Educerie · IB Diploma · Mathematics: applications and interpretation

Topic 1 — Number and algebra

Topic 1 on this course is the practical one: how accurate is a number, how does a quantity grow step by step, and what happens to money over time. Almost every question is set in a context, and the interpretation is usually worth a mark of its own.

1. Accuracy, rounding and error

Significant figures. Unless a question says otherwise, answers go to three significant figures. Count from the first non-zero digit: 0.004071 to 3 s.f. is 0.00407; 40710 to 3 s.f. is 40700.

Round once, at the end. Carry full calculator accuracy through every intermediate step. If you round 3.14159 to 3.14 in part (a) and use that in part (b), your part (b) answer will very often be wrong in the third figure — and that costs the accuracy mark even though your method was perfect. Store values in the calculator's memory rather than retyping them.

Percentage error compares an approximate value to the exact one:

ε = | (vA − vE) / vE | × 100 %

vA is the approximate value, vE the exact one. Two things go wrong here. First, students divide by the approximate value — the denominator is always the exact value. Second, they forget the modulus and report a negative percentage error; it is always positive.

Scientific notation is a × 10^k with 1 ≤ a < 10 and k an integer. a must be at least 1 and strictly less than 10 — 0.42 × 10⁵ is not in the required form, and neither is 42 × 10³.

2. Sequences and series

A sequence is a list of terms; a series is their sum. uₙ is the nth term on its own, Sₙ is the first n terms added together. Mixing them up is the commonest error in this section.

Arithmetic — a constant difference d is added each time:

• uₙ = u₁ + (n − 1)d
• Sₙ = (n/2)(2u₁ + (n − 1)d) or (n/2)(u₁ + uₙ)

Note (n − 1), not n: reaching the fifth term takes four steps.

Geometric — a constant ratio r multiplies each time:

• uₙ = u₁ r^(n−1)
• Sₙ = u₁(r^n − 1)/(r − 1)

On this course geometric sequences almost always model growth or decay. A quantity growing by 6% a year has r = 1.06; one falling by 6% a year has r = 0.94. Getting r from a percentage is where the marks are: an increase of p percent gives r = 1 + p/100, a decrease gives r = 1 − p/100.

The sum to infinity, S∞ = u₁/(1 − r), exists only when |r| < 1 — higher level only on this course, but the condition still has to be stated when it is used.

3. Financial mathematics

This is the section that distinguishes applications and interpretation, and it is worth learning the calculator's finance solver rather than the formulae.

Compound interest.

FV = PV × (1 + r/(100k))^(kn)

where PV is the present value, FV the future value, r the annual interest rate as a percentage, n the number of years and k the number of compounding periods per year.

k is the whole difficulty. Annually k = 1, half-yearly k = 2, quarterly k = 4, monthly k = 12. The exponent is kn — the total number of periods — while the rate is divided by k. Getting one of the two right and the other wrong is the standard mistake, and it produces an answer that looks plausible.

Real versus nominal. Inflation eats returns. If an investment earns 5% and inflation runs at 2%, the real rate is approximately 5 − 2 = 3 percent. A question that mentions inflation is asking for the real value, and simply reporting the nominal figure loses the interpretation mark.

Depreciation is compound decay: the same formula with a negative rate, or equivalently r = 1 − p/100.

Amortisation and annuities (standard level, section 1.7) are the same machinery run in reverse — a loan repaid in equal instalments, or a lump sum drawn down. Use the calculator's finance application; the sign convention is that money paid out is negative and money received is positive, and getting the signs wrong is what produces impossible answers.

4. Higher level extensions

Higher level adds, in outline: laws of logarithms and their use in solving growth equations; simplification with rational exponents; the sum of an infinite geometric sequence; complex numbers in Cartesian, polar and Euler form, used to model alternating current and other oscillations; and matrices — arithmetic, determinants, inverses, solving systems, and finally eigenvalues and eigenvectors used to describe the long-run behaviour of a system that steps forward repeatedly.

What actually loses marks in this topic

1. Rounding at an intermediate step, then being out in the third significant figure at the end.
2. Answering to 2 s.f. when 3 s.f. was the default.
3. Dividing by the approximate value in a percentage error calculation.
4. Reporting percentage error as negative.
5. u₁ + nd instead of u₁ + (n − 1)d.
6. Getting k wrong in compound interest, or dividing the rate but not multiplying the exponent.
7. Reading a growth rate of 6% as r = 0.06 rather than r = 1.06.
8. Giving a number where the question asked you to interpret it. The units and the context are the answer.

Educerie · written from the published IB syllabus structure for Mathematics: applications and interpretation Topic 1, first assessment 2021. Original text; formulae quoted are standard results that appear in the IB formula booklet. Last reviewed 5 September 2026.