Simple growth, compound growth and comparisons

Name the starting amount and the period of change

A growth rule needs more than a percentage. State the starting amount, what one period means, and which amount supplies the percentage base at each step. Twenty percent of the original amount each period and twenty percent of the current amount each period agree at the first step, but need not agree later. These are different update rules, not different methods of calculating one specified rule.

We use nonnegative amounts and count completed periods as 0, 1, 2 and so on. Period zero means before any update, so the value is the stated starting amount P. Write the per-period signed fractional change as r: +20% is r = 1/5, −10% is r = −1/10, and no change is r = 0. Keep the quantity’s unit and the period label visible. A mathematical amount may be continuous; a count of indivisible objects needs additional whole-count conditions.

Simple growth adds the same original-base amount

Start with 100 units and add 20% of that original amount each period. The increment is fixed at 20 units. The values at periods 0, 1, 2 and 3 are 100, 120, 140 and 160 units. Each new value is the previous value plus 20; the percentage base remains the original 100 even after the current amount changes.

In general, a fixed-increment rule adds an amount d each period: next amount = current amount + d. When d is specified as a fraction r of the original amount, d = P × r. This original-base percentage rule is the simple model compared here. A fixed negative d gives a fixed decline. More generally, an externally supplied fixed addition can be specified directly in quantity units, without being a percentage of the original; the zero-start case will make that distinction important.

Compound growth multiplies the current amount at every step

Now start with the same 100 units and increase the current amount by 20% each period. The factor is 1 + 1/5 = 6/5. The first value is 120; the second is 120 × 6/5 = 144; the third is 144 × 6/5 = 172.8 units. The increments are 20, 24 and 28.8, because the same percentage acts on a changing base.

The update rule is next amount = (1 + r) × current amount. For positive starting amounts and r > 0, repeated multiplication gives growth. For −1 < r < 0, the positive factor is smaller than one and gives decay. At r = 0 the factor is one, so the amount stays fixed. Compound describes the repeated current-base operation; it does not guarantee an increase or a particular result in the world.

Write the update rule before compressing it into a formula

A table keeps the period number beside its amount. For the 100-unit, +20% comparison, the first four simple values are 100, 120, 140, 160; the corresponding compound values are 100, 120, 144, 172.8. Write S(n) and C(n) for their values after n completed periods. The initial conditions are S(0) = C(0) = P. Their recursive rules use one known value to obtain the next: S(n + 1) = S(n) + P × r and C(n + 1) = (1 + r) × C(n).

After n additions, the simple formula is S(n) = P + n × P × r = P × (1 + n × r). After a positive whole number n of multiplications, the compound formula is C(n) = P × (1 + r)^n. The exponent counts repeated factors, not percentage points. If the factor is nonzero, its zeroth power is one, so C(0) = P agrees with the initial condition. If the factor is zero, define the value at period zero directly as P and use the update rule for later periods; no expression 0^0 is needed.

Compare models with the same start, period and horizon

With the same positive starting amount and positive per-period rate, the original-base and current-base models agree at periods zero and one. Later, compound increments are larger because the current base has grown, while the simple increment remains fixed. At period 2 in the +20% example, the amounts are 140 and 144; at period 3 they are 160 and 172.8. Their difference is an amount in the original unit, while each total gain as a percentage uses the original P as its denominator.

A fair numerical comparison must align its conditions: the same initial quantity, the same period length, the same rate interpretation and the same completed time horizon. Twenty percent per day is different from twenty percent per week. Counting two periods for each would compare two days with two weeks. If starts or rules differ, calculate each stated model and identify the difference rather than assuming the word compound decides the outcome. With zero rate or zero start in these percentage-based rules, the compared values can coincide throughout.

Fixed decline and repeated fractional decay have different limits

Starting from 100 units, a simple decline of 20% of the original removes 20 units each period: 100, 80, 60, 40, 20, 0. A sixth update would produce −20, which is outside an ordinary nonnegative-amount interpretation. For positive P and negative r, use the simple rule only while 1 + n × r is nonnegative. The arithmetic formula can still produce a signed value; that does not make it a valid remaining volume or count.

Compound decay at the same −20% rate multiplies the current amount by 4/5: 100, 80, 64, 51.2, 40.96, 32.768. Each removed amount becomes smaller. With positive P and a factor strictly between zero and one, a finite number of exact multiplications remains positive. A −100% rate has factor zero, reaching zero at the first update and staying zero afterward. Rates below −100% give a negative factor and are not used here for ordinary nonnegative amounts.

Zero starts allow update rules, but no relative change against zero

If P = 0, the original-base percentage increment P × r is zero, and current-base multiplication also keeps zero at every period. The resulting values and absolute differences are defined. However, total relative growth (final − P)/P is undefined, because the original denominator is zero. A configured multiplier can still be part of the rule; it cannot be recovered from an observed quotient 0/0.

A different fixed-increment model can start from zero and add 5 units each period: 0, 5, 10, 15. Its absolute rule is fully defined, but those additions are not 5% of the zero original. The amount d = 5 was supplied independently, rather than derived from P × r. External additions, removals or other inputs must appear in the rule if they are part of the question; pure current-base multiplication cannot create a positive amount from zero by itself.

Changed rates multiply their factors; reversals use reciprocals

If successive current-base changes have rates r₁ and r₂, apply their factors in order: final = P × (1 + r₁) × (1 + r₂). Starting with 100 units, +20% followed by −20% gives 100 × 1.2 × 0.8 = 96 units. The percentage numbers sum to zero, but the factors multiply to 0.96 because the second percentage has a different base. This is the same changed-base principle used in the previous lesson, now expressed as an update sequence.

To undo a nonzero factor f, divide by f or multiply by 1/f. A +20% step uses 6/5, whose inverse is 5/6; reversing it therefore decreases the new amount by one sixth, not by one fifth. To recover P from a positive-period compound value with a nonzero factor, divide by the complete factor (1 + r)^n. A zero factor loses the original information and has no reciprocal. For different rates, retain the complete product rather than replacing it with an unsupported average percentage.

Specify rounding and test the requested period boundary

Keeping exact fractions during the updates and rounding only the final report gives the ideal mathematical sequence. Rounding each period’s stored amount before using it as the next base gives a different update rule. Small changes from each rounding can accumulate, so the single final-rounding error bound does not describe the whole difference between those rules. When a question requires per-period rounding or whole items, state that requirement; otherwise preserve the exact values until reporting.

A threshold question needs both the inequality and the allowed times. Starting with 100 units at +20% per completed period, the compound amounts at periods 3 and 4 are 172.8 and 207.36, so period 4 is the first to reach at least 200. The simple amounts reach exactly 200 at period 5. If the question instead requires more than 200, the simple model needs period 6. Check the candidate and its preceding period. These rules are defined at whole completed periods; interpolation does not by itself supply a rule for a fractional period.

Keep the starting amount and period fixed while comparing bases

Begin with 100 units, a +20% rate and period 2 in the explorer. Follow both update rules in the table: the fixed original-base increments give 140 units, while repeated current-base multiplication gives 144. Match these values to the selected graph points and compare the increments from period 1. Change the period while keeping the preset fixed; at period zero both values must return to the stated original 100.

Try the −10% preset, no change, and the zero original. For the preset with a decline of one sixth of the relevant base per period, inspect period 6: fixed original-base decline reaches zero, while current-base decay remains positive. The explorer keeps the arithmetic exact and labels a rounded decimal summary when an exact fraction does not fit a short terminating display. Its graph shows periods 0 through 6 on one fixed scale. Distinguish the model’s stipulated rate, its actual amounts and any relative comparison that would require a nonzero original.

A formula applies within its stated period and conditions

The amount P has a quantity unit; the multiplier 1 + r is dimensionless; n counts completed periods of a specified length. A per-period percentage cannot be transferred to a different period without stating the conversion model. An arithmetic sequence has a fixed difference between consecutive values, while a geometric update uses a fixed multiplier. Those structures describe how values are generated, not an automatic claim that a physical process follows them.

Real processes may have capacity limits, changing inputs, changing rates or indivisible counts. A growing volume model needs enough room; an actual count of indivisible objects or people must be a whole number; a fixed-percentage material loss may cease to represent the process at very small sizes. A finite table checks listed cases but cannot prove the rule for every later time. State the intended horizon and conditions, then treat the resulting calculations as conditional predictions.

Check the initial value, one update and the comparison

Check period zero first. Then calculate one update directly and confirm whether its base was the original or the current amount. For fixed increments, inspect consecutive differences; for repeated multiplication, inspect the factor where the previous value is nonzero, or use the update rule directly at zero. Retain the sign of decline and verify that every amount lies within the allowed domain.

For a final comparison, name whether you are reporting an amount difference, relative gain against the original, a growth factor or a first threshold period. Keep exact equalities separate from rounded approximations and match units, periods and horizons. A recovered original should reproduce the stated final value under the forward rule. The next chapter begins with quantities, units and dimensions, making the measurement structure of these and other mathematical models more explicit.

Worked examples

Try it yourself

1. An amount starts at 80 units. Compare adding 25% of the original each period with increasing the current amount by 25% each period. Find the period-0, period-1 and period-2 values and name what stays fixed in each rule.

   Show the answer and reasoning

   Both start at 80 and reach 100 at period 1. At period 2, the simple amount is 120 and the compound amount is 125. The simple increment is 20; the compound factor is 1.25.

   Period zero precedes all updates. The original-base increment remains 0.25 × 80 = 20. The current-base update multiplies the new 100 by 1.25 at its second step, giving 125 rather than adding another original-base 20.

2. For P = 100 units and r = 1/5 per period, calculate both models at period 3 using their update rules and compact formulas. Give their amount difference and total relative gains.

   Show the answer and reasoning

   S(3) = 160; C(3) = 172.8. The difference is 12.8 units, with total gains +60% and +72.8% against the original.

   The simple model adds 20 three times: 100 + 3 × 20 = 160. The compound model multiplies by 6/5 three times: 100 × (6/5)^3 = 172.8. Subtract the model amounts for the difference, and divide each final-minus-original difference by the positive original 100 for its relative gain.

3. One model reports +20% per day, another +20% per week. Would comparing their period-2 values compare the same elapsed time? What must be aligned when comparing two growth models, and what happens to the amount’s unit?

   Show the answer and reasoning

   No: two days and two weeks differ. Align the start, rate interpretation, period length and elapsed horizon. The amount keeps its quantity unit under a dimensionless multiplier.

   A percentage number alone does not specify time. The periods identify when updates occur; converting between period lengths requires a stated rule. Multiplication by a fraction such as 6/5 changes a length, volume or other quantity’s amount without changing its measurement unit.

4. Starting with 100 units, a simple model removes 20 units per period and a compound model multiplies by 4/5. Find their period-5 values. Is period 6 valid for the simple model as a nonnegative amount? Separately, state period-0 and period-1 values for a −100% compound update.

   Show the answer and reasoning

   At period 5, simple is 0 and compound is 32.768 units. Simple period 6 would be −20 and is outside that nonnegative interpretation. A −100% compound rule starts at 100 in period 0 and reaches 0 in period 1.

   Compute 100 − 5 × 20 and 100 × (4/5)^5. A further fixed subtraction exceeds the remaining amount. For a zero compound factor, use the initial condition and next-value rule: the initial 100 is specified directly, then 0 × 100 = 0, with later values also zero. No 0^0 calculation is required.

5. With original P = 0 and rate +20%, find both percentage-based models after three periods. Can their total relative gain be stated as 0%? Compare with an independently specified rule that starts at zero and adds 5 units per period.

   Show the answer and reasoning

   Both percentage-based models stay at zero. Their total relative gain against zero is undefined. The separate fixed-addition rule gives 15 units after three periods, also without a defined relative gain against its zero original.

   The original-base increment is 0 × 1/5 = 0, and current-base multiplication cannot change zero. Relative gain would divide by the zero original. An external addition of 5 is a valid absolute update, but it was not derived as a percentage of that original.

6. Starting with 100 units, apply +20% then −20% to the current amount. Find the result. If an amount instead reaches 144 after two +20% compound periods, recover its original. Why would a zero compound factor prevent that recovery?

   Show the answer and reasoning

   The changing-rate result is 96 units. The original for the second sequence is 144 ÷ (1.2)^2 = 100 units. A zero factor has no reciprocal and cannot retain one recoverable original.

   The two changed rates give factors 1.2 and 0.8 whose product is 0.96, not 1. For two +20% periods, divide by the whole nonzero factor 1.44. Multiplication by zero sends every original to zero, so a later zero does not identify which original existed.

7. From 100 units at +20% per completed period, find the first period reaching at least 200 in each model, and the first simple period exceeding 200. Does replacing every compound value by its rounded report preserve the exact compound rule?

   Show the answer and reasoning

   At least 200: simple period 5 and compound period 4. More than 200: simple period 6. Using a rounded stored value as the next base generally changes the exact update sequence.

   Simple periods 4, 5 and 6 give 180, 200 and 220. Compound periods 3 and 4 give 172.8 and 207.36. Check both the inequality and the preceding period rather than rounding a period number blindly. A reported rounding can stay separate from the stored exact value; feeding the rounded report back into multiplication is a different rule.

What to carry forward

• Simple original-base growth adds a fixed amount P × r each period; compound change multiplies the current amount by 1 + r.
• Build a table and recursive update first. Period zero is the initial P; compact formulas summarize repeated additions or multiplications under their stated conditions.
• Match starts, rate interpretations, units, period lengths and horizons before comparing. Negative rates describe decline, with different nonnegative limits for fixed and current bases.
• Zero starts allow absolute update rules but no relative gain against zero. Reversing compound factors requires nonzero factors, and changed rates multiply their complete factors.
• Keep exact values until reporting unless a different rounding rule is specified. Check neighboring completed periods for thresholds and limit model predictions to their stated domain.