Lesson 4: Growth and Depreciation

§Text Passage

1Summary 重點摘要

Besides discounts and interest, percentage is used for population growth, car depreciation, and multi-year growth. Learn the multiplier method (1 + r% or 1 - r%) and it's fast.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Growth Rate CalculationThe formula for a growth rate is: growth rate = (new − old) ÷ old × 100%. For example, with old value 50 and new value 60, growth rate = (60 − 50) ÷ 50 = 10 ÷ 50 = 0.2 = 20%, that is, an increase of 20%. Why is the denominator the old value? Because the growth rate measures the change relative to the original amount, and the original 50 is the base. For example, a shop's membership growing from 400 to 460 adds 60 members, a growth rate of 60 ÷ 400 = 0.15 = 15%. The order of work: find the difference first, divide by the old value next, and convert to a percentage last. A common mistake is to divide by the new value 60, getting about 16.7%; the base of a growth rate is always the old value from before the change, so mark the old value before dividing.
2. Block 2: Multiplier Method for New ValueGiven the growth rate, find the new value with the multiplier method: new value = old value × (1 + growth rate). For example, old value 80 growing by 25% gives new value = 80 × 1.25 = 100. Why is the multiplier 1.25? Because the new value has two parts: the original 100% plus the added 25%, together 125%, which is the multiplier 1.25. The method arrives in one step and agrees with 'find the increase, then add': 80 × 25% = 20, and 80 + 20 = 100. Another example: $500 growing by 8% becomes $500 × 1.08 = $540. A common mistake is to multiply by the growth rate alone, computing 80 × 0.25 = 20 and taking it as the new value; the multiplier must be 1 plus the growth rate, since it stands for 'the original plus the added', and without the 1 only the increase remains.
3. Block 3: Depreciation RateThe formula for a depreciation rate is: depreciation rate = (old − new) ÷ old × 100%. For example, a machine's value falling from 200 to 150 gives depreciation rate = (200 − 150) ÷ 200 = 50 ÷ 200 = 0.25 = 25%, a fall of 25%. Why the name depreciation? Because goods, machines and cars lose value as the years of use mount up, and the size of that fall is described by the depreciation rate; its arithmetic mirrors the growth rate, with the same old value as the denominator. For example, a phone dropping from $3000 to $2400 depreciates 600 ÷ 3000 = 20%. A common mistake is to write the numerator as (new − old), which comes out negative; subtract the smaller from the larger to get the fall, divide by the old value, keep the answer positive, and check that a depreciation rate stays below 100%.
4. Block 4: Multiplier Method for DepreciationGiven the depreciation rate, find the new value with the multiplier method: new value = original × (1 − depreciation rate). For example, original 1000 depreciating by 15% gives new value = 1000 × 0.85 = 850. Why is the multiplier 0.85? Because after a 15% depreciation what remains is 85% of the original, that is 100% − 15% = 85%, and the multiplier 0.85 stands for 'the part remaining'. Compare with the growth multiplier 1.25: growth adds and depreciation subtracts, the signs exactly opposite. Another example: a $200000 car depreciating 20% in a year is worth $200000 × 0.8 = $160000 after the year. A common mistake is to use 1 + 15% by slip, inflating the value; a depreciated value is always below the original, so after computing, check at once whether the result is smaller — a wrong direction shows itself immediately.
5. Block 5: Compound GrowthRepeated growth cannot be found by adding the yearly rates; use a power: value after n years = original × (1 + r%) to the power n. For example, growth of 10% a year for 3 years gives 100 × (1.1)³ = 100 × 1.331 = 133.1, an increase of 33.1%, not 30%. Why not 30%? Because compound growth is interest on interest: the new value after the first year becomes the base of the second, so the second year's 10% works on 110 and adds 11 rather than 10, and three years of accumulation exceed 30%. Depreciation works the same way, multiplying by (1 − r%) year after year. A common mistake is to take 10% × 3 = 30% as the three-year growth; whenever growth compounds by the year, the multipliers must be applied year by year, or a power used to compute it in one go.
6. Block 6: Find Original ValueGiven the new value together with a growth or depreciation rate, find the original by division: original = new ÷ multiplier. For example, a number that has grown by 20% to become 120 has multiplier 1.2, so the original = 120 ÷ 1.2 = 100; another example: a machine worth $1500 after 25% depreciation has multiplier 0.75, so the original = $1500 ÷ 0.75 = $2000. Why divide? Because new value = original × multiplier, and with the product and the multiplier known, division restores the multiplicand. Check: 100 × 1.2 = 120 and $2000 × 0.75 = $1500, both matching the questions. A common mistake is to remove the grown percentage first, computing 120 × 0.8 = 96; to reverse a 20% increase you must divide by 1.2, because 120 is already the enlarged number, and subtracting only shrinks it further.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Growth rate = (new - old) ÷ old × 100%
• New value = old × (1 + growth rate)
• After depreciation = original × (1 - depreciation rate)
• Compound growth: (1 + r%)^n
• Find original: divide, don't multiply