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Mathematics 5.

Table of contents
Natural numbers
Sets
Natural numbers
The decimal system
The binary numeral system
Roman numerals
The number line
Comparing numbers
Rounding numbers
Adding natural numbers
Subtracting natural numbers
Multiplying natural numbers
Multiplication by 10; 100 and 1,000
Changes of the product
Multi-digit multiplication
Dividing natural numbers
Changes of the quotient
Dividing numbers by 10; 100 and 1,000
Dividing numbers by a multi-digit divisor
Factor and multiple
The order of operations
Mixed exercises
Geometry
Measurement, statistics
Angles
Fractions
The rectangle
The cuboid
Decimals
Integers
Orientation
Comparing numbers
When shopping or counting, we often need to compare natural numbers. In mathematics, we use relation symbols to make comparisons.
Relation symbol
How to read out
\latex{2} \latex{=} \latex{1}\latex{+} \latex{1}
\latex{2} is equal to \latex{1 + 1.}
\latex{b} \latex{\geq} \latex{4}
\latex{b} is greater than or equal to \latex{4.}
\latex{7} \latex{\lt} \latex{10}
\latex{7} is less than \latex{10}.
\latex{c} \latex{\leq} \latex{5}
\latex{c} is less than or equal to \latex{5.}
\latex{c} \latex{\gt} \latex{5}
\latex{c} is greater than \latex{5}.
\latex{5} \latex{\neq } \latex{7}
\latex{5} is not equal to \latex{7.}
Example 1
Place the following natural numbers on a number line.
a) greater than \latex{ 3 }
b) less than \latex{ 3 }
c) not greater than \latex{ 3 }
d) not less than \latex{ 3 }
e) greater than \latex{ 3 } but less than \latex{ 7 }
f) at least \latex{ 3 } but no more than \latex{ 7 }
Solution
Look for the natural numbers on the coloured part of the number line.
"You can't have more than \latex{ 3 } scoops of ice cream." "But I want at least \latex{ 3 } scoops!"
a) greater than \latex{ 3 } ( \latex{\color{88449a}\blacksquare} > \latex{ 3 } )
b) less than \latex{ 3 } ( \latex{\color{2dabe2}\large\bull} < \latex{ 3 } )
c) not greater than \latex{ 3 } ( at most; \latex{\color{ec2227}\blacktriangle} ≤ \latex{ 3 } )
\latex{\color{88449a}\blacksquare} = {\latex{ 4 }; \latex{ 5 }; \latex{ 6 }; ...}
\latex{\color{2dabe2}\large\bull} = {\latex{ 0 }; \latex{ 1 }; \latex{ 2 }}
\latex{\color{ec2227}\blacktriangle} = {\latex{ 0 }; \latex{ 1 }; \latex{ 2 }; \latex{ 3 }}
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
d) not less than \latex{ 3 } ( at least \latex{ 3 }; \latex{\color{07a64f}\blacktriangledown} ≥ \latex{ 3 } )
e) greater than \latex{ 3 } but less than \latex{ 7 } ( \latex{ 3 } < \latex{\color{ec2227}\large\bull} < \latex{ 7 } )
f) at least \latex{ 3 } but no more than \latex{ 7 } ( \latex{ 3 } ≤ \latex{\color{2dabe2}\blacktriangle} ≤ \latex{ 7 } )
\latex{\color{07a64f}\blacktriangledown} = {\latex{ 3 }; \latex{ 4 }; \latex{ 5 }; ...}
\latex{\color{ec2227}\large\bull} = {\latex{ 4 }; \latex{ 5 }; \latex{ 6 }}
\latex{\color{2dabe2}\blacktriangle} = {\latex{ 3 }; \latex{ 4 }; \latex{ 5 }; \latex{ 6 }; \latex{ 7 }}
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
A number marked with
\latex{\circ} -is not in the solution set, a number marked with \latex{\bullet} -is in the solution set.
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How do you compare natural numbers with multiple digits?
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Example 2
Arrange the following numbers in ascending order.
a) \latex{ 1,848 }; \latex{ 2,004 }; \latex{ 896 }
b) \latex{ 1,848 }; \latex{ 1,867 }; \latex{ 1,830 }
Solution
Show the numbers on a properly chosen number line, then arrange them in ascending order.
\latex{\textcolor{#ff0000}{0}896\lt \textcolor{#ff0000}{1},848\lt \textcolor{#ff0000}{2},004}
\latex{1,8\textcolor{#ff0000}{3}0\lt 1,8\textcolor{#ff0000}{4}8\lt 1,8\textcolor{#ff0000}{6}7}
\latex{ 500 }
\latex{ 1,000 }
\latex{ 1,500 }
\latex{ 2,000 }
\latex{ 1,820 }
\latex{ 1,840 }
\latex{ 1,860 }
\latex{ 1,880 }
\latex{ 896 }
\latex{ 1,848 }
\latex{ 2,004 }
\latex{ 1,830 }
\latex{ 1,848 }
\latex{ 1,867 }
You can decide which number is greater by comparing the digits in the same place value. You always start the comparison with the number in the greatest place value.
Exercises
{{exercise_number}}. Five children collected walnuts into baskets and then counted them. Zoe collected \latex{ 342 }, Martha \latex{ 348 }, Greg \latex{ 402 }, Frank \latex{ 384 } and Helga \latex{ 420 }. Write the number of walnuts in each child's basket in descending order.
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{{exercise_number}}. At the market, you notice that \latex{ 1 } \latex{ kg } of lemons costs no more than €\latex{ 2 } but at least €\latex{ 1 }. How much does \latex{ 1 } \latex{ kg } of the cheapest lemon cost, and how much does \latex{ 1 } \latex{ kg } of the most expensive lemon cost?
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{{exercise_number}}. A bag can hold at least \latex{ 3 } but no more than \latex{ 12 } books. How many books can be in the bag?
{{exercise_number}}. No less than \latex{ 930 } and no more than \latex{ 945 } pupils went on a school trip. How many pupils could have gone on the trip?
{{exercise_number}}. How many natural numbers are there, that are
a) less than \latex{ 17 };
b) no more than \latex{ 17 };
c) at most \latex{ 17 };
d) greater than \latex{ 17 } but less than \latex{ 25 };
e) at least \latex{ 17 } but at most \latex{ 25 };
f) greater than \latex{ 17 } but no less than \latex{ 25 }?
{{exercise_number}}. Describe the size of the numbers on the number line below.
\latex{ 18 }
\latex{ 19 }
\latex{ 20 }
\latex{ 21 }
\latex{ 22 }
\latex{ 23 }
\latex{ 10 }
\latex{ 19 }
\latex{ 14 }
\latex{ 16 }
\latex{ 18 }
\latex{ 20 }
{{exercise_number}}. Put the prices of the items in ascending order. (→)
Fabatka Store
Zsivány str. \latex{ 11 }., Szeged
Peach \latex{ 1 } \latex{ kg }
Plum \latex{ 1 } \latex{ kg }
Apple pie
Croissant
Milk \latex{ 1 } \latex{ litre }
Canned fish
Chocolate

Total:
€\latex{ 2.5 }
€\latex{ 4.5 }
€2.57
€\latex{ 1.99 }
€\latex{ 1.89 }
€\latex{ 0.99 }
€\latex{ 1.96 }

€\latex{ 16.4 }
\latex{ 03 }/\latex{ 02 }/\latex{ 24 } \latex{ 17:32 }
Thank you for your custom.
649649
{{exercise_number}}. In a school, each class has a different number of pupils. Each class has at least \latex{ 20 } and at most \latex{ 35 } pupils. How many classes can this school have?
{{exercise_number}}. At least how many children are in a family where each child has at least one brother and at least one sister?
{{exercise_number}}. Put Europe's peaks in ascending order by their height. Where are these peaks located?
A: Carrauntoohil, \latex{ 1,041 } \latex{ metres }
B: Grossglockner, \latex{ 3,798 } \latex{ metres }
C: Mount Olympus, \latex{ 2,919 } \latex{ metres }
D: Ben Nevis, \latex{ 1,344 } \latex{ metres }
E: Mont Blanc, \latex{ 4,810 } \latex{ metres }
F: Mount Elbrus, \latex{ 5,642 } \latex{ metres }
G: Kékes, \latex{ 1,014 } \latex{ metres }
H: Monte Rosa, \latex{ 4,634 } \latex{ metres }
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{{exercise_number}}. Which number is greater: one containing \latex{ 13 } thousands, \latex{ 8 } tens and \latex{ 9 } units, or one containing \latex{ 1 } ten thousand, \latex{ 7 } hundreds, \latex{ 3 } thousands and \latex{ 19 } units?
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{{exercise_number}}. How many digits can go in the ⬜ to make the relations true?
a) \latex{21 ,342 \lt 21 ⬜ 42}
b) \latex{7 ⬜ 832 \gt 72,632}
c) \latex{95 ⬜ 48 \leq 9 ⬜ 148}
{{exercise_number}}. Draw appropriate number lines and write the following natural numbers on them.
a) \latex{n \lt 5}
b) \latex{n \geq 86} and \latex{n} has two digits
c) \latex{200 \leq n \lt 1,000} and \latex{n} is a round hundred
d) \latex{79 \lt n \leq 88} and \latex{ n} is odd
Read the relations out loud using the following expressions: less than, no less than, greater than, not greater than, at least, at most.
{{exercise_number}}. Write down all the possible three-digit numbers where the unit place is occupied by the digit \latex{ 2 }, and the hundreds place by its double. The tens place should be determined by a roll of the dice. Draw a number line and then try to find the place of each number.
Quiz
A baker came up with three new biscuits. He named them rull, kump and son.
\latex{ 7 } rulls weigh as much as \latex{ 4 } kumps. \latex{ 5 } sons weigh the same as \latex{ 6 } kumps.
Put the biscuits in ascending order by their weight.
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