# How many significant numbers are in 550

## Curso de alemán para principiantes con audio / Lección 061b

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Lección 060b ← Lección 061b → Lección 062b
Lección 061
Mathematics in German - 11

### BM501 - BM510

BM501

Round
Several basic digits should now be replaced by zeros.
---
It should be rounded 5,728 to a multiple of 1,000.
In this case, what is of interest is the base number, which is the number of multiples of 100. It's a 7. So it has to be rounded up.
5.728 ≈ 6.000
---
It is to be rounded to a multiple of 1,000 in 5,028.
The basic number for the number of multiples of 100 is 0. So it has to be rounded down.
5.028 ≈ 5.000

BM502

The 5th deserves special attention again.
It should be rounded to a multiple of 1,000.
---
876.523
It is rounded up; because 5 is followed by other non-zero basic digits.
876.523 ≈ 877.000
---
876.500
Only zeros follow 5. In this case the even number rule is applied. Round off because 6 is an even number.
876.500 ≈ 876.000
---
877.500
Only zeros follow 5. round up, since 7 is an odd number.
877.500 ≈ 878.000

BM503

Repetition:
Round
---
Numbers are often rounded in everyday life so that we can memorize them more easily - often you don't want to know exactly.
Most numbers, however, result from measurements or calculations with finite precision, where it would be misleading to state more digits than the precision. In such cases, rounding to the accuracy of the calculation or measurement is necessary in any case.
There are also cases where only a certain number of decimal places is allowed, for example, you can only issue amounts of money in euros to whole cents, but VAT or discounts are usually in percent. The result is then to be rounded to whole cents.
The same applies when money is to be exchanged from one currency into another.

BM504

Repetition:
Round
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When rounding, it is first determined (measurement, approximate calculation) or determined to how many digits should be rounded. The digits following to the right then decide how the rounding is done:
• Numbers 0, 1, 2, 3 and 4 are rounded down.
• Numbers 6, 7, 8 and 9 are rounded up.
• With number 5, the rule is a little more complicated. Any further digits that follow, one of which is not 0, are rounded up. On the other hand, if the 5 is the last digit, rounding takes place so that the last digit is even after rounding.
---
The result is so statically neutral that if you round a lot of numbers according to the rule, there is on average no deviation of the values ​​after rounding from those before. Incidentally, this would also apply if you were always rounding in such a way that the last digit after rounding five is odd, but this is not common.

BM505

Repetition:
Round
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Because the rule with the number 5 is complicated, some people always round up on the number 5. This is also called commercial rounding.
If so many numbers are rounded off, the mean after rounding is slightly larger than before. In the case of monetary amounts, this is advantageous for those who receive the money - maybe that's why it's called commercial rounding ...
---
When it comes to measurements and calculations, it is often said to how many valid digits ('significant' digits - without preceding zeros) should be rounded, so 0.000127 has three significant digits.
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Even if a number is involved in an invoice that has already been rounded, the final result must be rounded again accordingly. So the number of significant digits of the result is the same as that of the number of the calculation with the fewest significant digits. In this context, it can also be important to note trailing zeros to indicate how many digits are significant, i.e. 0.00012700 has five significant digits, 0.0001270 only four and 0.000127 only three.
---
When rounding you use the symbol ~, it means "approximately" the same.
Examples:
rounded upunrounded numberrounded number
Tens479764479760
Hundreds479764479800
Thousands479764480000
Money / Euro on Cent17,12517,12
Money / Euro on Cent17,13517,14
Measurement, three significant digits7,2857,28
Measurement, three significant digits0,005145235310,00515
Calculation 1.20 * 11.1 / 2 + 2.10508,76509

BM506

Round
Summary:
Rounded numbers are approximate.
---
Rounding to a multiple of 10:
Round to a multiple of 10.
Even number rule: It will rounded upif the digit before 5 denotes an odd number. It will roundedif the digit before 5 denotes an even number.
---
Rounding to a multiple of 100:
Round to multiples of 100.
To multiples of 1,000; 10,000 etc. are rounded accordingly.

BM507

beam
---
In geometry, a ray or a half-line is - to put it clearly - a straight line that is limited on one side, but extends to infinity on the other side.
A straight line goes to infinity in both directions.
---
Figure 1 shows a number line on which the numbers 0, 1, 2, 3 ..., 7 have been entered.
You can enter more numbers if the distance between the points is shortened or if a longer piece of the number line is drawn. (Picture 2)
The illustration of numbers on number lines is often used for the clear presentation of numbers. (Picture 3)

BM508

Diamonds: one of the four colors in the French deck of cards: ♦ (Fig. 1)
The term "Karo" came into the German language in the 18th century via the French "carreau" and goes back to the Latin quadrum "Viereck, Quadrat".
Diamonds (rarely) or lozenges (or rhombus), equilateral parallelogram, mostly on the tip. (Picture 2)
Merkel diamond (picture 3 and 4) - The gesture is named after the German Chancellor Angela Merkel, for whom this form of holding the hand has become a characteristic part of her appearance in public.
---
checkered = provided with boxes or squares; diced (the dice)
checkered as pattern (pattern)
a plaid shirt (picture 5)
a plaid dress (picture 5a)
a yellow and black plaid Follow me car at the airport (picture 6)
yellow-black checkered signal flag in shipping (Fig. 6)
---
squared paper (picture 8 and 9)
Diamonds (marked red - in picture 8 and 9)
Checkered paper usually has square checks. But there is also squared paper with rectangular squares.
For math lessons, squared sheets or notebooks with squared paper are usually preferred.
Lined paper (pic 10)
Music paper (Fig. 11)
Graph paper (Fig. 12)

BM509

route
Illustration of routes with the help of a Route diagram. (Image 1)
---
pillar
Illustration of routes with the help of a Column chart. (Picture 2)
The column chart is a diagram that uses vertical, non-contiguous columns (rectangles with meaningless widths) to illustrate the frequency distribution of a variable.
The bar chart is particularly suitable to illustrate a few numbers (up to approx. 15).
With more categories, the clarity suffers and line diagrams are to be preferred.

BM510

bar graph
The bar chart is one of the most common types of charts. It is very similar to the bar chart, but shows the data series with horizontal bars. It is very well suited for illustrating rankings.
---
A diagram (from ancient Greek "geometric figure, outline") is a graphic representation of data, facts or information. Depending on the purpose of the diagram, very different types are used.

### BM511 - BM520

BM511

Column charts
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Explain the different bar charts!
Figure 2: Number of insurance members, divided into contributors and pensioners in a health insurance.
Image 3: Comparison of the aircraft ordered from Boeing and Airbus
Figure 4: Temperature fluctuations in the Arctic (deviations from a mean value)
Fig. 5: Public expenses of a Japanese city, which are composed of 4-5 different items. (e.g. local government, school, police, swimming pool, subsidies for social institutions, etc. etc.)
Figure 6: Distribution of men (blue) and women (red)
Figure 7: Age distribution (age groups: 0-17 years; 18-24 years, 25-44 years, 45-64, 65+)
Figure 8: Chemical weapons production (4 types of chemical warfare agents; [sarin, tabun, mustard gas, VX], amount in tons, per year)
Figure 9: Goods handling in the Port of Hamburg (3 types of goods [conventional general cargo, bulk cargo, general cargo container], in million tons, per year)
Figure 10: Results of the parliamentary elections in Ukraine 2007 (6 different parties, around 30 provinces, increase or decrease in votes)

BM512

Different types of diagrams:
---
Pie chart (pic 1)
Donut diagram (image 2)
Pie chart = pie chart (image 3)
Network diagram (Fig. 4)
Line diagram (picture 5)
Scatter diagram (image 6)
Quantity diagram (Fig. 7)
---

BM513

Hermann threw the ball 12 m further than Jörg. Thorsten managed 8 m less than Hermann and Bernd 5 m more than Hermann.
How far did Jörg throw the ball?
Which child threw the ball the furthest?
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Marlies dived 5 s longer under water than Petra. Gabi even 7 seconds longer. Simone dived 2 s less than Gabi.
How long did Petra dive underwater?
Which girl turned up the same time?
---
Rainer has already saved 40 euros, Lena 5 euros less. Karsten has 17 euros more than Lena, and karin is still 2 euros missing, then she saved 50 euros.
How many euros does each child have?
---
The father is 35 years old, Ingrid is only 5. Her sister Kerstin is 3 years older. Her brother is 25 years younger than the father.
How old are Kerstin and her brother?
How old is the mother

BM514

The height of a rectangular box cannot always be selected as the unit for a line diagram. There is often not enough space for this.
---
Draw a line diagram to illustrate the following values:
 Time 6:00 9:00 12:00 15:00 18:00 21:00 temperature 17 ° C 21 ° C 26 ° C 28 ° C 24 ° C 19 ° C
A distance of 17 mm could be drawn for 17 ° C.
Solution BM514
Here 4 boxes were chosen for 10 °.
Checkered paper usually has squares with an edge length of 5 mm.
So 10 ° equals 2 cm or 20 mm.
So 10 ° corresponds exactly to 20 mm.
For 17 ° C a distance of 34 mm is drawn.
How many millimeters are the other stretches?
Temperature measurements during the day

BM515

When creating a diagram, the following sub-steps must be observed.
1.) Find the smallest and the largest number to be displayed in each case!
2.) Round both numbers and multiples of the same power of ten!
3.) Determine the length of the line for the selected power of ten so that it can be shown on the drawing sheet.
4.) Draw the line on which the lines should stand vertically and determine the starting points of the lines! Note the space for the lettering!
5.) Calculate the length of the route and draw the routes!
6.) Label the diagram!
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z. E.g .: 30,232; 142,162; 79,833; 135,820; 52,965
around: 30,000; 142,000; 79,000; 135,000; 52,000
Length in the diagram: 3 mm; 14 mm; 8 mm; 14 mm; 5 mm

BM516

Draw a diagram with the values: 300; 150 and 70
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You can change the diagram in such a way that the assigned number is not indicated above every segment.
For this purpose, the diagram is supplemented by a second number line.
The dashed auxiliary lines (Fig. 2) clarify the drawing.
The number 300 is assigned to route A. These guidelines will be left out later.

BM517

scale
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Scale: the relationship between a graphic representation and reality
The city map was drawn on a scale of 1: 25,000.
Image 1: Drawing of a house and floor plan of the house with the individual rooms.
---
1 cm on the drawing corresponds to 100 cm in reality
1 cm (on the card) ≙ 1m (in reality)
Scale: 1 to 100
Scale: 1: 100
---
Scale = map route through natural route
Scale = map route to natural route
Scale = map route: natural route
scaleMap routeNatural routeTypical application
1:1.0001 cm 10 mBuilding or land register plan
1:5.0001 cm 50 mBasemap
1:25.0001 cm250 mhiking map
1:50.0001 cm500 mCycling map
1:100.0001 cm 1 kmCar card
1:200.0001 cm 2 km
1:250.0001 cm 2.5 km
1:500.0001 cm 5 kmGeneral Staff Map
1:1.000.0001 cm 10 kmInternational world map
1:80.000.0001 cm800 kmWorld map (whole world)
The common scale in the USA is the 1: 63,000 scale

BM518

Image 1: Measuring table sheet on a scale of 1: 25,000
1 cm on the map corresponds to 25,000 cm in reality
1 cm 25,000 cm
1 cm 250 m
4 cm ≙ 1 km
4 centimeters on the map corresponds to one kilometer in the field
at the top right of the card it says: "4 cm card"
at the bottom right of the map is 1: 25,000
In Germany, a measuring table sheet is a topographic map on a scale of 1: 25,000. Here, 4 cm on the map corresponds to 1 km in nature ("four-centimeter map"), which makes this map type particularly popular with hikers - also because of the associated precision.
---
Image 2: World map on a scale of 1 to 35 million (1: 35,000,000)
1 m on the map corresponds to 35 million m in reality
1 m ≙ 35,000 km
1 cm ≙ 350 km

BM519

large scale
small scale
---
Map of Europe
small map scale
map
large map scale
Depending on the wealth of content and the level of detail of the cards great standards, medium standards and small standards differentiated. The adjectives “large” and “small” refer to the size of an object on the map and not to the scale number.
What is called a large or small scale is relative and depends largely on the subject or state. For example, for engineering geology a map of 1: 200,000 is already considered small-scale, for a geographer, however, an overview map from around 1: 2,000,000. In a large country such as Russia, 1: 200,000 can still be considered a large scale, while in a small country such as Switzerland this is already considered a small scale.

BM520

Image 2: Scale: One Inch to One Statute Mile (at the bottom of the map)
metric scale
non-metric scale
---
Round scales are mostly used in cartography because they are easier to calculate with. Sometimes, for example, for reasons of space, maps are published on the map sheet or with non-metric systems of measurement in non-circular scales (examples: city map of Zurich 1: 12,600; topographic map of Great Britain 1: 63,360 corresponding to 6 inches by 1 mile).
---
1 inch = 25.4 mm = 1 inch (English; in)
1 mile (country mile) = 63,360 inches = 1 mile (statute mile; sm)
---
For these cards you would need a ruler with inch graduation (Fig. 1) in order to be able to easily measure the distance between two points. Otherwise you have to calculate a little more.
---
Picture 2: at the bottom of the picture: scale: one inch to a mile = 1: 63360

### BM521 - BM530

BM521

Scale 1: 100,000
---
Because: 1 km = 1,000 m = 100,000 cm, they say: 1 cm on the map corresponds to 100,000 cm in nature.
One writes: 1 cm 100,000 cm.
1 cm 1,000 m
1 cm ≙ 1 km
A scale applies to a map or a diagram.
---
How long are the following routes in reality if they have the following lengths on a map on a scale of 1: 100,000?
1 cm
5 cm
8 cm
10 centimeters
1 mm
4 mm
8 mm
5 mm

BM522

Draw the floor plan of a room on a scale of 1: 100 on squared paper (edge ​​length of a box 5 mm).
The room is 4.30 x 6.50 m.
How many boxes do you draw in width and how many in length?

BM523

Coordinate system
---
Coordinate systems are aids in mathematics for specifying positions.
Image 1: Coordinates on a chess board. Example: the white field at the bottom right has the coordinates "h1"
---
A coordinate is one of several numbers that are used to indicate the position of a point in a plane or in space. Each of the dimensions required for description is expressed by a coordinate. If a place is described by two coordinates, for example on a map, one speaks of a “coordinate pair”.
---
The most frequently used coordinate system - this is especially true for school mathematics - is the Cartesian coordinate system, which is named after René Descartes (1596-1650).

BM524

Cartesian coordinate system with points P (5; 3) and Q (-4; 2)
Cartesian coordinate system
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A Cartesian coordinate system is an orthogonal coordinate system. It is named after the Latinized name Cartesius of the French mathematician René Descartes, who made the concept of "Cartesian coordinates" known.In two- and three-dimensional space, it is the most frequently used coordinate system, as many geometrical facts can be described clearly and clearly in this.
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The two directional axes are at right angles (= orthogonal) to each other, i.e. they intersect at a 90 ° angle.
The horizontal axis is called the x-axis (= abscissa axis).
The vertical axis is the y-axis (= ordinate axis).
Sometimes the coordinate axes (x-axis and y-axis) are also called the abscissa or ordinate for short.
The point O (0∣0), where the two axes meet, is called the origin or coordinate origin or origo (Latin for "origin").
For one point with the coordinates and one writes or .

BM525

In the coordinate system, points can be specified with the help of numbers
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You draw two number lines that are perpendicular to each other and name them x and y, respectively. (Image 1)
In addition, draw a point and designate it with P.
From point P one draws straight lines perpendicular to the number lines.
These straight lines intersect the number line x in 6 and the number line y in 3. (Fig. 2)
The numbers 6 and 3 belong to the point P.
One says: The number pair (6; 3) is assigned to the point P.
For this one writes: P (6; 3) or
P (6 | 3) (image 3)
The number that is read from the number line marked "x" is written first. Then read the number on the y-axis. This order must always be adhered to.
first x, then y

BM526

Each pair of numbers has a very specific point in the coordinate system.
and vice versa:
Each point in the coordinate system has a pair of numbers.
---
Each point has a different pair of numbers.
---
Assign the corresponding pairs of numbers to points A to H!
---
Solution BM526
Image 1: A (4; 2.5); B (2; 4)
Image 2: C (1; 1.5); D (5; 3.5); E (7; 4)
Image 3: F (1; 4); G (4; 3); H (7; 1)

BM526

Shield citizens
---
The Shield citizens, living in the fictional town of Schilda, are the main characters in a whole series of short stories, the Schildbürgerstenzen.
---
Well-known shield pranks:
The shield citizens build a town hall: When the people from the shield built a new, pompous town hall, the architect forgets to include windows, and the town hall is pitch black inside. Thereupon the shield citizens try to catch the sunlight with buckets and carry it inside.
---
The sunken bell: In order to protect the valuable town hall bell from the enemy, the shield citizens decide to sink it into the lake. In order to remember where in the lake they can get the bell out again after the end of the war, the resourceful citizens carve a notch in the edge of the boat. When they realize after the war that they won't find the bell again, they are furious and cut the notch out of the edge of the boat, which of course only makes it bigger.
---
Explain why they didn't find the bell with this notch?

BM526

A sum of two even numbers is always an even number.
Example: 62 + 74 = 136
Why is that? Can you prove that?
Is that true for all the numbers?
Give a counterexample!

BM527

Commutative law:
a + b = b + a
Even if you swap the order of the summands, you get the same sum.
---
Associative law: For all natural numbers a, b and c the following applies:
a + (b + c) = (a + b) + c
When adding three summands, these summands can be combined as desired.

BM528

Calculate the sum of five consecutive numbers! The smallest number is 175,998 million.
---
Calculate in writing: 15,679.35 euros minus 3,837.29 euros!
Do the math in writing: 688,883 plus 326,513

BM529

Replace the letter "G" with the correct numbers!
GG4G4 + 9G5G ------ 46537 G2065 + 8G7G ------ 5G4G7 8G6G3 -17581 ------ G5G6G G93G2 -25G6G ------ 2G515

BM530

Subtract two subtrahends
---
Exercise: 658 - 234 - 112
---
There are several ways of solving the problem.
---
1st way
---
2nd way
---
3rd way
  3 1 2 6 5 8 - 2 3 4 - 1 1 2
All three ways lead to the same result.
The third way is short and cheap. The following individual tasks are calculated
3rd way
   3 1 2 6 5 8 One: 2 + 4 = 6; 6 + 2 = 8 - 2 3 4 Tens: 1 + 3 = 4; 4 + 1 = 5 - 1 1 2 Hundreds: 1 + 2 = 3; 3 + 3 = 6
So you add up the two subtrahends for each digit individually before subtracting them from the minuends.
Do the following exercises using this 3rd way. He's the most effective.

### BM531 - BM540

BM531

738 - 316 - 285
The following individual tasks are calculated:
   1 3 7 7 3 8 One: 5 + 6 = 11; 11 + 7 = 18 - 3 1 6 Tens: 1 + 8 + 1 = 10; 10 + 3 = 13 - 2 8 5 Hundreds: 1 + 2 + 3 = 6; 6 + 1 = 7 OT 1 1
---
Check the correctness of your solutions!
Always check the correctness of your solutions in the future!

BM532

Calculate!
---
a)
---
b)
---
c)
---
d)
 6 3 8 5 9 9 - 2 0 3 1 7 3 - 3 1 2 3 0 5

BM533

24,840 tons of fish were caught. The 4th part was frozen, the 8th part was canned and the rest was sold as fresh fish.
Calculate the proportions!

BM534

Subtract more than two subtrahends
---
  1 4 7 3 4 9 5 7 8 3 - 1 2 5 1 9 - 3 1 8 2 7 - 2 4 3 5 8 - 1 2 3 4 5
One: 5 + 8 + 7 + 9 = 29; 29 + 4 = 33
Tens: 3 + 4 +5 +2 + 1 = 15; 15 + 3 = 18
Hundreds: 1 + 3 + 3 + 8 + 5 = 20; 20 + 7 = 27
Thousands: 2 + 2 + 4 + 1 + 2 = 11; 11 + 4 = 15
Tens of thousands: 1 + 1 + 2 + 3 + 1 = 8; 8 + 1 = 9

BM535

Before the result of a task is determined, a Rollover be performed. The result is roughly known from the rollover.
Approximate values ​​are used for the rollover.
---
Example:
5,901-5,047-568 = x
Rollover:
5.900 - 5.000 - 600 = 300
x ≈ 300
---
300 is an approximation for x.
Now you can determine the exact difference:
   2 8 5 5 9 0 1 - 5 0 4 7 - 5 6 9
Then you compare the result with the estimate:
300 ≈ 285
The result is close to the rollover.
A gross mistake would be noticed.

BM536

Compare the results with the rollover!
---
a)
---
b)
 7 8 6 0 - 3 2 4 5 - 1 0 1 2 - 2 3 0
---
c)
 3 5 7 2 4 - 1 2 6 3 2 - 5 7 0 9 - 1 6 8 4 6
---
d)
 8 4. 7 5 0. 0 0 0 - 5. 9 5 0. 0 0 0 - 5 7 5. 0 0 0 - 6. 3 5 0 - 8 5

BM537

The difference between the largest and the smallest five-digit number is given.
Enter the successor of this difference!
Solution BM537
99.999 - 10.000 = 89.999
89.999 + 1 = 90.000

BM538

Multiplying natural numbers
with single digit numbers
---
7 * 6
7 * 30 = 7 * 3 * 10
7 * 400 = 7 * 4 * 100
7 * 13 = 7 * (10 + 3)
7 * 206 = 7 * (200 + 6)
---
3 * 60.000 = 3 * 60 * 1.000
3 * 60.000 = 180 * 1.000
3 * 60.000 = 180.000
---
3 * 300.000 = 3 * 3 * 100.000
3 * 300.000 = 9 * 100.000
3 * 300.000 = 900.000

BM539

Written multiplication
Written procedure
---
The following individual tasks are calculated:
3 * 2 = 6
Further one calculates:
3 * 1 = 3
At the end you calculate:
3 * 3 = 9

BM540

Written multiplication
417 * 5
5 * 7 = 35
5 * 1 = 5; 5 + 3 = 8
5 * 4 = 20
---
Here is the same task again with a written carry-over:

### BM541 - BM550

BM541

Multiply in writing!
5.429 * 5
Solution BM541
Method:
5 * 9 = 45; the 5 write and note the 4
5 * 2 = 10; 10 + 4 = 14; the 4 write and note the 1
5 * 4 = 20; 20 + 1 = 21; the 1 write and note the 2
5 * 5 = 25; 25 + 2 = 27; the 7 write and note the 2
The 2 write at the beginning.

BM542

Multiply in writing!
28.468 * 8
Solution BM542
Method:
8 * 8 = 64; the 4 write and note the 6
8 * 6 = 48; 48 + 6 = 54; the 4 write and note the 5
8 * 4 = 32; 32 + 5 = 37; the 7 write and note the 3
8 * 8 = 64; 64 + 3 = 67; the 7 write and note the 6
8 * 2 = 16; 16 + 6 = 22; the 2 write and note the 2
The noted 2 write at the beginning

BM543

For a steak house, 125 kg of beef are bought every week. With one delivery, the restaurant receives the amount for 5 weeks.
How many kilograms of beef are delivered?
125 kg * 5 625 kg
A quantity (product of measure and unit) is multiplied by a natural number by assigning the corresponding unit to the product of measure and natural number:
a kg * b) (a * b) kg
---
6.90 EURO * 5
For the written multiplication, the amount of money can be converted into cents. Then you carry out the written multiplication and finally you convert the cent amount back into euros.
6.90 EURO ➸ 690 cents
690 cents * 5 3450 cents
3450 cents ➸ 34.50 EUR
You could have multiplied without converting to cents:
6.90 EURO * 5
You calculate as with cents and consider the result that you had an indication in EURO. So you put the comma in the appropriate place and specify EURO as the name.
6.90 EURO * 5 34.50 EURO

BM544

Multiply in writing
---
1.242 * 8
2.413 * 6
3.112 * 9
---
212 * 7
313 * 6
241 * 8
---
115 * 4
208 * 8
411 * 7
---
145 * 7
263 * 6
419 * 8

BM545

Multiply in writing
---
224 EURO * 2
321 kg * 8
3,122 m * 3
4,211 g * 6
---
12.80 EURO * 7
32.23 EURO * 8
64.71 EURO * 3
92.23 EURO * 4
87.50 EURO * 3
76.64 EURO * 9

BM546

A common beech can be 900 years old. How old can 7 beeches be?

BM547

Multiply all sizes by 7! Enter the results in the next larger unit!
---
375 kg
554 kg
207 kg
892 g
257 g
803 g
849 mg
492 mg
658 mg
396 kg

BM548

Commutative law
The following applies to all natural numbers a, b:
a * b = b * a
Even if you swap the sequence of factors, you get the same product.
---
Associative law
For all natural numbers a, b, c the following applies:
(a * b) * c = a * (b * c)
When multiplying three factors, the factors can be combined in any way you like.

BM549

Show that the following equations are true without multiplying!
Example:
15 * 7 = 5 * 21
is true because:
5 * 3 * 7 = 5 * 3 * 7
---
125 * 7 = 5 * 175
96 * 6 = 3 * 192
400 * 7 = 5 * 560
270 * 6 = 9 * 180
480 * 4 = 8 * 240
150 * 9 = 3 * 450
Solution BM549
125 * 7 = 5 * 175 (one calculates: 125: 5 = 25; 175: 7 = 25; so divide crosswise)
5 * 25 * 7 = 5 * 7 * 25
---
96 * 6 = 3 * 192 (one calculates: 96: 3 = 32; 192: 6 = 32; so divide crosswise)
3 * 32 * 6 = 3 * 6 * 32
OR:
96 * 3 * 2 = 3 * 2 * 96
---
400 * 7 = 5 * 560 (one calculates: 400: 5 = 80; 560: 7 = 80; so divide crosswise)
5 * 80 * 7 = 5 * 7 * 80
---
270 * 6 = 9 * 180
9 * 30 * 6 = 9 * 6 * 30
---
480 * 4 = 8 * 240
8 * 60 * 4 = 8 * 4 * 60
---
150 * 9 = 3 * 450
3 * 50 * 9 = 3 * 9 * 50

BM550

Calculate!
---
600 + 5 * 40
3 * 20 + 140
600 - 5 * 40
3 * 200 - 300
---
If a product appears in a sum (or difference) as a summand (or minuend or subtrahend), the product must first be calculated.
In problems without brackets, first multiply, then add (or subtract).
---
short:
Point calculation before line calculation
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