Percentages and Compound Change

Repeated percentage change — interest, depreciation and population growth — and the reverse question that asks what something was worth before.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ต่อยอดจาก ม.1 บทที่ 7 ร้อยละ ไปยัง ม.5 ดอกเบี้ยและมูลค่าของเงิน
Common Core (US): 7.RP, F-LE
Cambridge: Lower Secondary Stage 9 — Number: fractions, decimals and percentages

3.1 Repeated percentage change

Compound growth and decay

A = A_{0} (1 + \frac{r}{100})^{n}

$r$ negative gives decay. The power is the number of periods — this is the same multiplier as Stage 8, applied repeatedly.

ทำไมถึงเป็นแบบนั้น

Compound and simple interest differ because compound interest is charged on the amount you owe NOW, which already includes earlier interest. That is the whole difference, and it is why credit-card debt grows faster than people expect.

ตัวอย่าง 1

$100, 000$ baht is invested at $6%$ per year, compounded annually. What is it worth after $10$ years, and how much more than simple interest?

วิธีทำ

1.Compound: $100, 000 \times 1.0 6^{10} \approx 100, 000 \times 1.7908$
2. $\approx 179, 080$ baht
3.Simple interest: $100, 000 \times (1 + 0.06 \times 10) = 160, 000$

ตอบ About $179, 080$ baht — roughly $19, 000$ more than simple interest

ตัวอย่าง 2

A motorbike bought for $60, 000$ baht loses $15%$ of its value each year. What is it worth after $3$ years?

วิธีทำ

1.Losing $15%$ leaves $85%$ , so the multiplier is $0.85$
2. $60, 000 \times 0.8 5^{3} = 60, 000 \times 0.614125$

ตอบ About $36, 848$ baht

เคล็ดลับ

The rule of $72$ estimates doubling time: divide $72$ by the percentage rate. At $6%$ that is about $12$ years, which matches the calculation above closely enough to catch an error.

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