5.3 Multi-Step Numerical Reasoning

What “Multi-Step” Means on the JOA

A multi-step numerical word problem is still natural ability under time, not a homework write-up. The stem simply forces two or three linked operations before the final answer — for example: convert a unit, apply a rate, then take a percentage of the result; or find two leg times and then an average speed; or scale a ratio and then apply a discount.

The official assessment mixes these with simpler numerical items, verbal stems, and abstract figures inside one 20-minute block. Your job is to extract, plan a short chain, compute, and decide whether to stay or flag — not to show full algebraic derivations.

The extract → plan → compute → check loop

1. Extract: List only the quantities you need. Circle the question sentence (“How many hours remain?”, “What is the average speed?”, “What percent is left?”).
2. Plan: Name the 2–3 steps in order before calculating hard. Example plan: “time leg 1 → time leg 2 → total distance ÷ total time.”
3. Compute: One step at a time; keep units beside each intermediate result.
4. Check: Does the magnitude make sense? Did you answer in the requested unit?

Skipping the plan step is how candidates multiply the wrong pair of numbers from a busy stem.

Data Extraction: Find the Signal, Ignore the Noise

Multi-step stems often include extra numbers that belong to context but not to the calculation path.

Worked extraction example

Stem: A logistics run covers 120 km on day one at 60 km/h. On day two the same truck travels for 2 hours at 70 km/h to a depot that sits 15 km from a town of 4,200 people. What is the truck’s average speed over the two driving days?

Signal: 120 km, 60 km/h, 2 hours, 70 km/h — needed for average speed. Noise: 15 km from town; 4,200 people — irrelevant to average driving speed.

Chain:

• Day-one time = 120 ÷ 60 = 2 hours
• Day-two distance = 70 × 2 = 140 km
• Total distance = 120 + 140 = 260 km
• Total time = 2 + 2 = 4 hours
• Average speed = 260 ÷ 4 = 65 km/h

If you accidentally fold 15 km or 4,200 into the arithmetic, you invent a fourth step the stem never asked for. Underline the actual question before touching the calculator of your mind.

Read the last line first (when stems are long)

For paragraph-length items, glance at the final question first, then harvest numbers with that target in mind. Knowing you need “remaining litres” stops you from computing total cost when price data was only flavour text.

Table and multi-figure stems

When numbers appear in a small table (two vehicles, three days, opening/closing stock), copy only the cells that feed the plan. Mentally label columns: rate, time, quantity. JOA numerical reasoning may present compact data; treat the table as a stem, not as a spreadsheet project.

Building 2–3 Step Chains Cleanly

Pattern A: Rate then percent

A machine produces 400 units in 5 hours. Output then rises by 25%. What is the new hourly rate?

• Step 1: Old hourly rate = 400 ÷ 5 = 80 per hour
• Step 2: New rate = 80 × 1.25 = 100 per hour

Alternative order: total rises to 400 × 1.25 = 500 in 5 hours → 100 per hour. Same chain depth.

Pattern B: Ratio then remaining quantity

A fuel tank holds 240 litres, mixed resin : solvent = 1 : 5. After using 60 litres of mixture (same ratio), how much solvent remains?

• Step 1: Solvent originally = 5/6 × 240 = 200 litres
• Step 2: Mixture used 60 litres; solvent used = 5/6 × 60 = 50 litres
• Step 3: Solvent left = 200 − 50 = 150 litres

You could also note that remaining mixture is 180 litres and solvent is still 5/6 of mixture → 150. Two valid chains; pick the clearer one.

Pattern C: Two rates and a comparison

Walker A covers 9 km in 1.5 hours. Walker B covers 10 km in 2 hours. Who is faster, and by how many km/h?

• A speed = 9 ÷ 1.5 = 6 km/h
• B speed = 10 ÷ 2 = 5 km/h
• Difference = 1 km/h (A faster)

Three tiny steps; each is a single division or subtraction.

Unit attention across steps

Label every intermediate:

Unit slips compound: converting only the first leg to hours and leaving the second leg in minutes before adding times is a classic multi-step failure.

When intermediate rounding is dangerous

If an intermediate is 1/3 hour, prefer 20 minutes or keep the fraction 1/3 through the next step rather than rounding to 0.33 and then multiplying. Friendly JOA numbers often cancel exactly; ugly rounding is a self-inflicted trap.

Time Management: When to Stay, When to Skip

The scored JOA budget averages about 23.5 seconds per question across numerical, verbal, and abstract items. Multi-step numerical stems legitimately need longer than a short verbal analogy — but they do not deserve unlimited time.

Stay when

• You can state a 2–3 step plan within a few seconds of reading.
• Numbers are clean (halves, quarters, multiples of 5 and 10).
• Units already match or need one obvious conversion.

Flag / skip when

• After one full read you still cannot name step 1.
• Three different unit systems collide and the options are close together.
• You have already spent ~30–40 seconds with no intermediate result in hand.
• The stem looks like a mini case study with many unused figures and high confusion.

Flagging is not failure. Official guidance emphasises completing as many as possible accurately; it is rare to finish all 51. Returning later with a calmer brain often solves a chain that felt impossible on first contact.

Micro-pacing template for multi-step items

Typed answers and multi-select awareness

Official-style numerical examples can use typed responses (enter the hours, not pick a letter). Multi-step chains ending in a typed value need the same unit discipline as multiple choice — an extra zero or a minutes/hours mix-up is fatal. Also note how many answers a multi-choice stem wants if you see multi-select wording elsewhere in the session; do not assume every numerical item is single-select.

Putting Chapter 5 together under the real clock

• 5.1 gives you rate and average-speed engines.
• 5.2 gives you ratio, proportion, and percent engines.
• 5.3 teaches you to sequence those engines without drowning in extra data.

A realistic JOA numerical word problem might ask: two journey legs at different speeds (5.1), then what percent of the total time was the slower leg (5.2), with an irrelevant altitude figure in the stem (5.3 extraction). Practise that blend in short timed sets so the process becomes automatic.

Final practical standard

Before you invest a full minute: “Can I name my next two operations?” If yes, execute. If no, flag. Multi-step numerical reasoning on the JOA is disciplined short-chain thinking under recruitment timing — display that ability, protect the rest of your 51-question attempt, and leave school-length methods where they belong: outside this assessment.