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Difficulty: MediumPercent Change and Interest

At the beginning of the year, a cloud storage service had a total of SS active subscribers. During the first quarter, the number of subscribers increased by 20%20\%. In the second quarter, the number of subscribers decreased by 10%10\%. In the third quarter, the number of subscribers increased by 25%25\%. If the total net increase in subscribers from the beginning of the year to the end of the third quarter was 70007{}000, what was the value of SS?

Answer: 20000 subscribers

Answer

The initial number of subscribers SS was 2000020{}000.
Each quarter's percent change applies to the balance at the start of that quarter. An increase of 20%20\% multiplies the quantity by 1.201.20. A decrease of 10%10\% multiplies it by 0.900.90. A subsequent increase of 25%25\% multiplies it by 1.251.25. The cumulative factor is 1.20×0.90×1.25=1.351.20 \times 0.90 \times 1.25 = 1.35. Since the final subscriber total is 1.35S1.35S, the net gain is 0.35S0.35S. Given that 0.35S=70000.35S = 7{}000, dividing 70007{}000 by 0.350.35 yields S=20000S = 20{}000.

Step-by-Step Solution

1
Calculate the composite growth factor after three quarters of successive percentage changes.
Growth factor =(1.20)×(0.90)×(1.25)=1.35= (1.20) \times (0.90) \times (1.25) = 1.35
Successive percentage changes multiply together, each using the previous period's value as its new base.
2
Express the net increase in terms of the initial amount SS.
Net increase =1.35S−S=0.35S= 1.35S - S = 0.35S
The net change is the final value minus the starting value.
3
Set the net increase equal to 70007{}000 and solve for SS.
S=70000.35=20000S = \frac{7{}000}{0.35} = 20{}000
Dividing the absolute net increase by the net percentage increase rate yields the original base value.

Key Concept

Successive Percent Change and Base Value Calculation
Estimated Time:1m 30s
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Topics

Percent Change and Interest

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