 1.20N1.20N. Decreasing that intermediate total by 15%15\% multiplies 1.20N1.20N by 0.850.85, giving 1.02N1.02N. Setting 1.02N=12,2401.02N = 12,240 gives N=12,000N = 12,000.

Step-by-Step Solution

1
Express the mid-year subscription total in terms of the initial subscription count NN.
Mid-year subscriptions =N×(1+0.20)=1.20N= N \times (1 + 0.20) = 1.20N
A 20%20\% increase scales the original quantity by a factor of 1.201.20.
2
Express the end-of-year subscription total in terms of NN after a 15%15\% decrease from the mid-year total.
End-of-year subscriptions =1.20N×(1−0.15)=1.20N×0.85=1.02N= 1.20N \times (1 - 0.15) = 1.20N \times 0.85 = 1.02N
A 15%15\% decrease reduces the new base quantity to 85%85\% of its mid-year value.
3
Set the algebraic expression for the final total equal to the given numerical total and solve for NN.
1.02N=12,240  ⟹  N=12,0001.02N = 12,240 \implies N = 12,000
Dividing the final amount by the net multiplier 1.021.02 yields the original starting value.

Key Concept

Successive Percentage Changes and Base Shift
Estimated Time:1m 30s
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Topics

Percentages, Percent Change, and Interest

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