The rule of three

1. Problems of reduction to unity

Here's a problem: 3 boxes of chocolates cost 6 Euros. How much money cost one box?
One box will cost 6 Euros: 3 boxes = 2 Euros each box. Such problems are called "reduction to unity"
From the previous problem, we might ask something else: How many boxes can I buy with one euro?
In this case, we must divide the boxes among the Euros. 3 boxes: 6 Euros = 0.5 (the number of boxes that I can buy with one euro).

See how to solve problems.
Do these problems on a paper

 1. If 8 kilos of apples cost 16 Euros. How much money will one kilo cost? 2. From the previous problem. How many kilos can I buy with one euro? 3. I have 12 bottles of wine that cost me 120 Euros. How much money will one bottle cost? 4. From the previous problem. How many bottles can I buy with one euro? 5. If 500 metal wheels is 3000 kilos. How many kilos is each wheel? 6. From the previous problem. How many wheels can I do with 1 kilo?

2. The rule of three direct

After we know what the value of a unit is, we can also know what the values of other units are. In these cases, we must apply the rule of three.

For example: 3 packs of cigarettes cost 6 Euros. How much money will 10 packs cost?
We have three numbers and we need another one that is the unknown quantity.

If 3 packets (A) cost 6 Euros (B)
10 packs (C) will cost x (D)

A package will cost 2 Euros (6:3) and 10 packs will cost 20 Euros (2 x 10). We also multiply (6 x 10) and we divide it by 3. The result is 20 Euros.
When we solve the problem by the rule of three, first we must multiply the numbers B and C and the we divide it by A (6 x 10): 3 = 20 Euros.
We must check that the quantities A and C are of the same species. In this case they are packets.

For example: We have driven 560 kilometers in 8 hours. How many kilometers will we go over in 12 hours?

In 8 hours (A) ------ > 560 km (B)
In 12 hours (C) ------ > x (D)

x = (560 x 12) : 8 = 6720 : 8 = 840 kilómetros.

In general, the rule of three with directly proportional magnitudes is solved by multiplying B and C and by dividing it by A.

Solve these problems::

 1. 6 kilos of chocolates cost 6.3 Euros. How much money will 12 kilos cost? 2. A worker makes 200 pieces in five hours. How many pieces can he make in 48 hours? 3. An artist draws 30 paintings in 3 hours. How long will it take him to paint 200 paintings? 4. An assembler earns 72 Euros for 40 hours. How much money will he charge for 80 hours? 5. With 12 kilograms of apples, we will get 7 liters of cider. How many liters will we obtain with 48 kg? 6. If 8 meters of wire-cable cost 13 Euros. How much money will 16 meters cost?

3. The rule of three inverse

In inversely proportional magnitudes, when one amount increases, the other one decreases.
Example: The speed of a car and the time it takes to travel a distance. The more speed, the less time it takes.

For example: 12 builders build a house in 60 days. How long will it take the two builders build it?

12 builders (A) take 60 days (B)
2 builders (B) take x (D)

One builder will only take (12 x 60) = 720 days. Two builders will take 720:2 = 360 days.
With the rule of three, we multiply A and B and then we divide it by C.
D = (A x B): C

In general, the rule of three with inversely proportional magnitudes is solved by multiplying A and B and by dividing it by C.

Solve these problems:

 1. 30 soldiers dig a trench in five days. How many days will 15 soldiers take? 2. A car traveling at 80 kilometers per hour takes 3 hours to go from Teruel to Zaragoza. How many hours will it take to a speed of 120 km per hour? 3. 5 builders take 45 days to make a chalet. How many days will 15 builders take? 4. I finished a book in 33 days by reading 20 pages per day. How many days will I take if I read 30 pages per day?

4. Review of the rule of three

Do these problems:
 1. If 4 meters of telephone cable cost 32 Euros. How much money will 7 meters cost? 2. With 38 kilos of barley, we get 3 beers. How many beers will we get from 114 kilos? 3. A high-speed train traveling at 150 kilometers per hour takes 2 hours to go from Madrid to Seville. How many hours will it take to a speed of 200 kilometers per hour? 4. If 3 pairs of shoes cost 360 Euros. How much money will 5 pairs cost? 5. I can read a novel in 7 hours by reading 120 words per minute. How many hours will I take if I read 84 words per minute? 6. If 12 electricians do an installation in 60 days. How many days will 3 electricians take?

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®Arturo Ramo García.-Record of intellectual property of Teruel (Spain) No 141, of 29-IX-1999
Plaza Playa de Aro, 3, 1º DO 44002-TERUEL