Money, Tax & Conversions

Tax brackets, currency conversion, and unit conversions appear in roughly 13% of QR questions combined. None of it is mathematically difficult - the challenge is following the right procedure without missing a step. These questions reward systematic execution, not clever shortcuts.

The Core Principle

Tiered calculations apply different rates to different portions of a value. The most common form is income tax brackets, but the same logic applies to tiered pricing, graduated fees, and postage rates.

The critical rule: apply each rate only to the portion within that bracket. Never apply the highest rate to the entire amount.

Tax brackets: first £10,000 at 0% · £10,001-40,000 at 20% · above £40,000 at 40%. Income: £55,000.

Wrong: 55,000 × 0.40 = £22,000 (applied top rate to everything)

Right:

The correct answer is almost half the wrong one.

Worked Example: Three Tax Brackets

Question: A country has the following income tax rates. Bill earns $52,000. How much tax does he pay?

1. Split income into brackets:
Bracket 1: `$12,000` (first $12k)*
* Bracket 2: $45,000 − $12,000 = $33,000
* Bracket 3: $52,000 − $45,000 = $7,000
2. Calculate tax for each: $12,000 × 0 = $0, $33,000 × 0.15 = $4,950, $7,000 × 0.30 = $2,100
3. Total tax = `$7,050`

Calculator: 33000 * 0.15 = 4950, press P. 7000 * 0.30 = 2100, + C = 7050.

Worked Example: Tiered Pricing

Question: A water company charges: first 10 m³ at £1.20, next 15 m³ at £1.80, anything above 25 m³ at £2.50. A household uses 32 m³. What's their bill?

Calculator with memory: 10 * 1.2 = 12 (P) → 15 * 1.8 = 27 (P, memory 39) → 7 * 2.5 = 17.5 + C = 56.5. Time: ~25 sec.

Same logic as tax brackets - different rates for different portions.

Calculator Strategy for Brackets

Use the memory function (P to store, C to recall):

1. Calculate bracket 1 tax, press P (stores to memory)
2. Calculate bracket 2 tax, press P (adds to memory)
3. Calculate bracket 3 tax, + C = (adds memory to current)
4. Display shows total tax

The Setup

Currency conversion is multiplication or division by an exchange rate. The confusion comes from which direction to go.

Given: 1 GBP = 1.50 USD

GBP → USD: multiply by rate. 100 × 1.50 = 150 USD.
USD → GBP: divide by rate. 150 / 1.50 = 100 GBP.

Sense check: if 1 GBP = 1.50 USD, GBP is stronger. Converting to the weaker currency = more units. Converting to the stronger currency = fewer units.

The Fraction Method

Set up the exchange rate as a fraction so the units cancel:

Problem: convert 240 EUR to GBP. Exchange rate: 1 GBP = 1.20 EUR.

Setup: 240 EUR × (1 GBP / 1.20 EUR) = 200 GBP. The EUR units cancel.

Calculator: 240 / 1.20 = 200.

Sense check: EUR is weaker than GBP (you need 1.20 EUR to buy 1 GBP), so converting EUR to GBP should give a smaller number. 200 < 240. ✓

The Fraction Cancellation Method

Unit conversion is just multiplying by a fraction where the top and bottom are equal, expressed in different units. This solves the "do I multiply or divide?" confusion every time.

Problem: convert 50 km/h to m/s.

(50 km / 1 hr) × (1000 m / 1 km) × (1 hr / 3600 s) = (50 × 1000) / 3600 = 13.89 m/s

Each fraction equals 1 (1000 m = 1 km, 1 hr = 3600 s). km cancels km, hr cancels hr - you're left with m/s.

If you're unsure whether to multiply or divide, write the conversion as a fraction with the unit you want to cancel on the opposite side. The units do the thinking for you.

Example: convert 5000 metres to km.

5000 m × (1 km / 1000 m) = 5 km. The "m" cancels.

Must-Know Conversions

The critical one: `1 ml = 1 cm³`. This converts between capacity (litres) and volume (cubic centimetres) and appears constantly in water tank and container questions.

Squared and Cubed Unit Conversions

This is where students lose marks. Converting area or volume units means squaring or cubing the linear conversion factor.

Convert 3.5 m² to cm²: 3.5 × 10,000 = 35,000 cm²
Convert 2,400,000 cm³ to m³: 2,400,000 / 1,000,000 = 2.4 m³

QR questions sometimes require rounding based on real-world context, not mathematical rules.

"How many lorries are needed to transport 850 boxes if each lorry holds 120?"
850 / 120 = 7.083 → round up to 8 (can't send 0.083 of a lorry).

"How many complete glasses can be filled from a 1.5 L bottle if each glass holds 200 ml?"
1500 / 200 = 7.5 → round down to 7 (the 8th glass wouldn't be full).

These combine multiplication, addition, and percentage operations. Follow the right order:

1. Base cost (unit price × quantity)
2. Discount (if any - multiply by `1 − discount%`)
3. Delivery (if any - add flat fee)
4. VAT (if any - multiply by `1 + VAT%`)

Order matters: discount usually applies before VAT. Read the question to confirm.

Example Sequence on Calculator

Problem: 25 items at £12 each, 10% discount, then 20% VAT.

25 * 12 = 300 (base) → − 10 % = 270 (discount) → + 20 % = 324 (VAT).

Using the % key makes this a single chain - no clearing needed.

Trap 1: Applying top tax rate to entire amount

Split into brackets. Apply each rate to its portion only.

Trap 2: Currency direction

Converting to a stronger currency = smaller number. Converting to a weaker currency = larger number. If your answer goes the wrong direction, you multiplied instead of dividing (or vice versa).

Trap 3: Area/volume unit conversion

1 m = 100 cm, but 1 m² = 10,000 cm² (not 100). Square (or cube) the conversion factor for area / volume.

Trap 4: Discount then VAT, not VAT then discount

Most questions apply discount first, VAT last. Read carefully - the order changes the answer.

Trap 5: Rounding direction

"How many buses needed?" → round up. "How many can fit?" → round down. Mathematical rounding (0.5 up) is rarely the right rule.

Next lesson: 3.7 Complex Problems - breaking multi-step problems into manageable sub-calculations, covering the 30-40% of QR that chains multiple skills together.