# Rules of three

In a problem of proportionality, if we don’t know one datum in a proportion and we know the other three, we can calculate the unknown datum using cross-product. We call it **rule of three**.

If the proportionality is direct, we call it **direct rule of three**, and if it is inverse we call it **inverse rule of three**.

**A. DIRECT RULE OF THREE**

1. Adrian finds that in each delivery of 500 bricks there are 20 broken bricks. How many bricks are broken in a delivery of 7500?

2. In a drink of 200 ml of fruit juice there are 140 ml of water. How many litres of water are there in 3 litres of that drink?

**B. INVERSE RULE OF THREE**

1. A truck that carries 3 tons needs 15 trips to carry a certain amount of sand. How many trips are needed to carry the same amount of sand with another truck that carries 5 tons?

2. Two hydraulic shovels make the trench for a telephone cable in ten days. How long will it take to make the trench with 5 shovels?

**C. COMPOUND RULE OF THREE**

We say that we have compound proportionality when we have more than 2 magnitudes.

- We have to identify the magnitudes

- We have to order magnitudes and data.

- We have to identify the proportionality relationship between the magnitude to the unknown datum and the others.

NOTE: It is useful to put the magnitude with the unknown datum in the last position.

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2. A team of workers will build a wall of 400 m^{2} in 15 days if they work 8 hours everyday. How long will they take if the wall has 600 m^{2} and they work 10 hours everyday?

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**Exercises**:

1.- I paid 30 euros for 12 chairs. If my friend Charles want 5 chairs as mine. How much does he have to pay?

2.- Three gardeners mow the grass of a park in 12 hours. How long will it take if one of them have to go?

3.- If 5 trucks transport 120 tons of goods in 2 days: what goods quantity will 7 trucks transport in 3 days?

4.- It takes 15 days for a team of 10 workers working 8 hours a day to finish an order. How many people with part time jobs (half the day) will be needed to do the same work in 10 days?

Solutions: 1) 12,50 €; 2) 18 hours; 3) 252 tons; 4) 30 persons

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