# Work and Rate Problems

> More workers means less time. Use person-hours, rates per hour and filling and emptying to solve job problems.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/work-and-rates/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The garden shed
It takes 3 people 4 hours to build a garden shed. Next weekend only 2 people are free. How long will it take them?

Liam thinks, "Fewer people, so it should take longer, not shorter." He works out the total **amount of work**: 3 people working 4 hours is $3 \times 4 = 12$ **person-hours**. The shed always needs 12 person-hours, however you split it.

With 2 people: $12 \div 2 = 6$ hours. Liam's common sense was right, and the person-hours told him exactly how much longer.

**In short:** In work problems, the total work is workers × time, and it stays the same for the same job. So more workers finish faster: if 4 people take 6 hours, 8 people take 3 hours.

## Before you start

- Multiplying and dividing, including fractions like $3\frac{1}{2} \times 3$.
- Adding and subtracting fractions with different denominators (see the Fractions unit).
- Minutes and hours: $\frac{2}{3}$ h $= 40$ min.

## Person-hours

### Key idea: Total work stays the same
$$\text{work} = \text{number of workers} \times \text{time}$$

Find the total work first. Then divide by the new number of workers, or the new time.

### Worked example: One girl alone
3 girls take $3\frac{1}{2}$ hours to paint an attic. How long would 1 girl take?

Work $= 3 \times 3\frac{1}{2} = 10\frac{1}{2}$ girl-hours. One girl takes $10\frac{1}{2}$ hours.

### Worked example: How many workers?
26 road workers take 15 days to build a road. The job must be finished in 13 days. How many workers are needed?

Work $= 26 \times 15 = 390$ worker-days. In 13 days: $390 \div 13 = 30$ workers.

### Watch out: More workers, less time
Doubling the workers **halves** the time. Do not multiply the time by the workers' ratio. Always ask: should my answer be bigger or smaller? (This is called **inverse proportion**.)

## When the job changes size

If the job is bigger or smaller, scale the work too.

### Worked example: Half the road
6 men pave 2 km of road in 5 hours. How long do 5 men take to pave 1 km?

2 km takes $6 \times 5 = 30$ man-hours, so 1 km takes 15 man-hours. With 5 men: $15 \div 5 = 3$ hours.

### Worked example: Time saved
4 men take 4 hours to dig a drain. If 6 men had been available, how much time would they save?

Work $= 16$ man-hours. With 6 men: $16 \div 6 = 2\frac{2}{3}$ h $= 2$ h 40 min. Time saved: 1 h 20 min.

## Rates: filling and emptying

When things work at the same time, think about **how much of the job is done in 1 hour (or 1 minute)**.

### Key idea: Rates add up
If A does a job in 3 hours, A does $\frac{1}{3}$ of the job each hour. If B takes 6 hours, B does $\frac{1}{6}$ each hour.

Together: $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$ of the job each hour, so the whole job takes 2 hours.

### Worked example: The leaky bathtub
A tap fills a bath in 12 minutes. With the plug out, a full bath empties in 15 minutes. With the tap on and the plug out, how long to fill it?

Each minute: the tap adds $\frac{1}{12}$, the drain removes $\frac{1}{15}$.

$\frac{1}{12} - \frac{1}{15} = \frac{5}{60} - \frac{4}{60} = \frac{1}{60}$ of the bath per minute. So it takes 60 minutes.

### Exam tip: Check with sense
Two people working together must finish **faster** than the faster one alone. If your answer is longer, something is wrong.

### Challenge: Selective level: sharing seats
5 people get into a train carriage with only 2 empty seats. They take turns so each person sits for the same time. The trip takes 2 hours 10 minutes. How long does each person sit?

Total "seat time" $= 2 \text{ seats} \times 130 \text{ min} = 260$ seat-minutes. Shared by 5 people: $260 \div 5 = 52$ minutes each.

24 men build a kit home in 18 days. How many men are needed to finish in 14 days? Work $= 432$ man-days. $432 \div 14 = 30.86$, so 30 men is not quite enough. You need **31** men. When people are involved, round **up**.

**Mind map**

```text
Work problems
  Person-hours
    Workers x time
    Stays the same for the same job
  Inverse
    More workers, less time
  Job size changes
    Scale the work
  Rates
    Fraction of the job per hour
    Working together: add
    Draining: subtract
  Sense check
    Together is faster than alone
```

## Quick summary

- Total work $=$ workers $\times$ time. It stays the same for the same job.
- More workers means less time (inverse proportion).
- If the job size changes, scale the total work.
- A rate is the fraction of the job done per hour or minute. Add rates for things working together, subtract for things working against.
- When you need a whole number of people, round up.

## Quiz

Try these questions from the chapter quiz.

1. 3 people take 4 hours to build a shed. How long do 2 people take?
   - A. $2\frac{2}{3}$ hours
   - B. 3 hours
   - C. 6 hours
   - D. 8 hours
   - Answer: C. 6 hours. Work $= 12$ person-hours. $12 \div 2 = 6$.

2. 2 people paint a fence in 6 hours. How long would 4 people take?
   - A. 3 hours
   - B. 12 hours
   - C. 6 hours
   - D. 1.5 hours
   - Answer: A. 3 hours. Work $= 12$ person-hours. $12 \div 4 = 3$.

3. 3 girls take $3\frac{1}{2}$ hours to paint an attic. How long would 1 girl take?
   - A. $1\frac{1}{6}$ hours
   - B. 7 hours
   - C. $10\frac{1}{2}$ hours
   - D. $6\frac{1}{2}$ hours
   - Answer: C. $10\frac{1}{2}$ hours. $3 \times 3\frac{1}{2} = 10\frac{1}{2}$ hours.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/work-and-rates/#quiz