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Work and Rate Problems

Learn to solve problems where people or machines work together or alone using proportional reasoning and rate formulas.

SSAT Middle Level: Quantitative

What Is a Rate?

A rate is a ratio that compares two different quantities with different units. When you say "60 miles per hour" or "5 pages per minute," you're describing a rate.

Speed
\( \frac{\text{miles}}{\text{hour}} \)
Typing
\( \frac{\text{words}}{\text{minute}} \)
Painting
\( \frac{\text{rooms}}{\text{hour}} \)

The Work Formula Triangle

Three quantities are always connected in work and rate problems. Knowing any two lets you find the third.

Work Rate Time
\( W = R \times T \)
\( T = W \div R \)
\( R = W \div T \)
\( W = R \times T \)

Understanding Unit Rates

A unit rate expresses how much work one person or machine completes in exactly one unit of time. This is the building block for solving all work problems.

1 hour 2 hours 3 hours Painter A: paints 1 room in 3 hours 1 hour 2 hours Painter B: paints 1 room in 2 hours

The Combined Work Rule

When two workers team up, their rates add together. This is the most important rule for work problems on the SSAT.

Why add?
Each worker contributes their fraction of the job per hour. Together, those fractions combine.
Always faster
Working together is always faster than either person alone: \( T_{\text{together}} < T_A \) and \( T_{\text{together}} < T_B \).

Visualizing Combined Work

1 Complete Job Worker A alone: 6 hours
\(\frac{1}{6}\)
 Worker B alone: 3 hours
\(\frac{1}{3}\)
 Together: each hour they finish...
\(\frac{1}{6}\)
\(\frac{1}{3}\)
\(= \frac{1}{6} + \frac{1}{3} = \frac{1}{2}\) per hr
 Together they finish in 2 hours

Rate vs. Time: The Key Distinction

Many students confuse rate and time. Remember: rates add, but times do not.

Rate Time
What it measures Work done per unit time Total time to finish
Units \( \frac{\text{jobs}}{\text{hour}} \) hours
When combining workers Add rates together Do NOT add times
Faster worker has... Higher rate Lower time
Relationship \( \text{Rate} = \frac{1}{\text{Time}} \)

How to Solve Work Problems

1
Identify each worker's time. How long does each person take alone to finish the whole job?
2
Convert to rates. Each rate is \( \frac{1}{\text{time}} \). If it takes 5 hours, the rate is \( \frac{1}{5} \) job per hour.
3
Add the rates. Find a common denominator and add the fractions.
4
Flip to find time. The combined time is the reciprocal of the combined rate: \( T = \frac{1}{\text{combined rate}} \).
5
Check your answer. The combined time must be less than the shortest individual time.

Example 1: Two Painters

Problem: Alex can paint a fence in 4 hours. Ben can paint the same fence in 12 hours. How long will it take them working together?

1
Find rates: Alex: \( \frac{1}{4} \) fence/hr. Ben: \( \frac{1}{12} \) fence/hr.
2
Add rates: \( \frac{1}{4} + \frac{1}{12} = \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \)
3
Flip for time: \( T = \frac{1}{\frac{1}{3}} = 3 \) hours

Example 2: Filling a Pool

Problem: Pipe A fills a pool in 6 hours. Pipe B fills the same pool in 3 hours. How long to fill it with both pipes open?

Pipe A
rate = \(\frac{1}{6}\)
 Pipe B
rate = \(\frac{1}{3}\)
Combined: \(\frac{1}{6} + \frac{1}{3} = \frac{1}{2}\)

Example 3: Three Workers

Problem: Machine A finishes a job in 4 hours, Machine B in 6 hours, and Machine C in 12 hours. How long working together?

1
Rates: A = \( \frac{1}{4} \), B = \( \frac{1}{6} \), C = \( \frac{1}{12} \)
2
Common denominator (12): \( \frac{3}{12} + \frac{2}{12} + \frac{1}{12} = \frac{6}{12} = \frac{1}{2} \)
3
Flip: \( T = \frac{1}{\frac{1}{2}} = 2 \) hours

The Proportion Shortcut

Many SSAT problems don't ask about combined work. Instead, they test proportional reasoning: if you know one rate, scale it up or down.

1
Find the unit rate: \( \frac{15 \text{ widgets}}{3 \text{ hours}} = 5 \text{ widgets/hour} \)
2
Multiply: \( 5 \times 8 = 40 \text{ widgets} \)

Finding Unknown Time

Problem: Sarah reads 42 pages in 3 hours. How long will it take her to read a 98-page book at the same rate?

Known 42 pages in 3 hours Unknown 98 pages in ? hours
\( \frac{42}{3} = \frac{98}{t} \;\rightarrow\; 42t = 294 \;\rightarrow\; t = 7 \) hours

More Workers = Less Time

When you double the number of identical workers, you halve the time. This is called inverse proportion: as one quantity goes up, the other goes down.

Number of Workers Time (hours) 12h 1 6h 2 4h 3 3h 4 2h 6

Common Mistakes

✕ Adding Times
If A takes 3 hours and B takes 6 hours, the answer is NOT \( 3 + 6 = 9 \) hours. You must add rates, not times.
✕ Averaging Times
The combined time is NOT \( \frac{3+6}{2} = 4.5 \) hours. Averaging ignores how rates actually combine.
✕ Forgetting to Flip
After adding rates \( \frac{1}{3} + \frac{1}{6} = \frac{1}{2} \), the answer is \( T = 2 \) hours, not \( \frac{1}{2} \) hour. Remember to take the reciprocal.

Summary of Problem Types

Problem Type Method Key Formula
Simple rate Find unit rate, multiply \( W = R \times T \)
Proportion Set up equal ratios, cross-multiply \( \frac{a}{b} = \frac{c}{d} \)
Two workers together Add rates, take reciprocal \( \frac{1}{T} = \frac{1}{T_A} + \frac{1}{T_B} \)
Inverse proportion Product is constant \( W \times T = k \)
Partial work Rate \( \times \) Time = fraction done \( \text{Fraction} = R \times T \)

Question 1 of 5

A printer prints 120 pages in 4 minutes. At this rate, how many pages will it print in 7 minutes?

Question 2 of 5

Tom can mow a lawn in 3 hours. Jerry can mow the same lawn in 6 hours. Working together, how long will it take?

Question 3 of 5

It takes 8 workers a total of 6 hours to build a wall. How long would it take 12 workers to build the same wall?

Question 4 of 5

Pipe A fills a tank in 5 hours and Pipe B fills the same tank in 10 hours. If both pipes are open, what fraction of the tank is filled after 2 hours?

Question 5 of 5

Emma types 45 words per minute. She needs to type a 2,700-word essay. How many minutes will it take?