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Growth-rate questions in plant physiology are short, but they combine biology and a bit of maths, and many students lose marks on the maths. The biology tells you which equation to use; the maths is one line of algebra or one logarithm. This post explains where each equation comes from, shows how to recognise each pattern from a table of readings, and solves for the rate and the time with worked numbers.
Where is this in the syllabus?
The NEET (UG) 2026 syllabus lists "Phases of Plant growth and plant growth rate" under plant growth and development, and the CBSE Biology curriculum for 2026-27 lists "phases of plant growth and plant growth rate" in Chapter 13. The text is section 13.1.4, Growth Rates, of the 2026-27 NCERT Class 11 chapter Plant Growth and Development. Check the syllabus for your exam year.
What makes growth arithmetic or geometric?
NCERT ties each pattern to what happens after a cell divides:
Arithmetic growth: after mitosis, "only one daughter cell continues to divide while the other differentiates and matures." The number of dividing cells stays the same, so the organ adds the same amount in each interval. NCERT's example is a root elongating at a constant rate.
Geometric growth: "both the progeny cells following mitotic cell division retain the ability to divide." The number of dividing cells keeps multiplying, so the organ adds the same proportion in each interval.
A quick count shows the difference. Start with one dividing cell and allow ten rounds of division. With one daughter dividing each time, you gain one cell per round: 11 cells. With both dividing, the count doubles each round: 2¹⁰ = 1,024 cells.
What are the two equations?
Arithmetic: Lt = L0 + rt, where L0 is the length at time zero, Lt the length at time t, and r the elongation per unit time. A plot of length against time is a straight line.
Geometric (exponential): W1 = W0 e^(rt), where W0 is the initial size, W1 the final size, t the time and e the base of natural logarithms. NCERT calls r the relative growth rate and says it is "also the measure of the ability of the plant to produce new plant material, referred to as efficiency index."
The idea is old. In 1919 V. H. Blackman published The Compound Interest Law and Plant Growth in Annals of Botany, comparing plant growth to money growing at compound interest. Plant scientists still analyse growth relative to size: a 2015 review by Pommerening and Muszta describes relative growth rates as tools for assessing "growth performance and growth efficiency of plants".
Worked examples
All readings below are sample numbers for practice.
1. Which pattern is it? Two seedlings are measured daily.
Root length (cm): 2.0, 2.8, 3.6, 4.4. Each day adds 0.8 cm. Equal differences, so arithmetic, with r = 0.8 cm per day.
Shoot fresh mass (g): 2.0, 3.0, 4.5, 6.75. Each day multiplies by 1.5. Equal ratios, so geometric. Since e^r = 1.5, r = ln 1.5 ≈ 0.405 per day.
Check the second: 2.0 × e^(0.405 × 3) ≈ 2.0 × 3.375 = 6.75 g.
2. Arithmetic: predict a length. A root is 3.5 cm long and elongates at 0.8 cm per day. After 4 days, Lt = 3.5 + 0.8 × 4 = 6.7 cm.
3. Geometric: find r from two readings. A callus grows from 3 g to 12 g in 10 days. Then 12 = 3e^(10r), so e^(10r) = 4 and r = ln 4/10 ≈ 0.139 per day.
4. Geometric: find the doubling time. With the same r, the mass doubles when e^(rt) = 2, so t = ln 2/r = 0.693/0.139 ≈ 5 days. That matches the data: 3 to 6 g in 5 days, 6 to 12 g in the next 5.
5. Absolute against relative rate. In one day, leaf P grows from 4 to 6 cm² and leaf Q from 40 to 42 cm². Both gain 2 cm², the same absolute growth. Relative to starting size, P grew by 2/4 = 50 per cent and Q by 2/40 = 5 per cent. NCERT's Figure 13.7 asks you to make exactly this kind of comparison.
How does the S-shaped curve fit in?
Geometric growth cannot go on forever. NCERT describes a slow lag phase, a rapid exponential (log) phase, and then a stationary phase as nutrients become limited. Plotted against time, this gives a sigmoid or S-curve, which NCERT says is typical of cells, tissues and organs growing in a natural environment. The equation W1 = W0 e^(rt) describes only the middle, exponential part. For the population version of the same idea, see our post on organisms and populations.
Which mistakes cost the most marks?
Using the arithmetic equation for proportional data. Check differences against ratios first.
Taking log base 10. r comes from the natural log, ln. If you use log₁₀, multiply by 2.303.
Mixing time units. If r is per day, t must be in days.
Calling r the absolute growth rate. In W1 = W0 e^(rt), r is the relative rate.
Reading the stationary phase as no life. Growth slows because of limits, not because the cells are dead.
For the hormones that control these phases, see our post on plant growth regulators.
A practice routine
Day 1: Write both equations from memory and say, in one sentence each, what the daughter cells do.
Day 2: Make up two four-reading tables, one arithmetic and one geometric, and have a friend classify them.
Day 3: Three "find r" and two "find the doubling time" sums with natural logs.
Day 4: Sketch the sigmoid curve and label which part each equation describes.
If the logarithm step is what slows you down, the Fermi Concept Library covers every concept in mathematics and biology, each with its own lesson, so you can revise natural logs before returning to the biology.
Before your next biology session, work out how long the callus in example 3 takes to grow from 3 g to 100 g, and check that you get about 25 days.
