Guide · Advanced
Compound Rule of Three
Updated: June 2026
Real problems often have more than one moving part. How long will a job take if you change both the number of workers and the amount of work? The compound rule of three — sometimes called the double rule of three — handles exactly this, by treating each factor separately and then combining them. It looks intimidating but breaks down into the same simple steps you already know.
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What makes a problem “compound”
A simple rule of three links one cause to one result. A compound problem has two or more causes acting at the same time. Workers and quantity of output both affect the number of days needed; distance and speed and the number of vehicles might all affect fuel used. The trick is to never juggle them all at once. Instead, look at each factor on its own — what would it do to the result if it were the only thing changing — and let the method recombine them.
The method, one factor at a time
Start from the known result and multiply it by one ratio per factor:
For each factor decide direct or inverse, then build the ratio so it pushes the result the right way:
- Direct factor — use
new ÷ old. More of it should give more result. - Inverse factor — use
old ÷ new. More of it should give less result.
A full worked example
If 4 workers build 3 walls in 12 days, how many days do 6 workers need for 5 walls?
Two factors affect the days: workers and walls.
- Workers — more workers means fewer days, so this is inverse:
4 ÷ 6. - Walls — more walls means more days, so this is direct:
5 ÷ 3.
days = 12 × 0.6667 × 1.6667
days = 13.33
So six workers need about 13⅓ days for five walls. Notice each ratio was written to nudge the answer in the sensible direction — fewer days for more hands, more days for more walls.
Using the calculator's compound mode
Switch the tool to Compound and choose how many factor columns you need. Fill the “known” row with the situation you already understand, fill the unknown row with the new values, set each factor column to direct or inverse, and leave the result cell blank. The tool multiplies the base result by every ratio and prints the full line of working, so you can check each factor pushed the answer the way you expected.
Tips to avoid mistakes
- Judge each factor in isolation. Pretend the others are frozen while you decide direct or inverse.
- Keep columns consistent. The known and unknown values in a column must be the same kind of quantity and the same unit.
- Sanity-check the size. If the answer moved in a direction that makes no physical sense, one of your factors is set the wrong way round.
- Round only at the end. Keep full precision through the multiplications, then apply your chosen number of decimals.
Frequently asked questions
What is the compound rule of three?
It extends the rule of three to problems where the result depends on more than one factor. You build a direct or inverse ratio for each factor and multiply them together.
How do I decide if a factor is direct or inverse?
Look at it alone, holding the others fixed. If increasing it increases the result, it is direct; if increasing it decreases the result, it is inverse.
Can it handle three or more factors?
Yes — one ratio per factor, each set to direct or inverse, all multiplied by the starting value. The calculator supports up to three factor columns.