Question 9·Hard·Percentages
| Year | Subscriptions sold |
|---|---|
| 2012 | 5,600 |
| 2013 | 5,880 |
The manager of an online news service received the report above on the number of subscriptions sold by the service. The manager estimated that the percent increase from 2012 to 2013 would be double the percent increase from 2013 to 2014.
How many subscriptions did the manager expect would be sold in 2014?
For problems involving year-to-year changes and percent relationships, first compute the actual percent change between the years given using . Then carefully interpret any wording about “double,” “half,” or other comparisons of percents to adjust that percent correctly. Finally, apply the resulting percent to the correct base year by multiplying the starting value by , and watch out for common traps like doubling the raw increase instead of the percent or applying the percent to the wrong year’s value.
Hints
Start with the change from 2012 to 2013
How many more subscriptions were sold in 2013 than in 2012? Subtract the 2012 value from the 2013 value.
Turn that change into a percentage
Use the formula to find the percent increase from 2012 to 2013. What percent is the increase of 280 out of 5,600?
Relate the two percent increases
The problem says the 2012–2013 percent is double the 2013–2014 percent. If you know the first percent, how do you get a percent that is half as large?
Apply the smaller percent to the 2013 value
Once you know the 2013–2014 percent increase, remember that it should be applied to the 2013 subscriptions (5,880). How do you express “increase a number by a certain percent” as a multiplication?
Desmos Guide
Find the 2012–2013 percent increase as a decimal
In Desmos, type (5880 - 5600) / 5600 and note the decimal result. This is the percent increase from 2012 to 2013 expressed as a decimal.
Halve that percent for 2013–2014
Take that decimal and divide by 2 (for example, type ((5880 - 5600) / 5600) / 2). This new decimal is the percent increase from 2013 to 2014.
Apply the 2013–2014 increase to the 2013 value
Now compute 5880 * (1 + ((5880 - 5600) / 5600) / 2). The value Desmos displays for this expression is the expected number of subscriptions in 2014.
Step-by-step Explanation
Find the increase from 2012 to 2013
Use the numbers in the table:
- 2012: 5,600 subscriptions
- 2013: 5,880 subscriptions
The increase is:
So there were 280 more subscriptions in 2013 than in 2012.
Convert that increase to a percent increase
Percent increase is
Here, the original amount is the 2012 value, 5,600:
So the percent increase from 2012 to 2013 is .
Use the relationship between the two percent increases
The problem says: the percent increase from 2012 to 2013 would be double the percent increase from 2013 to 2014.
That means:
- 2012→2013 percent =
- 2013→2014 percent is half of that, because the earlier one is double the later one.
So the 2013→2014 percent increase is:
Apply the 2013–2014 percent increase to the 2013 value
To find the expected 2014 subscriptions, increase the 2013 value, 5,880, by :
- A increase means multiplying by .
Compute the product:
So the manager expected 6,027 subscriptions in 2014 (choice B).