IMAT Logical Reasoning · Numerical Reasoning

Repeated Growth, Half-Life and Checking by Estimate

Repeated growth multiplies by the same factor each step, a half-life halves the amount each period, and a rough estimate removes most wrong options before any exact calculation.

All Numerical Reasoning notes

Why this matters

The 2023 paper asked a breeding-population count month by month and a ratio after two different half-lives. Estimation and bounds (the fewest, the most, at least) run through many Cambridge items on costs and quantities.

Concept 1 of 2: Repeated growth, doubling and half-life

When something grows by the same factor in every time step, the factors multiply: doubling three times is

\times 8

, not

\times 6

. A half-life is the same idea run backwards: each half-life multiplies what is left by one half. When the rule is a story (each pair breeds once a month, newborns mature after a month), a small table with one row per step is safer than any formula.

Definition

Growth by a factor $r$ each step: after $n$ steps, $N = N_0 \times r^n$ . A 10% rise per year is $r = 1.1$ .
Half-life $T$ : the time for half of the substance to remain. After $t$ the fraction left is $\left(\tfrac{1}{2}\right)^{t/T}$ .
After 1, 2, 3, 4 half-lives: $\tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \tfrac{1}{16}$ remains. It never quite reaches zero.
Two substances with different half-lives each follow their own halving; compare them only after working each out.
Step rules: make a table (step, what is there, what is added) and fill it row by row.

Growth and decay

N = N_0\, r^{\,n} \qquad N = N_0 \left(\tfrac{1}{2}\right)^{t/T}

$N_0$ starting amount
$r, n$ factor per step, number of steps
$T$ half-life, in the same unit as the time $t$

Worked example

A culture of 500 bacteria doubles every 20 minutes. How many are there after 2 hours? Separately, a medicine has a half-life of 6 hours. How much of a 400 mg dose is left after 24 hours?

Practice this conceptself-check · 3 quick reps

Repeated percentages compound

10% a year for 2 years is a 21% rise, not 20%, because the second 10% is taken on the larger amount. Use the multiplier raised to the number of years.

Two half-lives leave a quarter, not nothing

After one half-life half remains; after two, a quarter; after three, an eighth. An option saying the substance is gone after two half-lives treats the halving as a straight-line fall.

Concept 2 of 2: Estimation, bounds and checking the options

Five options usually differ by much more than rounding error: often by a power of ten, or by a simple slip. Rounding every number to one significant figure gives an estimate in seconds that rules most of them out. For questions about the fewest or the most, push every quantity to its extreme in the same direction.

Definition

Estimate: round each number to 1 significant figure, calculate, and pick the option of the right size. Rounding one number up and another down helps the errors cancel.
Quick checks: units; odd or even; last digit; whether the answer must be smaller than one of the given numbers.
Bounds: the most items fit when each item is as small as allowed; the fewest when each is as large as allowed.
Round up when the question asks how many tins, coaches or packs are needed: 4.2 tins means 5 tins.

Worked example

Estimate

\dfrac{4.92 \times 0.0198 \times 612}{3.1}

. Separately, a box holds 3 kg of apples, and each apple weighs between 150 g and 250 g. What are the fewest and the most apples the box can hold?

Practice this conceptself-check · 3 quick reps

"How many are needed" rounds up

If a wall needs 4.2 litres of paint and paint comes in 1 litre tins, you must buy 5 tins. Rounding to the nearest whole number gives 4, which leaves part of the wall unpainted. "How many fit" rounds down instead.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Repeated growth, doubling and half-life
Growth and decay
$N = N_0\, r^{\,n} \qquad N = N_0 \left(\tfrac{1}{2}\right)^{t/T}$

Watch out for (3)

Repeated percentages compound
→ Repeated growth, doubling and half-life
Two half-lives leave a quarter, not nothing
→ Repeated growth, doubling and half-life
"How many are needed" rounds up
→ Estimation, bounds and checking the options

Drill the past-year questions on this topic

Open the bank with this topic pre-filtered.

Open in the bank