| Month | Percent change in fund value |
|---|---|
| January | +4% |
| February | −5% |
A mutual fund experienced the percent changes shown above during the first two months of a year. The fund manager wants the value at the end of March to represent an overall increase of 3% compared with its value at the start of January.
By approximately what percent must the fund's value change in March to meet the manager's goal? (Round your answer to the nearest tenth of a percent.)
The table shows successive percent changes: each month starts with the value left by the previous month. Convert each rate to a multiplier, then use a Desmos regression to find the March rate that produces the desired overall increase. The tempting shortcut is to add the percentages, but the changes act on different starting values. Round only at the end.
Hints
- Hint 1
A multiplier includes the whole value you started with. An increase of uses ; a decrease uses . Write a multiplier for each table row before finding March's change.
- Hint 2
February's decrease is taken from January's ending value, not January's starting value. That's why monthly factors multiply. What multiplier describes the manager's goal relative to the start of January?
- Hint 3
Let be March's unknown percent increase. Its multiplier is . Set the product of the three monthly factors equal to the goal's multiplier, then replace with in Desmos to solve for .
Step-by-step
Approach 1: Multiply the monthly factors
Step 1Convert the changes to multipliers
A multiplier tells you how much of a month's starting value remains after its change. The table's January increase gives , and February's decrease gives . The manager wants the March ending value to be times January's starting value.
- Step 2
Match the monthly product to the overall goal
Let be March's percent increase, so its multiplier is . Each month's percent applies to the value entering that month, so multiply the three monthly factors. Both sides measure the March ending value relative to January's start, giving . Adding the percentages would wrongly give every month the same starting value.
- Step 3
Solve for March's percent change
Type in Desmos. The asks Desmos to find the unknown rather than make a slider. Under PARAMETERS, it shows . Round only now: the fund must increase by approximately in March. Choice B.
Approach 2: Compare values using a $100 start
Step 1Find the value entering March
Only percentages are given, so choosing a starting value won't change the percent March needs. Type in Desmos. It displays : January's gain is included before February's decrease is applied.
- Step 2
Find the desired ending value
The manager wants 3% more than January's start. Type on the next line; Desmos displays as the desired value at the end of March.
- Step 3
Divide the gain by March's starting value
A percent change is the change divided by the value immediately before it. Type ; Desmos returns . The denominator is March's starting value, not January's . Rounded to the nearest tenth, the fund needs a increase in March. Choice B.