6.3 Multi-Step Calculation Without a Calculator

Why multi-step, no-calculator fluency matters

Official NBT guidance is explicit: Quantitative Literacy is written without a calculator, and formulae are provided only when a specialised relationship is required. Many QL items are not single-operation prompts — they are multi-step. You might convert a unit, apply a percent, then compare a ratio, all inside one multiple-choice question.

This section trains the exam skill of chaining operations accurately: track units, respect order of operations, and use estimation to catch slips before you pick an option.

A reliable multi-step workflow

1. Underline the ask — what quantity and unit must the final answer report?
2. List knowns with units (R/month, L/day, hours/week).
3. Convert so every quantity you will add or multiply shares compatible units.
4. Estimate a ballpark (round numbers).
5. Compute exactly on paper, one operation per line.
6. Check that the exact answer sits near the estimate and answers the original ask.

Worked example — residence cost chain

Monthly residence rent is R2 850 for 10 months. There is a once-off deposit of R1 200, and an early-payment discount of R500. What is the total amount paid for the year?

Step A — rent for the year:

• 2 850 × 10 = R28 500

Step B — add deposit:

• 28 500 + 1 200 = R29 700

Step C — subtract discount:

• 29 700 − 500 = R29 200

Estimate check: 3 000 × 10 = 30 000, plus about 1 000, minus 500 ≈ 30 500 — close enough to 29 200 that the arithmetic is plausible (we rounded rent up, so the estimate sits slightly high, as expected).

Worked example — load-shedding share of a week

Over 6 days a suburb loses power for 2.5 hours each day. What percentage of a full 7-day week (24 hours per day) is that?

Step A — total outage hours:

• 6 × 2.5 = 15 hours

Step B — hours in a full week:

• 7 × 24 = 168 hours

Step C — fraction and percent:

• 15 ÷ 168
• Simplify: divide numerator and denominator by 3 → 5 ÷ 56
• 5 ÷ 56: 56 × 0.08 = 4.48, remainder 0.52; 0.52 ÷ 56 ≈ 0.009 → about 0.089
• As a percent: ≈ 8.9% (exactly 5/56 × 100%)

Exact fraction form often appears as an option: 5/56 of the week. Estimate: 15/168 is a bit under 15/150 = 0.10, so roughly 9% — reject any option near 15% or 25%.

Worked example — water tank with unit conversion

A tank holds 2.4 kL. A residence uses 45 L per day for 30 days. What fraction of the tank is used, and how many litres remain?

Step A — convert capacity to litres:

• 1 kL = 1 000 L, so 2.4 kL = 2 400 L

Step B — volume used:

• 45 × 30 = 1 350 L

Step C — remaining:

• 2 400 − 1 350 = 1 050 L

Step D — fraction used:

• 1 350 / 2 400 = 135 / 240 = 27 / 48 = 9/16
• 9/16 = 0.5625 = 56.25%

If you forget to convert 2.4 kL and treat “2.4” as litres, every later step collapses — unit conversion must come first.

Order of operations inside a chain

Some stems bury a mixed expression inside a larger story. Evaluate the expression with the standard order, then continue the story arithmetic.

Expression drill: 18 − 6 ÷ 2 × 3^2 + 4

1. Exponent: 3^2 = 9
2. Division and multiplication left to right: 6 ÷ 2 = 3, then 3 × 9 = 27
3. Subtraction and addition left to right: 18 − 27 = −9, then −9 + 4 = −5

If a campus “adjustment factor” equals that expression and you must then take 10% of an enrolment of 2 000, you would continue: 10% of 2 000 = 200, then apply the signed factor only if the stem defines how — always finish the pure arithmetic before inventing an interpretation.

Estimation table for common campus numbers

Worked estimate vs exact: 487 × 19.

• Estimate: 500 × 20 = 10 000, then adjust down (slightly smaller factors) → expect a bit under 10 000.
• Exact: 487 × 20 − 487 = 9 740 − 487 = 9 253.

The estimate correctly ruled out any option like 4 870 (forgot a factor of about 2) or 19 000 (added instead of multiplying).

Units that must match before combining

Common multi-step traps

Putting Chapters 6.1–6.3 together

A single QL item might ask you to convert mixed forms (6.1), interpret a ratio phrase (6.2), and finish a three-step cost or volume calculation (6.3). Practise writing every intermediate line. Under timed QL conditions, neat paper work is faster than mental juggling — and it is the only reliable substitute for a calculator.