5.1 Rates, Distance & Averages

Word Problems on the JOA — Reasoning, Not a Maths Curriculum

On the Job Opportunities Assessment (ADF-RECREF132, revised 27 June 2025), numerical items sit inside one mixed 51-question / 20-minute assessment with verbal and abstract reasoning. The official guide’s numerical examples include word-problem styles such as rate and distance — for instance, a cyclist covering a stated distance at a stated speed, with a typed answer for how many hours the trip takes. That design samples how you extract quantities and apply a simple relation under time, not whether you can sit a separate school Mathematical Ability paper.

This chapter deliberately avoids algebra curricula, geometry proofs, trigonometry, Pythagoras, and SOH-CAH-TOA as JOA “domains.” Those belong to older two-test formats that are retired. What you need for modern JOA word problems is fluent quantity reasoning: read the stem, identify what is asked, map the numbers onto a small relation, compute carefully, and move on.

Why rate problems appear so often

Rate problems are ideal for a natural-ability test because they require:

1. Reading discipline — which number is distance, which is speed, which is time?
2. Relation selection — which form of the rate equation fits the unknown?
3. Unit attention — hours vs minutes, km vs m, per hour vs per minute.
4. Clean arithmetic under a ~23.5-second average budget (1,200 seconds ÷ 51).

None of that requires advanced school theory. All of it rewards calm process.

The Core Rate Family: Distance, Speed, Time

The single relation that powers most JOA-style rate items is:

[ \text{distance} = \text{speed} \times \text{time} ]

Rearranged for the two common unknowns:

Official-style worked example (typed hours)

Stem (paraphrased style): Abby cycled 70 km at 20 km/h. How many hours did the ride take?

Setup: Unknown is time. Units already match (km and km/h).

[ \text{time} = \frac{70}{20} = 3.5 \text{ hours} ]

If the interface wants a typed number, enter 3.5 (or the exact format the on-screen instruction allows). Do not invent a multiple-choice option that is not there. Do not convert to minutes unless the stem asks for minutes (3.5 hours = 210 minutes — a classic distractor if you misread the unit of the answer).

Worked example: find speed

A patrol covers 45 km in 1.5 hours at constant speed. What is the speed in km/h?

[ \text{speed} = \frac{45}{1.5} = 30 \text{ km/h} ]

Quick check: 30 × 1.5 = 45 — closes the loop.

Worked example: find distance

A vehicle travels at 80 km/h for 2.25 hours. How far does it go?

[ \text{distance} = 80 \times 2.25 = 180 \text{ km} ]

Tip under time: 2.25 = 2 + 1/4, so 80 × 2 = 160 and 80 × 0.25 = 20; total 180. Decomposition beats long multiplication when the clock is running.

Unit consistency before any division

Most rate errors are unit errors, not “bad maths.”

Micro-example: 90 km at 30 km/h takes 3 hours. If the stem asked for minutes, the answer is 180 — not 3. Always re-read the unit of the unknown in the last line of the stem.

Average Rate Traps (Not the Mean of Speeds)

Candidates often average two speeds with ((v_1 + v_2) / 2). That is correct only in special cases (for example, equal time at each speed). When a journey has legs with different speeds and different times, or equal distances at different speeds, the honest average speed is:

[ \text{average speed} = \frac{\text{total distance}}{\text{total time}} ]

Worked trap: equal distances, different speeds

A runner covers 10 km at 10 km/h, then another 10 km at 20 km/h. What is the average speed for the full 20 km?

Wrong instinct: ((10 + 20) / 2 = 15) km/h.

Correct process:

• Time leg 1: (10 / 10 = 1) hour
• Time leg 2: (10 / 20 = 0.5) hour
• Total distance = 20 km; total time = 1.5 hours
• Average speed = (20 / 1.5 ≈ 13.33) km/h

The average is below the midpoint of 10 and 20 because more time was spent on the slower leg. JOA-style options may include 15 as a deliberate distractor for candidates who average speeds.

Worked case: equal times

If someone drives 1 hour at 60 km/h and 1 hour at 100 km/h:

• Total distance = 60 + 100 = 160 km
• Total time = 2 hours
• Average speed = 80 km/h

Here the arithmetic mean of speeds matches because times are equal — but you still arrived via total distance ÷ total time, which is the safe habit for every stem.

Partial-rate and “how much longer” stems

Some items give a rate and a completed fraction of work or distance, then ask for remaining time.

Example: A tank fills at 200 litres per hour. After 1.5 hours it holds 300 litres and is still filling at the same rate toward a 500-litre capacity. How many more hours to full?

• Remaining volume = 500 − 300 = 200 litres
• Remaining time = 200 ÷ 200 = 1 hour

Alternatively: full time would be 500 ÷ 200 = 2.5 hours; already used 1.5; remaining 1 hour. Same answer, two setups — pick the one that clicks faster.

JOA pacing for rate items

Remember the official goal: complete as many as possible, as fast and accurately as possible — finishing all 51 is rare. A rate item you can close in 20 seconds is high value; a unit jungle that eats 70 seconds is a flag candidate.

Process checklist (use until automatic)

1. Underline the unknown and its unit (hours? km/h? minutes?).
2. List given quantities with units.
3. Pick the formula form (time / speed / distance).
4. Align units before arithmetic.
5. Compute, then sense-check (is 3.5 hours plausible for 70 km at 20 km/h? Yes.).
6. Submit or type the value in the requested unit; move on.

Practice this checklist on short mixed drills so that under JOA pressure the steps collapse into one smooth pass — natural ability displayed cleanly, not school-exam theatrics.