Why do this problem?

This problem offers students to explore ideas involving percentage change, in particular using the link between multiplication and percentage change. In particular, students explore equivalence of change (ie. if a $10\%$ decrease isn't the opposite of a $10\%$ increase, what is?). The problem is designed to move students towards the Stage 5 concepts of geometric sequences and exponential growth and decay.

Possible approach

You could warm up by finding some repeated 10% increases or decreases, for example “The value of a car decreases by 10% each year. If the original value was £5000, how much is it worth after 3 years?”
“The annual interest on a savings account is 10%. By what percentage does the balance increase over four years?”

Discuss the methods that students have been using. The “multiplier method” (representing a 10% increase as 110% of the original quantity, so multiplication by $\frac{110}{100} = \frac{11}{10}$ or by 1.1) will hopefully emerge.

Ask the students, If a sum invested gains 10% each year how long will it be before it has doubled its value? They could use trials, or a table, to come to the answer which should be 8.

Then ask the second question, If an object depreciates in value by 10% each year how long will it take until only half of the original value remains? The answer to this should be 7.

This can now be a discussion point. Are they surprised the answers are different? Why don’t they get the same answer? If you changed the 10% to a different number, would the answers be the same? Could they ever be the same?

Allow students to explore, using different numbers if they want to. You could encourage students to use spreadsheet and graphical software. You could encourage students to represent the problem algebraically. Graphically, they should see that the curves are not the same. Algebraically, the sticking point is that the expressions are not one another’s reciprocal, which ultimately provides a proof (that is, $\frac12$ and $2$ are reciprocals but $1.1^n$ and $0.9^n$ are not).

Invite students to share their work and encourage them to make connections between the different representations that have been used. Can they explain why the two answers are different? Has anybody found a number such that the increase and decrease do give the same answer? Or, better, has anybody proved that that would be impossible?

Key questions

Is a 10% increase the opposite of a 10% decrease? What happens if you do one and then the other?

Is mutliplying by 1.1 the opposite of mltiplying by 0.9? Why?

Possible support

If students haven’t seen it before, you might want to give them a chance to practice finding some percentages using the multiplier method.

Students may find it difficult to calculate percentage changes when a starting value is not given. If this is the case, tell them to choose a starting value and use that one. Then choose another starting value and check they get the same answer. Then they could use a letter of their choice (which could be any number) as a starting value and check they still get the same answer. For the rest of the problem, you could let students use a letter as their starting value, or a number of their choice (you might recommend using 1).

Possible extension

The problem The Legacy explores the idea of exponentials in the context of long term investments.