# Exponential growth and decay · Grade 11 · 3 units

> Exponential growth and decay, Grade 11 · 3 units: Using the formula and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Exponential growth and decay (Grade 11 · 3 units) you practise 8 skills: Spotting an exponential process · Step by step, without a formula · Percentages and the growth factor · Using the formula · The rate from two amounts · How many time units · Comparing processes · Reading a graph.

In a linear process the amount changes by the same number in every time unit. In an exponential process it is multiplied by the same number in every time unit, or changes by the same percent. In growth the amount keeps getting larger, and in decay it keeps getting smaller.

The number you multiply by in every time unit is called the growth factor. Growth of 5 percent is multiplying by 1.05, and decay of 10 percent is multiplying by 0.9. To go back from the factor to the percent, take away 1 and multiply by a hundred.

The amount after some time units is the starting amount times the factor to the power of the number of units. In short: s = a × kᵗ, where a is the starting amount, k is the factor and t is the number of time units. To find an amount from some time units before, divide by the factor once for each time unit.

To find after how many time units an amount is reached, work step by step until you reach it or pass it. From the graph of an exponential process, read the starting amount, and get the factor by dividing a value by the one before it. To compare two processes, work out both after the same number of time units.

**Tip:** Before working anything out, turn the percent into a factor: growth adds to 1, and decay takes away from 1.

## What you learn

### Spotting an exponential process

- Example: Today the number of fish in a lake is 2,000. Each year it grows by 90. What kind of process is this?
- Options: An exponential process · A linear process
- Answer: A linear process
- Solution: Each year the amount changes by the same number, so this is a linear process.

### Step by step, without a formula

- Example: Today the number of people in a town is 60. Each year it is multiplied by 2. What will the number of people in a town be in 3 years?
- Answer: 480
- Solution: Multiply by 2 3 times: 60 × 2 = 120 120 × 2 = 240 240 × 2 = 480 In 3 years, the number of people in a town will be 480.

### Percentages and the growth factor

- Example: Each year the debt on a loan grows by 5 percent. By what number is the debt on a loan multiplied each year?
- Answer: 1.05
- Solution: As a decimal, 5 percent is 0.05, so the number is 1 + 0.05 = 1.05 .

### Using the formula

- Example: Each year the number of fish in a lake grows by 10 percent. Today the number of fish in a lake is 7,000. What will the number of fish in a lake be in 3 years?
- Answer: 9317
- Solution: The factor is 1.1, so work it out: 7,000 × 1.1^3 = 9,317 So in 3 years, the number of fish in a lake will be 9,317.

### The rate from two amounts

- Example: At the start the debt on a loan was 1,875 shekels, and 2 years later it was 2,028 shekels. Each year it grew by the same percent. By what percent did it grow each year?
- Answer: 4%
- Solution: Divide: 2,028 ÷ 1,875 = 1.0816 Try a factor: 1.04^2 = 1.0816 The factor is 1.04, so each year the debt on a loan grew by 4 percent.

### How many time units

- Example: Each year the number of people in a town is multiplied by 2. At the start it is 5. After how many years will it be 160?
- Answer: 5
- Solution: The amounts one after another: 5, 10, 20, 40, 80, 160. The amount 160 comes after 5 years, that is 5 years.

### Comparing processes

- Example: Today the amount in Plan A is 2,400 shekels, and each year it grows by 5 percent. Today the amount in Plan B is 4,500 shekels, and each year it grows by 2 percent. In which will the amount be greater in 4 years?
- Options: Plan A · Plan B
- Answer: Plan B
- Solution: Work out both amounts: 2,400 × 1.05^4 ≈ 2,917.22 4,500 × 1.02^4 ≈ 4,870.94 So in 4 years, the amount in Plan B will be greater.

### Reading a graph

- Example: The graph shows the number of fish in a lake every year. (A graph of the number of fish in a lake over three years. The points are labelled 64, 32, 16 and 8.) What was the number of fish in a lake at the start?
- Answer: 64
- Solution: At the start the time is 0, and there the value is 64.

## Not on the site yet

- Solving an exponential equation algebraically
- Explaining in words why a process is exponential

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 3 units (grades 10 and 11).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 11 · 3 units topics](https://kixplo.com/il/en/math/grade-11-3u/)