The rule of three |
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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? |
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2. From the previous problem. How many kilos can I buy with one euro? |
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3. I have 12 bottles of wine that cost me 120 Euros. How much money will one bottle cost? |
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4. From the previous problem. How many bottles can I buy with one euro? |
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5. If 500 metal wheels is 3000 kilos. How many kilos is each wheel? |
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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? |
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2. A worker makes 200 pieces in five hours. How many pieces can he make in 48 hours? |
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3. An artist draws 30 paintings in 3 hours. How long will it take him to paint 200 paintings? |
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4. An assembler earns 72 Euros for 40 hours. How much money will he charge for 80 hours? |
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5. With 12 kilograms of apples, we will get 7 liters of cider. How many liters will we obtain with 48 kg? |
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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? |
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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? |
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3. 5 builders take 45 days to make a chalet. How many days will 15 builders take? |
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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? | Educational applications |
Mathematics |
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