The compound three rule is used for problems with more than two quantities, directly or inversely proportional.

## Examples

1) In 8 hours, 20 trucks unload 160m^{3} of sand. In 5 hours, how many trucks will need to unload 125m^{3}?

*Solution:* setting up the table, placing in each column the quantities of the same species and, in each row, the quantities of different species that correspond:

Hours | Trucks | Volume |

8 | 20 | 160 |

5 | x | 125 |

*Identification of relationship types:*Initially we put a down arrow on the column containing x (2nd column).

Next, we must compare each quantity with that where x is. Notice that,** increasing** the number of working hours we can **decrease** The number of trucks. So the relationship is *inversely proportional* (**up arrow in 1st column**).

**Increasing** the volume of sand we should **increase** The number of trucks. So the relationship is *directly proportional* (**down arrow in 3rd column**). We must equate the ratio containing the term x with the product of the other ratios according to the direction of the arrows.

*Assembling the ratio and solving the equation, we have*:

Therefore, it will be necessary **25 trucks**.

2) In a toy factory, 8 men assemble 20 strollers in 5 days. How many carts will be assembled by 4 men in 16 days?

*Solution:* setting up the table:

Men | Carts | Days |

8 | 20 | 5 |

4 | x | 16 |

Notice that, **increasing** the number of men, the production of strollers **increases**. So the relationship is *directly proportional* (we don't need to reverse the reason).

**Increasing** The number of days, the production of carts **increases**. So the relationship is also *directly proportional* (we don't need to reverse the reason). We must equate the ratio containing the term x with the product of the other ratios.

*Assembling the ratio and solving the equation, we have*:

Soon they will be assembled **32 carts**.

3) Two masons take 9 days to build a 2m high wall. Working 3 masons and increasing the height to 4m, how long will it take to complete this wall?

Initially we put a down arrow on the column containing x. Then put matching arrows for the quantities **directly proportional** with the unknown and discordant to the **inversely proportional**as shown below:

*Assembling the ratio and solving the equation, we have*:

Therefore, to complete the wall will require **12 days**.

## Complementary Exercises

Now it's your turn to try. **Practice** trying to do these exercises:

**1)** Three taps fill a pool in 10 hours. How many hours will it take 10 taps to fill 2 pools?

Answer: 6 hours.

**2)** A team of 15 men extracts in 30 days 3.6 tons of coal. If increased to 20 men, how many days can they extract 5.6 tons of coal? *Answer: 35 days.*

**3)** Twenty workers, working 8 hours a day, spend 18 days to build a 300m wall. How long will it take a class of 16 workers working 9 hours a day to build a 225m wall?

Answer: 15 days.

**4)** A truck driver delivers a load in a month, traveling 8 hours a day at an average speed of 50 km / h. How many hours a day should he travel to deliver this load in 20 days at an average speed of 60 km / h? * Answer: 10 hours a day.*

**5)** With a certain amount of yarn, a factory produces 5400m of 90cm wide fabric in 50 minutes. How many meters of cloth, 1 meter and 20 centimeters wide, would be produced in 25 minutes? * Answer: 2025 meters.*