KBAT Problems
KBAT Problem: Repeated Growth with Indices
A KBAT problem where a quantity grows by the same factor each period, indices turn repeated multiplication into one clean expression.
Recognise the repeated factor
A population that rises by the same percentage each year multiplies by the same factor repeatedly, which is a power. Writing it with an index instead of multiplying step by step is the efficient, mark-earning method.
Understand the problem
An original growth problem. RM2000 is placed in an account paying 5% interest per year, compounded, meaning each year's interest is added before the next year is worked out.
The value after 3 years is wanted. Because the same factor multiplies the balance every year, indices turn three separate multiplications into one expression.
Plan and solve
- Each year the balance is multiplied by 1 + 0.05 = 1.05.
- After 3 years: value = 2000 × (1.05)³.
- Evaluate the power: (1.05)³ = 1.157625.
- Value = 2000 × 1.157625 = RM2315.25 (to 2 decimal places).
Check and a variant
Check year by year: 2000 → 2100 → 2205 → 2315.25, which matches the index method. A twist toward decrease: a car worth RM48 000 loses 12% of its value each year.
The yearly factor is 1 − 0.12 = 0.88, so after 2 years the value is 48 000 × (0.88)² = 48 000 × 0.7744 = RM37 171.20. Growth uses a factor above 1, depreciation a factor below 1, the index method handles both.
Frequently asked questions
Why use a power instead of adding the interest each year?
Because the same factor multiplies the balance every period, and repeated multiplication is exactly what an index records. Writing 2000 × (1.05)³ is shorter and less error-prone than three separate steps, and it scales easily, twenty years is just the power 20.
Adding a flat amount each year would be simple interest, a different model.
What is the multiplier and how do I get it?
For growth it is 1 plus the rate as a decimal, so 5% gives 1.05; for a decrease it is 1 minus the rate, so a 12% fall gives 0.88. That single number is what you raise to the power of the number of periods.
Getting the multiplier right is the step most answers turn on.
How is this different from simple interest?
Simple interest adds the same fixed amount each year, based on the original sum, so it grows in a straight line. Compound interest applies the rate to the latest balance, so interest earns interest and growth speeds up.
Compound problems use a power; simple-interest ones use multiplication by the number of years. Read which the question means.