Averages, Simple & Compound Interest

1Work with the total, not the average

The average of n values is their sum divided by n, so the sum is the average times n — and the sum is what you should actually carry through a calculation. Almost every average question is easier read as a statement about totals, because totals add and averages do not.

Two consequences fall straight out. Add the same k to every value and the average rises by exactly k; multiply every value by m and the average is multiplied by m. And the average always lies between the smallest and largest value, which rules out a wrong option faster than any arithmetic.

Figure. The horizontal line is the average of 12, 18, 9, 21 and 15. Above it sit surpluses of 3 and 6; below it, deficits of 3 and 6. They cancel exactly, and they must, because the average is the level at which the material above the line would fill the gaps below it. Reading the average as a levelling operation is what makes "one value changes by 10, so the average moves by 10/n" obvious rather than memorised.

How it works

1. Convert to a total at once"The average of 8 numbers is 20" is really "eight numbers summing to 160".
2. Do the arithmetic on the totalAdding, removing or replacing a value changes the total by a known amount.
3. Divide back at the endOnly the final line divides by the count — and check the count has not changed.

2Evenly spaced numbers average themselves

When the values are equally spaced — consecutive integers, an arithmetic progression, every fourth number — the average is the middle term, which is also the mean of the first and the last. No summing is needed at all.

The reason is the pairing: the first and last differ from the centre by the same amount in opposite directions, and so do the second and second-last, all the way in. So the average of the first n natural numbers is (n+1)/2, and the average of the first n odd numbers is n itself.

Figure. Five equally spaced values on a line. The outer pair sits 8 either side of the centre and the inner pair 4 either side, so each pair contributes nothing net and the middle term is left holding the average by itself. With an even count there is no middle term to hold it, and the average falls in the gap between the two central values.

How it works

1. Confirm the spacing is constantThe gaps must all be equal. If they are not, the shortcut is simply false.
2. Average the two ends(first + last)/2. With an odd count this is the middle term you could have read off.
3. Count the terms if you need the sum(last − first)/gap + 1, and the sum is that count times the average.

3When someone joins, leaves or is replaced

Every one of these questions is answered by the same two lines: write the old total, write the new total, and take the difference. The difference is the value that came in, or went out, or the change between the two that swapped.

There is a shortcut worth having. When one member joins and the average rises by d, the newcomer's value is the new average plus the old count times d — because the newcomer must both match the new average and supply the rise for everyone already there.

Figure. Ten students average 15, so their total is 150. Including the teacher lifts the average to 16 across 11 people — total 176. The single difference 26 is the teacher's age. Same picture for a departure or a swap.

How it works

1. Old totalOld count × old average. Write the number, not the expression.
2. New totalNew count × new average. Remember the count has changed if someone joined or left.
3. SubtractThe difference is exactly the value that entered or left. For a replacement, it is the gap between the two.

4Weighted average

Two groups of sizes n₁ and n₂ with averages A₁ and A₂ combine to (n₁A₁ + n₂A₂)/(n₁ + n₂), which is not (A₁ + A₂)/2 unless the groups happen to be the same size. Thirty students averaging 60 and twenty averaging 70 give 64, not 65 — the larger group pulls the answer towards itself.

The alligation reading of the same fact is faster and worth having: the combined average divides the gap between A₁ and A₂ in the ratio n₂ : n₁, inversely to the group sizes. Here it sits 4 above 60 and 6 below 70, and 4 : 6 is the inverse of 30 : 20.

Figure. The combined average is a balance point on the segment joining the two group averages, and the heavier group sits closer to it. The distances come out in the inverse ratio of the weights — 4 to the crowd of 30 and 6 to the crowd of 20 — which is why the answer is 64 and not the midpoint 65.

How it works

1. Weight each average by its countn₁A₁ and n₂A₂ are the two group totals; add them for the combined total.
2. Divide by the combined countn₁ + n₂, not 2. Dividing by 2 is what turns a weighted average into a plain one.
3. Or use the distancesThe answer sits between the two averages, nearer the bigger group, in the inverse ratio.

5Simple interest grows in a straight line

SI = PRT/100. The interest is charged on the original principal every single year, so the same amount is added each year and the total grows linearly. Nothing the interest earns ever earns anything itself.

That linearity makes the multiple-of-the-principal questions trivial. A sum doubles when the interest equals the principal, so RT = 100; it trebles when the interest is twice the principal, so RT = 200. No principal ever appears in either answer.

Figure. ₹8000 at 5% simple interest. The steps are identical — ₹400 every year, for ever — because each year's interest is charged on the ₹8000 and never on anything the earlier years produced. A straight line is the signature of simple interest, and any curvature at all means the question is about compounding.

How it works

1. Interest per year is fixedPR/100, the same every year. Multiply by T for the total interest.
2. Amount is principal plus interestA = P(1 + RT/100). The principal is never touched.
3. For multiples, set SI against PDoubling needs SI = P, trebling needs SI = 2P. The P cancels and only RT survives.

6Compound interest is successive percentage change

A = P(1 + R/100)^T, and CI = A − P. What that formula says is that each year's interest joins the principal and earns in its turn, so the yearly interest is not constant — it grows.

The useful way to hold it is as the percentage chapter's successive-change rule applied over and over. Two years at 10% is not 20% but 10 + 10 + 100/100 = 21%; three years is 33.1%. Once you see compound interest as a chain of percentage increases, the whole of it reduces to something you already know.

Figure. The three bars are the amount at the end of each year, drawn to scale, and the line across them is the original ₹10000. What sits above that line grows by more each year — ₹1000, then ₹1100, then ₹1210 — because the part of the bar above the line has itself started earning. Under simple interest the three bars would rise in equal steps and the line would stay where it is.

How it works

1. Find the growth factor1 + R/100. At 10% that is 1.1, and it multiplies the amount once per period.
2. Raise it to the number of periodsThree years at 10% is 1.1³ = 1.331, so the amount is 133.1% of the principal.
3. Subtract the principal for CIThe formula gives the amount. The question usually wants the interest — read which.

7The gap between compound and simple

For one year the two are identical — nothing has had time to compound. From two years on, compound interest is ahead, and the gap has a closed form: P(R/100)² over two years, and P(R/100)²(3 + R/100) over three.

The two-year formula is not something to take on trust. The gap is exactly one year's interest on the first year's interest: the first year earns PR/100, and the second year that sum itself earns R% of it, which is P(R/100)². Reading it that way makes the three-year version plausible too, since by then two earlier years' interest are earning.

Figure. ₹8000 at 5%, five years. The two lines leave the axis together and stay together for a full year — there is nothing to compound yet. They separate at the end of year two by ₹20 and the gap then widens every year, because the compound curve bends upwards while the simple one stays straight. By year five the ₹8000 has become ₹10000 under simple interest and ₹10210 under compound.

How it works

1. Check the periodOne year: no difference at all. Two or three years: use the closed form.
2. Use the difference directlyNever compute CI and SI separately and subtract; the formula is one multiplication.
3. Invert it if the gap is givenGiven the difference and the rate, P falls out by dividing — that is the usual question.

8Compounding more often

If interest compounds k times a year at an annual rate R, each period carries R/k and there are kT periods in T years: A = P(1 + R/(100k))^(kT). Half-yearly at 10% means 5% twice, not 10% twice, and quarterly means 2.5% four times.

Compounding more often always earns more, but by less and less. At 10% nominal, annual gives 10%, half-yearly 10.25%, quarterly 10.38% and monthly 10.47% — the effective rate creeps up and flattens. The effective annual rate is what makes two differently-compounded offers comparable.

Figure. Bar heights are the extra effective annual return above the 10% nominal — the part the table's amounts hide in their last digits. Annual compounding adds nothing beyond the nominal; halving the period buys +0.25 percentage points, halving again only +0.13 more, and going monthly only +0.09 more. More often always earns more, by less and less — the flattening is the lesson.