MATHEMATICAL STUDIES — PAPER 1 REVISION

AQA Mathematical Studies

Paper 1 Revision

Chapters 1–4 ·60 Marks ·1h 30min ·Calculator Allowed

Ch 1: Analysis of Data

Ch 2: Personal Finance

Ch 3: Modelling & Estimation

Ch 4: Critical Analysis

AQA LEVEL 3

1 hour 30 minutes

60 marks

Calculator allowed

Chapters 1–4

Paper 1 — Structure: Skill → Past Paper

Each skill section is immediately followed by an AQA past paper question on that exact skill

Skill Explanation  →  Past Paper Question  →  Next Skill

CHAPTER 1

Analysis of Data

Covers data collection, sampling methods, measures of average and spread, and graphical representations.

Data Types & Sampling

Averages & Spread

Box Plots & CF

Histograms

4 skills → 4 past paper Qs

CHAPTER 2

Personal Finance

Real-world financial mathematics including income, borrowing, savings, and interest calculations.

Tax, NI & Loans

APR

Mortgages

Compound Interest

4 skills → 4 past paper Qs

CHAPTER 3

Modelling & Estimation

Using mathematical models and estimation techniques to solve real-world problems at scale.

Standard Form

Fermi Estimation

Scaling & Subdividing

3 skills → 3 past paper Qs

CHAPTER 4

Critical Analysis

Evaluating statistical claims, identifying misleading data, and understanding correlation versus causation.

Argument Structure

Misleading Data

Correlation

3 skills → 3 past paper Qs

Data Types & Sampling

Ch 1.1

Data Types

Qualitative

Descriptive — no numerical value

e.g. colour, gender, favourite subject

Quantitative Discrete

Exact countable values only

e.g. number of texts, shoe size, goals scored

Quantitative Continuous

Any value within a range

e.g. height, weight, time, temperature

Exam tip: Always state which type of data is being used and justify your answer — one mark is often awarded for this.

Sampling Methods

Random Sample

Every member of the population has an equal chance of selection.

Stratified Sample

Groups represented in the same proportion as the population.

Cluster Sampling

Random sample taken from naturally occurring sub-groups.

Quota Sampling

Interviewers given a quota of specific types to recruit.

STRATIFIED FORMULA

Number in sample = (Group size ÷ Total) × Sample size

Example: College has 1522 students. Female PT = 465 .

Sample of 20:  465 ÷ 1522 × 20 = 6  |  Male PT: 624 ÷ 1522 × 20 = 8  |  Check: 6+3+8+3 = 20 ✓

PAST PAPER

Stratified Sampling — AQA 2016

AQA Mathematical Studies 1350

Paper 1 · 2016

EXAM TIP Always show the formula (group ÷ total) × sample size to earn the method mark (M1) — even if your final answer is wrong, you can still score 1 mark for correct method.

QUESTION  [3 MARKS TOTAL]

A college wants to find out about students' opinions on the college canteen. The college has 1200 students . 720 are female and 480 are male . The college takes a stratified sample of 50 students .

(a)  How many female students should be in the sample?

2 marks

(b)  Describe one advantage of using a stratified sample rather than a simple random sample.

1 mark

MARK SCHEME

Part (a) — 2 marks

M1

Method:  720 ÷ 1200 × 50

A1

Answer:  30 female students

Check: 30 female + 20 male = 50 ✓

Part (b) — 1 mark

B1

Ensures both genders are represented in the correct proportion to the population.

CHAPTER 1 · AQA MATHEMATICAL STUDIES

Averages & Measures of Spread

KEY FORMULAE REFERENCE

OUTLIER RULE

Value > UQ + 1.5 × IQR  → upper outlier

Value < LQ − 1.5 × IQR  → lower outlier

WORKED EXAMPLE — MEAN FROM FREQUENCY TABLE

COMPARING DISTRIBUTIONS

Always compare one measure of average (mean or median) AND one measure of spread (IQR or SD). A single comparison earns 0 marks .

EXAM TIP

State which average you chose and why . Use median when data has outliers; use mean when all values should contribute. IQR is always preferred over range for spread comparisons.

EXAMPLE: FIND THE MEAN NUMBER OF GOALS SCORED

A football team recorded the goals scored in 20 matches:

①

Calculate Σfx: 0+7+8+9+4 = 28

②

Apply formula: x̄ = Σfx ÷ Σf = 28 ÷ 20 = 1.4 goals

⚠

Do NOT just average the frequencies — always use Σfx ÷ Σf

Past Paper — Averages & Spread

AQA PAPER 1 · 2020 & 2022

EXAM TIP

When comparing distributions , you MUST comment on BOTH a measure of average (median/mean) AND a measure of spread (IQR/SD). One comparison alone earns 0 marks for the comparison question.

Q1 — HISTOGRAM & FREQUENCY DENSITY (AQA 2020)

PAST PAPER QUESTION

4 marks

A histogram shows the times taken by 80 students to complete a puzzle.

The class 5 ≤ t < 10 has frequency density 3.2 .

The class 10 ≤ t < 20 has frequency density 2.4 .

(a) How many students took between 5 and 10 minutes? [2]

(b) How many students took between 10 and 20 minutes? [2]

MARK SCHEME

(a)

Frequency = FD × class width

3.2 × 5 = 16

→ 16 students

[2 marks: 1 method + 1 answer]

(b)

Frequency = FD × class width

2.4 × 10 = 24

→ 24 students

[2 marks: 1 method + 1 answer]

Key

Always: Frequency = Frequency Density × Class Width

Q2 — COMPARING DISTRIBUTIONS (AQA 2022)

PAST PAPER QUESTION

3 marks

Two classes sit the same test. Both have a median mark of 58 .

Class A : IQR = 12  |  Class B : IQR = 28

Write two comparisons between the distributions. [3]

MARK SCHEME

[1]

Both classes have the same median (58) , so the same average performance.

[1]

Class B has a larger IQR (28 vs 12) , so Class B's marks are more spread out / less consistent.

[1]

Class A is more consistent — marks are more tightly clustered around the median.

Key

Must reference both median AND IQR — not just one measure

CHAPTER 1 · SECTIONS 1.4–1.5

Box Plots & Cumulative Frequency

Exam Tip: When comparing two distributions using box plots, you must make one comment about the median (average) AND one about the IQR (spread)— a single comparison alone will not earn full marks.

Box & Whisker Plots

Five-number summary:

Min → LQ → Median → UQ → Max. The

box = IQR

(middle 50% of data).

Whiskers extend to the minimum and maximum values (excluding outliers).

Outlier rule:

A value is an outlier if it is

> UQ + 1.5 × IQR

or

< LQ − 1.5 × IQR

. Plot outliers as separate points (×).

Comparing box plots:

Always comment on

both

the

median

(average)

and

the

IQR

(spread).

Cumulative Frequency Graphs

Plot CF at the UPPER class boundary — NOT the midpoint. Draw a smooth S-shaped curve through the points.

Reading off values:

Median = n/2

 | 

LQ = n/4

 | 

UQ = 3n/4

Percentiles:

The

k

th percentile lies at position

k% × n

on the

cumulative frequency axis.

WORKED EXAMPLE — N = 92 STUDENTS

Median = 46th value  ·  LQ = 23rd value  ·  UQ = 69th value

90th percentile = 0.9 × 92 = 82.8th value → read from graph

AQA MATHEMATICAL STUDIES · CHAPTER 1.4–1.5

Past Paper — Cumulative Frequency (AQA 2014 Sample)

AQA 1350 · Paper 1 · 5 Marks

Source: AQA Mathematical Studies 1350 Sample Paper 2014 — Sports Club Ages Question

QUESTION

The table shows the ages of members of a sports club. There are 100 members in total.

(a) Calculate an estimate for the mean age .  [3 marks]

(b) In which class interval does the median lie?  [2 marks]

PART (a)

Mean from Grouped Data — Use Midpoints

3 marks

1

Find midpoints of each class:

20.5  |  30.5  |  40.5  |  50.5  |  60.5✓ M1

2

Calculate Σfx:

(20.5×18) + (30.5×24) + (40.5×30) + (50.5×16) + (60.5×12)

= 369 + 732 + 1215 + 808 + 726 = 3850✓ M1

3

Mean = Σfx ÷ Σf = 3850 ÷ 100 = 38.5 years✓ A1

PART (b)

Median Class — Cumulative Frequency

2 marks

1

Median position = n ÷ 2 = 100 ÷ 2 = 50th value✓ M1

2

Build cumulative frequencies:

16–25: 18  →  26–35: 42  →  36–45: 72

50th value falls between 42 and 72  ∴ Median lies in 36 – 45 class. ✓ A1

✓

Answer: The median lies in the 36 – 45 class interval.

EXAM TIPS

Always use midpoints for mean from grouped data — never class boundaries. Median position = n/2 = 50th value ; find which class contains it using cumulative totals.

Histograms & Frequency Density

AQA CH 1.6

EXAM TIP

Always calculate FD = Frequency ÷ Class Width before drawing. Check your answer: FD × Class Width must equal the original frequency . For unequal class widths, NEVER use a bar chart — examiners will penalise this. When asked to "compare", you MUST give one average AND one measure of spread.

CORE FORMULA

Frequency Density = Frequency ÷ Class Width

Frequency = FD × Class Width

Area of bar = Frequency  |  Height of bar = Frequency Density (NOT frequency)

WORKED EXAMPLE — CALCULATING FD

KEY RULES

Area = Frequency — the height (FD) alone does NOT give frequency.

Unequal class widths? Always use a histogram — never a bar chart.

Comparing distributions? State one average (mean/median) AND one spread (SD/IQR).

Skew: Positive skew → tail right. Negative skew → tail left. Symmetric → normal.

CHOOSING STATISTICAL METHODS

AQA Mathematical Studies · Chapter 1.6

Past Paper — Histogram (AQA 2020)

AQA Mathematical Studies 1350 · Paper 1 · 2020

EXAM TIP

Frequency = FD × Class Width — ALWAYS multiply the frequency density by the class width. Do NOT just read off the frequency density as the frequency. In a histogram, area = frequency , not height.

QUESTION — 4 MARKS

A histogram shows the times taken by 80 students to complete a puzzle. The histogram provides the following information:

(a)  How many students took between 5 and 10 minutes?  [2 marks]

(b)  How many students took between 10 and 20 minutes?  [2 marks]

MARK SCHEME — FULL WORKING

Part (a) — 5 ≤ t < 10

1

Method:

Frequency = FD × Class Width

M1

2

3.2 × 5 =

16 students

A1

Part (b) — 10 ≤ t < 20

1

Method:

Frequency = FD × Class Width

M1

2

2.4 × 10 =

24 students

A1

Check: 16 + 24 = 40 students in these two classes (out of 80 total) ✓

Chapter 2 — Income Tax, NI & Student Loans

AQA Mathematical Studies · 2015–16 Rates

Source: AQA Mathematical Studies Chapter 2.2–2.3 · 2015–16 tax rates

INCOME TAX

PERSONAL ALLOWANCE (TAX-FREE)

£10,600 per year

TAX BANDS

STEP-BY-STEP METHOD

1

Taxable income = Annual income − £10,600

2

If taxable ≤ £31,785: Tax = 20% × taxable

3

If taxable > £31,785: 20% × £31,785 + 40% ×remainder

NATIONAL INSURANCE & STUDENT LOANS

NATIONAL INSURANCE (MONTHLY)

NI CALCULATION

NI = 12% × (monthly earnings − £672)

STUDENT LOANS

9% × (gross earnings − £17,335)

Only on earnings above the threshold · Written off after 30 years

EXAMPLE

Earn £25,000 → Loan = 9% × (£25,000 − £17,335) = 9%× £7,665 = £689.85/yr

EXAM TIPS

Always subtract the personal allowance first (£10,600) before calculating tax — never apply rates to gross income.

NI uses MONTHLY earnings — if given an annual salary, divide by 12 before applying the NI bands.

Student loan: 9% on earnings ABOVE £17,335 — not on total earnings. Subtract the threshold first.

Show all steps clearly — marks are awarded for method. Write out each calculation separately.

KEY FORMULAE SUMMARY

📌 Taxable = Income − £10,600

📌 Tax = 20% × min(taxable, £31,785) + 40% × excess

📌 NI = 12% × (monthly − £672) if £672–£3,532

📌 Loan = 9% × (earnings − £17,335)

Past Paper — Income Tax & NI (AQA 2016)

AQA PAPER 1 · 2016 · 5 MARKS

QUESTION

In 2015–16: Personal allowance = £10,600 . Basic rate income tax = 20% on taxable income up to £31,785. Higher rate = 40% above £31,785. Sasha earns £21,588 per year.

(a)

Calculate Sasha's annual income tax.

3 marks

(b)

NI rate is 12% on monthly earnings between £672 and £3,532. Calculate Sasha's monthly NI contribution.

2 marks

EXAM TIP

Always subtract the personal allowance FIRST to find taxable income before applying any tax rate.  |  NI is calculated on MONTHLY earnings — divide the annual salary by 12 before applying the NI rate to the band above £672.

Part (a) — Annual Income Tax

3 marks

1

Find taxable income by subtracting personal allowance:

£21,588 − £10,600 = £10,988

[1 mark]

2

Taxable income (£10,988) is below £31,785 → apply basic rate only:

Tax = 20% × £10,988

[1 mark]

3

Calculate:

0.20 × £10,988 = £2,197.60

[1 mark]

Annual Income Tax =

£2,197.60

Part (b) — Monthly NI Contribution

2 marks

1

Convert annual salary to monthly earnings:

£21,588 ÷ 12 = £1,799

[1 mark]

2

£1,799 is between £672 and £3,532 → apply 12% on the band :

NI = 12% × (£1,799 − £672)

= 12% × £1,127 = £135.24

[1 mark]

Monthly NI =

£135.24

APR & Loan Calculations

AQA MATHEMATICAL STUDIES · CHAPTER 2.4–2.5

Exam Tips: Always use APR for comparison — never compare monthly rates directly. · Show full substitution into the formula for method marks. · For "show that" questions, substitute given values and show every step clearly. · Factorise A out of both terms before dividing.

APR Formula & Typical Rates

APR FORMULA

C = A₁/(1+i)^t₁ + A₂/(1+i)^t₂ + …

C

= loan amount borrowed

i

= APR as a decimal (e.g. 0.15 for 15%)

Aₖ

= repayment amount k

tₖ

= time of repayment k (in years)

Always compare loans using APR — lower APR = cheaper loan

TYPICAL APR RATES

Payday loans

> 1,000%

Credit cards

20 – 30%

Bank loan

~ 4%

Mortgage

~ 4%

Student loan

~ 5.5%

Worked Example (AQA 2012 — Rachael)

Scenario: Rachael borrows £4,000 . She repays in two equal instalments of £2,500 —one at end of Year 1, one at end of Year 2.

PART A — VERIFY APR = 16.26%

C = 2500 / 1.1626 + 2500 / 1.1626²

C = 2150.52 + 1849.44

C = £3,999.96 ≈ £4,000

✓

PART B — FIND INSTALMENT A WHEN APR = 15%

4000 = A / 1.15 + A / 1.15²

4000 = A(0.8696 + 0.7561)

4000 = 1.6257A

A = 4000 ÷ 1.6257 = £2,460.02

Check: total repaid (2 × £2,460.02 = £4,920.04) > £4,000 ✓

Source: AQA Mathematical Studies 1350 · Chapter 2.4–2.5

Past Paper — APR Formula

AQA Mathematical Studies · Sample Paper 2013 · 5 Marks

Source: AQA Mathematical Studies 1350 Sample Paper 2013 — Rachael APR Question

CHAPTER 2 · PERSONAL FINANCE · APR

QUESTION

Rachael borrows £4,000 . She agrees to repay the loan in two equal instalments , one at the end of Year 1 and one at the end of Year 2.

(a)

The APR is 16.26% . Show that the APR formula gives C ≈ £4,000 when A = £2,500.

2 marks

(b)

Rachael considers a different loan with APR 15% . She will still repay in two equal instalments. Find the value of each instalment.

3 marks

APR FORMULA

C = A₁/(1+i)^t₁ + A₂/(1+i)^t₂ + …

MARK SCHEME

Exam Tips: For 'show that' — substitute the given values and show every step clearly. For part (b) — factorise A out of both terms first, then divide both sides by the bracket.

PART (A) — SHOW THAT

2 marks

1

Substitute A = 2500 and i = 0.1626 into the APR formula:

2

C = 2500 / 1.1626 + 2500 / 1.1626²

3

= 2150.52 + 1849.44 = £3,999.96 ≈ £4,000✓

PART (B) — FIND EACH INSTALMENT

3 marks

1

Set up equation: 4000 = A/1.15 + A/1.15²

2

Factorise A: 4000 = A(1/1.15 + 1/1.15²) = A(0.8696 + 0.7561)

3

4000 = 1.6257A  →  A = £2,460.02

Mortgages & Compound Interest

AQA CH 2.6–2.8  |  PAPER 1 TOPIC

MORTGAGES

§ 2.6

Recurrence Relation

Balance after each period

KEY FORMULA

A n+1 = A n × (1 + r) − M

r = annual rate (decimal)  |  M = annual repayment

WORKED EXAMPLE

£150,000 mortgage, 5% p.a., £1,000/month

M = £12,000/yr  →  A 1 = 1.05 × £150,000 − £12,000

= £145,500

EXAM TIP

Multiply monthly repayments × 12 for annual M

(1 + r) = multiplier that adds interest each year

COMPOUND INTEREST

§ 2.7

Growth Formula

Interest on interest

KEY FORMULA

Amount = P(1 + r) n

P = principal  |  r = rate (decimal)  |  n = years

WORKED EXAMPLE

£5,000 at 3% p.a. for 4 years

= £5,000 × (1.03) 4 = £5,000 × 1.1255

= £5,627.54

EXAM TIP

Use P(1+r) n — NOT simple interest (P × r × n)

Total repaid must always exceed the original loan

AER & INFLATION

§ 2.8

True Annual Rate

Effective vs nominal

AER FORMULA

r = (1 + i/n) n − 1

i = nominal rate  |  n = compounding periods/year

WORKED EXAMPLE & INFLATION

Monthly rate 0.2% → AER

= (1.002) 12 − 1 = 2.43%

Real rate ≈ nominal rate − inflation rate

EXAM TIP

AER > nominal rate when compounding > once/year

Always use AER to compare savings/loan products

Past Paper — Mortgage Recurrence (AQA 2013 Sample)

AQA 1350 · Sample 2013 · 4 marks

Source: AQA Mathematical Studies 1350 Sample Paper 2013 — Harry mortgage question

QUESTION

Harry's Mortgage

Harry takes out a mortgage for £120,000 . He repays £920 per month . The interest rate is 0.6% per month .

The recurrence relation is given below, where A 0 = £120,000.

RECURRENCE RELATION

A n = 1.006 × A n−1 − 920

PARTS TO ANSWER

a

Explain the significance of

1.006

in the recurrence relation.

[1 mark]

b

Use the recurrence relation to find the amount outstanding at the end of

months 1 and

2

.

[2 marks]

c

After 6 months the balance is

£118,782.26

. Find the amount Harry will have

paid off

in

the first 6 months.

[1 mark]

MARK SCHEME — FULL WORKING

a

Significance of 1.006

[1 mark]

1.006 = 1 + 0.006

The 1 keeps the existing balance; 0.006 adds 0.6% monthly interest.

b

Balance at end of months 1 and 2

[2 marks]

Month 1: A₁ = 1.006 × 120,000 − 920

= 120,720 − 920

= £119,800

Month 2: A₂ = 1.006 × 119,800 − 920

= 120,518.80 − 920

= £119,598.80

c

Amount paid off in 6 months

[1 mark]

Amount paid off = £120,000 − £118,782.26

= £1,217.74

Exam Tip: 1.006 = 1 (keep balance) + 0.006 (add interest). Always show the full substitution for each month — do not skip steps. For part (c), simply subtract the final balance from the original loan amount.

Chapter 2 — Key Formulae Summary

PERSONAL FINANCE

EXAM TIPS

→

Use the multiplying factor approach for ALL percentage problems.

→

For VAT: divide by 1.2 to find price ex-VAT.

→

For reverse percentages: always divide by the multiplying factor.

→

Learn the APR formula — it appears regularly in Paper 1.

→

AER: check whether n = 12 (monthly) or n = 4 (quarterly).

INCOME TAX

Taxable = Income − £10,600

Tax = 20% × first £31,785

      + 40% × remainder

Personal allowance deducted first; two tax bands apply.

NATIONAL INSURANCE

NI = 12% × monthly income

      between £672 – £3,532

Only earnings within the band are taxed at 12%.

STUDENT LOAN

Repayment = 9% ×

  (Earnings − £17,335 )

Only repay when earnings exceed the threshold.

APR

C = Σ Aₖ / (1+i)^tₖ

C = credit advanced

Aₖ = repayments

Sum of discounted repayments equals credit advanced.

MORTGAGE RECURRENCE

Aₙ₊₁ = Aₙ(1+r) − M

r = monthly rate

M = monthly payment

Balance grows by interest then reduces by payment M.

COMPOUND INTEREST & AER

Amount = P(1+r)ⁿ

AER = (1 + i/n)ⁿ − 1

n = compoundings/year

AER compares accounts with different compounding periods.

PERCENTAGE CHANGE & VAT

New = Old × multiplier

Old = New ÷ multiplier

VAT: Price × 1.2

Reverse %: always divide by the multiplying factor.

CURRENCY EXCHANGE

Foreign = GBP × rate

GBP = Foreign ÷ rate

Check: buy vs sell rate

Identify which direction the exchange is going first.

AQA Level 3 Mathematical Studies — Chapter 2: Personal Finance

Chapter 3 — Standard Form

AQA MATHEMATICAL STUDIES

EXAM TIPS

Always check 1 ≤ a < 10 in your final answer

Multiply → add powers

Divide → subtract powers

Fermi problems: round to 1 s.f. first , then calculate

If result not in SF, adjust a and power accordingly

CORE RULES

DEFINITION

a × 10ⁿ   where   1 ≤ a < 10

Always check: is a between 1 and 10? If not, rewrite.

MULTIPLY

(a × 10ᵐ) × (b × 10ⁿ) = ab × 10ᵐ⁺ⁿ

Multiply the numbers, ADD the powers.

DIVIDE

(a × 10ᵐ) ÷ (b × 10ⁿ) = (a/b) × 10ᵐ⁻ⁿ

Divide the numbers, SUBTRACT the powers.

SMALL NUMBERS (NEGATIVE POWERS)

0.000 000 2 = 2 × 10⁻⁷

Count decimal places to the left of the first significant digit.

⚠ NOT STANDARD FORM CHECK

15 × 10¹⁰ → rewrite as 1.5 × 10¹¹

15 > 10, so adjust: move decimal, increase power by 1.

FERMI ESTIMATION

Round to 1 s.f. first, then calculate

Simplifies arithmetic; final answer in standard form.

WORKED EXAMPLES

AQA Chapter 3.2

Past Paper — Standard Form (AQA 2016)

AQA MATHEMATICAL STUDIES · PAPER 1 · 2016

Source: AQA Mathematical Studies 1350 — Paper 1, 2016

CHAPTER 3 — STANDARD FORM

📝 EXAM QUESTION — 4 MARKS TOTAL

(a)

Write

0.000 000 003 7

in standard form.

1 mark

(b)

The mass of a proton is

1.67 × 10⁻²⁷

kg. The mass of an electron is

9.11 × 10⁻³¹

kg. How many times heavier is a proton than an electron? Give your

answer to 3 significant figures.

3 marks

✅ MARK SCHEME — STEP-BY-STEP

Part (a)

Write 0.000 000 003 7 in standard form

1

Count decimal places to move: the digit 3 is in the 9th decimal place

2

Write as

a × 10ⁿ

where

1 ≤ a < 10

, so

a = 3.7

3

Power is negative (small number):

n = −9

Answer: 3.7 × 10⁻⁹ ✓

1 mark — correct answer

Part (b)

Proton ÷ Electron mass

1

Method:

1.67 × 10⁻²⁷ ÷ 9.11 × 10⁻³¹

— divide numbers, subtract powers

2

Calculate:

(1.67 ÷ 9.11) × 10⁽⁻²⁷⁺³¹⁾

=

0.1833 × 10⁴

3

Check standard form:

0.1833 × 10⁴ is NOT valid

→ rewrite as

1.833 × 10³ =

1833

Answer: 1830 (3 s.f.) ✓

M1 method

A1 calculation

A1 3 s.f.

💡

EXAM TIP

When dividing standard form: divide the numbers and SUBTRACT the powers (−27 − (−31) = +4). Always check your final answer satisfies 1 ≤ a < 10 . Here 0.1833 × 10⁴ fails — rewrite as 1.833 × 10³ before rounding to 3 s.f.

Fermi Estimation — Three Techniques

AQA CHAPTER 3.3–3.5  ·  MODELLING & ESTIMATION

TECHNIQUE 1

Scaling

Estimate a known quantity, then scale it up or down to reach the required answer.

Exam tip: Always show each scaling step clearly. Use standard form throughout.

WORKED EXAMPLE — UK HEARTBEATS PER YEAR

1

Measure pulse rate: 80 bpm

2

Scale to 1 year: 80 × 60 × 24 × 365 ≈ 4 ×10⁷ beats/person/yr

3

Scale to UK population (6 × 10⁷): 6×10⁷ ×4×10⁷ = 2.4 × 10¹⁵

Answer: 2.4 × 10¹⁵ beats/year

TECHNIQUE 2

Subdividing

Break a complex shape or quantity into smaller, manageable geometric parts.

Exam tip: Sketch the shapes. Use simple triangles, rectangles, or circles to approximate.

WORKED EXAMPLE — AREA OF BRITISH ISLES

1

GB ≈ triangle: ½ × 500 × 1000 = 2.5 × 10⁵ km²

2

Ireland ≈ square: 250² ≈ 6 × 10⁴ km²

3

Total: 2.5×10⁵ + 6×10⁴ ≈ 3 × 10⁵ km²

Answer: ≈ 3 × 10⁵ km²

TECHNIQUE 3

Stating Assumptions

Make reasonable assumptions explicitly, then build the calculation on those stated values.

Exam tip: State every assumption clearly —marks are awarded for valid assumptions, not just the final answer.

WORKED EXAMPLE — FISH IN THE OCEANS

1

Assume ocean volume = 10¹⁸ m³ ; fish live in top 20 m

2

Fish zone: 10¹⁸ × (20/4000) = 5 × 10¹⁵ m³

3

Territory per fish ≈ 10³ m³ → Number = 5 ×10¹²

Answer: ≈ 5 × 10¹² fish

Source: AQA Mathematical Studies — Chapter 3.3–3.5

Past Paper — Fermi Estimation (AQA 2019)

AQA Paper 1 · 2019

Source: AQA Mathematical Studies 1350 Paper 1, 2019 — UK water usage Fermi estimation question

QUESTION

Estimate the number of litres of water used in the UK per day. State any assumptions you make.

5 marks

A1

ASSUMPTION 1 — [1 MARK]

UK population ≈ 6.7 × 10⁷

Any reasonable population figure accepted by the examiner

A2

ASSUMPTION 2 — [1 MARK]

Average person uses ≈ 150 litres/day

Any reasonable per-person daily usage figure accepted

MARK SCHEME — CALCULATION

Method [1]

Multiply population by daily usage: 6.7 × 10⁷ × 150

Calc [1]

Rewrite 150 in standard form: 6.7 × 10⁷ × 1.5 × 10² = 1.005 × 10¹⁰

Std Form [1]

Final answer in standard form: ≈ 10¹⁰ litres per day

Note: Any reasonable assumptions are accepted — marks are awarded for method and consistency, not the exact numerical answer.

EXAM TIP

1

State assumptions clearly — each valid assumption earns a mark independently

2

Show your method — write out the multiplication step explicitly

3

Give answer in standard form — a × 10ⁿ where 1 ≤ a < 10

You do NOT need the exact answer — a reasonable estimate with clear working and standard form earns full marks.

Chapter 3 — Modelling & Useful Facts

AQA §3.6–3.7

THE MODELLING CYCLE

Represent mathematically → Use techniques →Interpret results → Compare with real data →Improve model.

A model improved later is still valuable.

HUMAN BODY FACTS

Weight 80 kg · Height 1.7 m · Lifetime 75 years

Heart rate 80 bpm · Blood volume 5 litres

Walking speed 3 mph · Water density 1 g/cm³

EARTH & UNIVERSE

Earth radius 6 000 km

Spins once per day · Orbits Sun once per year

Speed of light 3 × 10⁸ m/s

1 year ≈ 365 days · 1 day = 24 h

DATA & EQUIVALENCES

1 byte = 8 bits · 1 MB = 10⁶ bytes

1 litre = 1 000 cm³

1 mile ≈ 1 600 m

1 km² = 10⁶ m²

KEY FORMULAE

Speed = Distance ÷ Time

Density = Mass ÷ Volume

Sphere surface:

4πr²

Sphere volume:

(4/3)πr³

CRITICAL EVALUATION

Are the assumptions reasonable ?

Short-term regional changes are NOT evidence for/against global trends.

An improved model is still valuable — imperfect ≠ useless.

AQA Mathematical Studies — Chapter 3.6–3.7

AQA MATHEMATICAL STUDIES · CHAPTER 4.1–4.2

Argument Structure & Clarity

Exam Tip: For each article, identify the evidence used, the conclusion drawn, and any flaws. For clarity questions, give specific examples from the text. For self- contradiction, show exactly where the argument contradicts itself.

Argument Structure

Every argument has 3 parts

1

Evidence

The data or facts used to support the claim. Must be sound and relevant to the argument being made.

2

Reasoning

The logical steps connecting the evidence to the conclusion. Are there holes or leaps in logic?

3

Conclusion

The claim being made. Is it well-supported by the evidence and reasoning provided?

Four Clarity Problems

Common flaws to identify

1

Emotive Language

Words designed to provoke an emotional response rather than inform.

e.g. "blatantly obvious", "no sane person would…"

2

Vague Phrases

Imprecise language that could mean many different things.

e.g. "a large number", "in the next few years"

3

Unjustified Assumptions

Assuming the reader already knows things that have not been stated in the argument.

4

Self-Contradiction

The argument contradicts itself — show exactly where the contradiction occurs.

AQA Mathematical Studies · Chapter 4.1–4.2

Past Paper — Critical Analysis (AQA 2018)

AQA PAPER 1 · 2018 · 3 MARKS

QUESTION

An article states: "A new study has convincingly shown that children who eat more vegetables are better behaved."

Give three reasons why this claim may not be valid.

[ 3 marks — any three from the mark scheme below ]

MARK SCHEME — ANY 3 OF THESE 5 VALID ANSWERS [1 MARK EACH]

1

CORRELATION ≠ CAUSATION

A third factor (e.g. parenting style) may cause both more vegetable eating and better behaviour — the study does not prove vegetables cause the behaviour.

2

VAGUE / EMOTIVE LANGUAGE

"Convincingly shown" is vague and emotive — no detail is given on sample size, method, or statistical significance of the study.

3

SAMPLE REPRESENTATIVENESS

The sample may not be representative of all children — it could be biased by age, background, or selection method, limiting generalisability.

4

SUBJECTIVE / VAGUE MEASURE

"Better behaved" is subjective and vague — it is not clearly defined or objectively measured, so different observers may judge it differently.

5

REVERSE CAUSATION

Reverse causation is possible — better-behaved children may be more likely to eat vegetables (e.g. they follow parental instructions), not the other way around.

EXAM TIP — CRITICAL ANALYSIS CHECKLIST

For every claim, look for these five issues and quote specific words from the text:

Emotive / vague language

Correlation vs causation

Sample size / representativeness

Unjustified assumptions

Self-contradiction

Sampling, Trialling & Misleading Data

AQA CHAPTER 4.3–4.4  |  CRITICAL ANALYSIS SKILLS

Sampling & Trialling

Representative sample — reflects the population in all relevant ways; always report sample size n and selection method.

Confidence intervals — e.g. 38.1 ± 3% (n = 1811, 95%) means 95 out of 100 polls give a result within this range.

Placebo & control group — control group takes placebo to check that the drug effect exceeds the placebo effect.

Double-blind trial — neither patients nor doctors know who receives the real treatment, eliminating bias.

Exam tip: For sampling questions, identify who is NOT represented. For trialling, explain why a placebo is needed — to check drug effect > placebo effect.

Misleading Data

Selective data — reporting only data that supports the conclusion; ignores contradictory evidence.

Non-zero axes — y-axis starting above zero makes small changes appear dramatic; always check axes start at 0.

Truncated scales & inconsistent intervals — uneven axis intervals distort the visual impression of trends.

Absolute numbers vs percentages — both can mislead; a large absolute rise may be a tiny percentage, and vice versa.

Exam tip: For misleading graphs — always check axes start at 0 and use uniform scales. State how the graph misleads, not just that it does.

AQA Level 3 Mathematical Studies — Chapter 4.3–4.4

Past Paper — Misleading Data (AQA 2022 & 2016)

PAPER 1 · 2 + 3 MARKS

AQA 2022 — QUESTION (2 MARKS)

A graph shows the number of road accidents in 20 mph zones from 2010 to 2015. The y-axis starts at 50 rather than 0. Explain why this graph could be misleading.

MARK SCHEME

1 mark

The non-zero axis makes the increase appear much larger than it actually is.

1 mark

The visual impression exaggerates the change in the number of accidents.

BONUS CONTEXT — AQA 2016: "CASUALTIES SOAR IN 20 MPH ZONES"

(2010: 6 FATALITIES → 2011: 7 FATALITIES)

① Emotive language

"Soar" is misleading — only 1 extra fatality occurred.

② More zones introduced

More zones = more opportunities for accidents, even if each zone is equally safe.

③ Too small to be significant

Absolute numbers (6 and 7) are too small to draw statistically valid conclusions.

EXAM TIP

For misleading graph questions, always explain HOW the graph misleads — not just that it does.

1

Check the y-axis — does it start at 0? A non- zero axis exaggerates changes.

2

Check the scale — are intervals consistent? Uneven intervals distort trends.

3

Check for missing context — e.g. more 20 mph zones were introduced over time.

⚠ COMMON MISTAKE

Don't just write "the graph is misleading." State exactly which feature misleads and why it creates a false impression.

AQA MATHEMATICAL STUDIES · CHAPTER 4.5–4.6

Correlation, Causation & Models

Correlation ≠ Causation

4 POSSIBLE EXPLANATIONS

Coincidence — pure chance; no real link between A and B

A causes B — the first variable directly drives the second

B causes A — reverse causation; direction is opposite to assumed

Third factor C causes both A and B independently

Classic example: Ice cream sales & crime both rise in summer — hot weather (third factor) causes both. Ice cream does not cause crime.

Exam tip: Always consider all four explanations. State which applies and explain the mechanism clearly for full marks.

Climate Change Models

VOSTOK ICE CORE DATA — 420,000 YEARS

CO₂ lags temperature by ~800 years — temperature changes first, then CO₂ follows

Al Gore's flaw: claimed CO₂ causes warming — but data shows temperature leads CO₂

Current CO₂ ≈ 400 ppm — above the entire historic range, so past correlation may not apply

Both sides of the debate are selective with data; short-term regional changes are not global evidence

Critical analysis: Identify the evidence used, the conclusion drawn, and any flaws in reasoning — including selective use of data.

Exam tip: For climate questions — state CO₂ lags temperature in historic data, and note that 400 ppm is outside the historic range.

Past Paper — Correlation & Causation

AQA 2016 · PAPER 1 · 4 MARKS

PAST PAPER QUESTION

A newspaper article states: "Research shows that countries with more televisions per person have higher life expectancy. Therefore, buying more televisions will help people live longer."

(a) What is wrong with the conclusion? [2 marks]

(b) Suggest a more likely explanation for the correlation. [2 marks]

MARK SCHEME — PART (A)

(a) What is wrong with the conclusion?

2 marks

Correlation does not imply causation — just because two variables are correlated

does not mean one causes the other.

1 mark

Buying televisions does not cause people to live longer — the article incorrectly

assumes causation from correlation.

1 mark

MARK SCHEME — PART (B)

(b) More likely explanation

2 marks

A

third factor

— wealthier countries can afford more televisions

and

have better

healthcare and nutrition.

1 mark

Wealth causes

both

higher TV ownership

and

longer life expectancy — you must

explain both links.

1 mark

EXAM TECHNIQUE — 3-STEP METHOD FOR CORRELATION/CAUSATION QUESTIONS

1

State: Correlation ≠ causation

2

Identify: the third factor (e.g. wealth)

3

Explain BOTH links: third factor → Variable A and third factor → Variable B

Also consider: coincidental / A→B / B→A / third factor

Exam Technique — How to Score Maximum Marks

PAPER 1

Show All Working

Method marks are available even if your final answer is wrong — never write just the answer.

Use the Mark Allocation

1 mark = 1 distinct point. For 3 marks, plan and write 3 separate, clearly stated points.

Compare Distributions

Always comment on BOTH an average (mean/median) AND a measure of spread (SD/IQR)— never just one.

State Assumptions

In modelling questions, write your assumptions clearly before you begin any calculation.

Check Standard Form

Always verify 1 ≤ a < 10 in your final answer — a common error that loses easy marks.

Critical Analysis

For each article: identify the evidence used, the conclusion drawn, and any flaws in the reasoning.

COMMON ERRORS

✗

Do NOT average two means — use weighted mean

✗

Plot CF at upper class boundary, not midpoint

✗

Correlation ≠ causation

✗

Always state units in your answer

Complete Formulae Reference — All Chapters

AQA Mathematical Studies · Paper 1

EXAM TIP

Always show working — method marks are available even if the final answer is wrong. Check: 1 ≤ a < 10 in standard form. Compare distributions using BOTH an average AND a measure of spread.

CHAPTER 1 — STATISTICS

AVERAGES

MEAN (RAW DATA)

x̄ = Σx ÷ n

MEAN (FREQUENCY TABLE)

x̄ = Σfx ÷ Σf

SPREAD

INTERQUARTILE RANGE

IQR = UQ − LQ

OUTLIER (UPPER)

> UQ + 1.5 × IQR

OUTLIER (LOWER)

< LQ − 1.5 × IQR

HISTOGRAMS & SAMPLING

FREQUENCY DENSITY

FD = freq ÷ class width

STRATIFIED SAMPLE

(group ÷ total) × sample size

CHAPTER 2 — PERSONAL FINANCE

TAX & DEDUCTIONS

TAXABLE INCOME

income − £10,600

INCOME TAX

20% × first £31,785 + 40% × rest

NI CONTRIBUTIONS

12% on monthly £672–£3,532

STUDENT LOAN

9% above £17,335/year

LOANS & INTEREST

APR FORMULA

C = Σ Aₖ/(1+i)^tₖ

MORTGAGE RECURRENCE

Aₙ₊₁ = Aₙ(1+r) − M

COMPOUND INTEREST

P(1 + r)ⁿ

AER

(1 + i/n)ⁿ − 1

CHAPTER 3 — MODELLING

STANDARD FORM

STANDARD FORM RULE

a × 10ⁿ, 1 ≤ a < 10

MULTIPLY

Add the powers of 10

DIVIDE

Subtract the powers of 10

USEFUL FORMULAE

SPEED

Speed = Distance ÷ Time

DENSITY

Density = Mass ÷ Volume

SPHERE VOLUME

V = (4/3)πr³

ESTIMATION TIPS

FERMI ESTIMATION

State assumptions → calculate →check

MODELLING

Simplify real world → state assumptions

CHAPTER 4 — CRITICAL ANALYSIS

ARGUMENT STRUCTURE

VALID ARGUMENT

Evidence + Reasoning → Conclusion

EVALUATING ARTICLES

Identify: evidence, reasoning, flaws

CORRELATION VS CAUSATION

KEY RULE

Correlation ≠ Causation

EXPLANATION 1

Pure coincidence

EXPLANATION 2

A causes B

EXPLANATION 3

B causes A

EXPLANATION 4

Third factor causes both A & B

DATA PRESENTATION

MISLEADING GRAPHS

Check: axis scale, sample size, context

SELECTIVE DATA

Both sides may cherry-pick evidence

Practice Questions — Chapters 1 & 2

WITH WORKED ANSWERS

CHAPTER 1 — STATISTICS & DATA

Q1

A college has 800 students: 300 male, 500 female. A stratified sample of 40 is needed. How many females should be selected?

ANS

500 ÷ 800 × 40

= 25 females

Q2

Data set: 3, 5, 7, 7, 9, 11, 13. Find the median and interquartile range (IQR).

ANS

Median = 7  |  LQ = 5, UQ = 11

IQR = UQ − LQ = 11 − 5 = 6

Q3

The frequency density for class 10–20 is 2.5. What is the frequency for this class?

ANS

Frequency = FD × class width = 2.5 × 10

= 25

CHAPTER 2 — PERSONAL FINANCE

Q4

Pete earns £45,360/year. Personal allowance £10,600. Calculate his income tax (20% on first £31,785; 40% on remainder).

ANS

Taxable = £45,360 − £10,600 = £34,760

20% × £31,785 = £6,357  |  40% × £2,975 = £1,190

Total tax = £6,357 + £1,190 = £7,547

Q5

£2,000 is invested at 3% compound interest per year. How many years to double the investment?

ANS

Solve 1.03ⁿ = 2 by trial: n = 24 → 1.03²⁴ ≈ 2.033 ✓

Doubles after 24 years

Q6

Mortgage: £120,000 at 0.6% monthly interest, £920/month repayment. What is the balance after month 1?

ANS

A₁ = £120,000 × 1.006 − £920

= £120,720 − £920 = £119,800

AQA Mathematical Studies — Practice Questions Chapters 1–2

Practice Questions

CHAPTER 3 — MODELLING & STANDARD FORM

CHAPTER 4 — CORRELATION & ANALYSIS

CHAPTER 3 — STANDARD FORM & ESTIMATION

Q1

Write 85,000,000 in standard form.

ANSWER

8.5 × 10⁷  (check: 1 ≤ 8.5 < 10 ✓)

Q2

Calculate (3 × 10⁸) × (4 × 10⁵). Give your answer in standard form.

ANSWER

3 × 4 = 12; 10⁸ × 10⁵ = 10¹³ → 12 × 10¹³ = 1.2 × 10¹⁴

Q3

Gulf Stream: 100 km wide, 1 km deep, flows at 5 km/h. Find the volume of water per hour in m³.

ANSWER

Volume = 1 × 100 × 5 = 500 km³/h. Convert: 1 km³ = 10⁹ m³ → 5 × 10¹¹ m³/h

Q4

Great Britain modelled as a triangle: height 1000 km, base 500 km. Find the area in km².

ANSWER

Area = ½ × 500 × 1000 = 250,000 km² = 2.5 × 10⁵ km²

CHAPTER 4 — CRITICAL ANALYSIS & CORRELATION

Q5

An article claims "casualties soar in 20 mph zones" — 6 fatalities in 2010, 7 in 2011. Give two criticisms of this claim.

ANSWER

(1) "Soar" is emotive — only 1 extra fatality, a small increase.

(2) More 20 mph zones were introduced — more zones = more opportunities for fatalities even if each zone is equally safe.

Q6

Ice cream sales and crime both rise in summer. Does this mean ice cream causes crime? Explain fully.

ANSWER

No — correlation ≠ causation. This is a third factor : hot weather causes both higher ice cream sales AND higher crime rates. The two variables are not causally linked.

EXAM TECHNIQUE — CORRELATION QUESTIONS

Always consider all 4 explanations : (1) Coincidence  (2) A causes B  (3) B causes A  (4) Third factor causes both.

For full marks: name the third factor AND explain how it causes both variables.

KEY REMINDERS

Standard form: always check 1 ≤ a < 10 .  |  Multiply standard form: multiply coefficients, add powers .  |  Critical analysis: identify evidence, conclusion, and any flaws in reasoning.  | Correlation ≠ causation — always consider the third factor .

Interactive Revision Quiz — Paper 1

 TIMED QUIZ · 5 QUESTIONS

1

2

3

4

5

Click a number to navigate

CHAPTER 1 — STATISTICS

A distribution has LQ = 45 and UQ = 65. (a) Calculate the IQR. (b) Is a value of 90 an outlier? Show your working.

[3 marks]

Press

Reveal Answer

to see the worked solution

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 EXAM TIPS

Show ALL working — method marks available even if final answer wrong

Outlier check: > UQ + 1.5×IQR or < LQ − 1.5×IQR

Standard form: always check 1 ≤ a < 10 in final answer

Correlation ≠ causation — state all four possible explanations

Fermi: state assumptions clearly, use standard form

AQA MATHEMATICAL STUDIES — PAPER 1 REVISION

Good Luck in Paper 1!

Final Reminders

WORKING & METHOD

Show ALL working

Method marks are available even if the final answer is wrong — never just write the answer.

Compare distributions correctly

Always comment on BOTH an average (mean/median) AND a measure of spread (SD/IQR).

Standard form: check 1 ≤ a < 10

Verify your final answer satisfies this condition before writing it down.

CF graphs: plot at upper class boundary

Never plot cumulative frequency at the midpoint — always use the upper boundary.

FORMULAE & REASONING

Histograms: FD = frequency ÷ class width

Frequency density on the y-axis — never plot raw frequency in a histogram.

APR & Mortgages

APR:

C = Σ Aₖ/(1+i)^tₖ

 |  Mortgage:

Aₙ₊₁ = Aₙ(1+r) − M

Fermi estimation: state assumptions

Use standard form, show each step, and check your answer is reasonable.

Correlation ≠ causation

Always consider all four explanations: coincidence, A→B, B→A, or a third factor.

You've got this!

— Remember: show your working, check your units, and trust your revision.