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Dividing natural numbers
Divide \latex{ 24 } cards between \latex{ 3 } children, so that each of them gets the same number of cards.
Give each child \latex{ 1 } card at a time until there are no cards left.
\latex{ 24 } cards can be equally divided between \latex{ 3 } children by giving \latex{ 8 } cards to each.
Division without a remainder
When \latex{ 24 } cards are equally divided between \latex{ 3 } children, then each will get \latex{ 8 } cards.
\latex{\underbrace{ 24 \div 3} } = \latex{ 8 }
dividend
factor
quotient
quotient
Checking the division with
multiplication:
\latex{8\times3 = 24}
division:
\latex{24 \div 8 = 3}
quotient \latex{\times} factor = dividend
dividend \latex{\div} quotient = factor
Division with a remainder
When \latex{ 26 } cards are equally divided between \latex{ 3 } children, then \latex{ 2 } cards will be left out.
\latex{ 26 \div 3 } = \latex{ 8 }
\latex{ 2 }
remainder
Checking a division with a remainder:
\latex{8\times 3 + 2 = 26}
quotient \latex{\times} factor + remainder = dividend
\latex{(26- 2)\div8 = 3}
(dividend − remainder) \latex{\div} quotient = factor
The remainder is always smaller than the factor.
The properties of division
Observe what happens when the dividend and the factor are switched.
What happens when you group them differently?
What happens when you group them differently?
\latex{36 \div 3 = 12}
the quotient is a natural number
\latex{3 \div 36 = ?}
there is no such natural number
\latex{36 \div 3 ≠ 3 \div 36}
In the case of division, the dividend and the factor cannot be switched because the quotient may change.
\latex{a\div b\neq b\div a}
\latex{(36 \div 6) \div 3 = 6 \div 3 = 2}
\latex{36 \div (6 \div 3) = 36 \div 2 = 18}
\latex{(36 \div 6) \div 3 \neq 36 \div (6\div3)}
In the case of division, brackets cannot be placed randomly because the quotient may change.
\latex{(a\div b)\div c\neq a\div(b\div c)}
Zero in a division
Can the dividend be zero?
\latex{0 \div7 = 0},--because--\latex{7\times0 = 0}
\latex{0 \div 100 = 0},--because--\latex{100\times0 = 0}
\latex{0 \div 100 = 0},--because--\latex{100\times0 = 0}
When \latex{ 0 } is divided by any natural number (other than \latex{ 0 }), the quotient will be \latex{ 0 }.
Can the factor be 0?
If you would like to interpret \latex{7 \div 0 = \textcolor{#E3004F}{\blacktriangle}} , then the \latex{\textcolor{#E3004F}{\blacktriangle}} should be a number, which multiplied by \latex{ 0 } equals \latex{ 7 }, that is, \latex{\textcolor{#E3004F}{\blacktriangle} \times 0 = 7}.
When learning about multiplication, you saw that multiplying any number by \latex{ 0 } equals \latex{ 0 }. Therefore, no natural number can be written in place of \latex{\textcolor{#E3004F}{\blacktriangle}}.
Can the factor and the dividend both be 0?
If you would like to interpret \latex{0 \div 0 = \textcolor{#009EE0}{\blacksquare}} , then the \latex{\textcolor{#009EE0}{\blacksquare}} should be a number, which multiplied by \latex{ 0 } equals \latex{ 0 }, that is, \latex{\textcolor{#009EE0}{\blacksquare} \times 0 = 0}.
Since multiplying any number by zero equals zero, thus \latex{\textcolor{#009EE0}{\blacksquare}} could be any natural number.
The quotient cannot be determined, so \latex{0 \div 0} is undefined.
The quotient cannot be determined, so \latex{0 \div 0} is undefined.
Division by zero is undefined.
Dividing a sum or a difference
Example
Mum bought \latex{ 6 } red apples and dad bought \latex{ 8 } green apples for Adam and David. If the children divided them equally, how many apples did each get?
Solution 1
The children did not take into account the colour of the apples.
Adam
David
\latex{ 6 } from mum
\latex{ 8 } from dad
\latex{(6 + 8) \div 2 = 14 \div 2 = 7}
Each child got \latex{ 7 } apples.
Solution 2
The children took into account the colour of the apples.
Adam
David
\latex{ 6 } from mum
\latex{ 8 } from dad
Adam
David
\latex{6 \div 2 + 8 \div 2 = 3 + 4 = 7}
Each child got \latex{ 7 } apples.
You get the same result in both cases:
\latex{(6 + 8) \div 2 = 6 \div 2 + 8 \div 2}
When dividing sums, the addends can be divided separately and then the quotients must be added.
\latex{(a+b)\div c=}
\latex{=a\div c+b\div c}
\latex{=a\div c+b\div c}
Dividing a difference is similar to dividing a sum.
Example:....\latex{(21 − 9) \div 3 = 12 \div 3 = 4}....and...\latex{21 \div 3 − 9 \div 3 = 7 − 3 = 4}.
You get the same result, that is,
\latex{(21 − 9) \div 3 = 21 \div 3 − 9 \div 3}
When dividing a difference, the minuend and the subtrahend can be divided separately, and the quotients can be subtracted.
\latex{(a-b)\div c=}
\latex{=a\div c-b\div c}
\latex{=a\div c-b\div c}
Exercises
{{exercise_number}}. What is the dividend if the factor is a single-digit number, the remainder is \latex{ 8 }, and the quotient is \latex{ 15 }?
{{exercise_number}}. Perform the calculations in the simplest possible way.
a) \latex{(130+39)\div13}
b) \latex{(1,717+3,434)\div17}
c) \latex{(6,622-5,544)\div11}
d) \latex{3,785\div11-3,763\div11}
e) \latex{2,713\div19+1,087\div19}
f) \latex{3,971\div100+4,029\div100}
{{exercise_number}}. Write down the letter-symbol combinations with equal solutions.
A \latex{=(42-24)\div6}
B \latex{=42\div6-24}
\latex{ \textcolor{#E3004F}{\blacksquare}=42-(24\div6)}
\latex{ \textcolor{#E3004F}{\blacktriangle}=42\div6-24\div6}
C \latex{=(7\times6)\div2}
D \latex{=42\div6-4}
\latex{ \textcolor{#E3004F}{\bullet}=42-24\div6}
\latex{ \textcolor{#E3004F}{\blacktriangledown}=42\div(6-24\div6)}
{{exercise_number}}. Divide numbers greater than \latex{ 50 } and smaller than \latex{ 60 } by \latex{ 4 }.
{{exercise_number}}. What are the possible remainders if the factor is
a) \latex{ 7 };
b) \latex{ 12 };
c) \latex{ 13 };
d) \latex{ 1,001 }?
{{exercise_number}}. You can see the first three numbers of a sequence. What are the next three members?
a) \latex{16,384}; \latex{4,096}; \latex{1,024;} ...
b) \latex{126\times243}; \latex{126\times81}; \latex{126\times27;...}
{{exercise_number}}. A cow gives \latex{ 4,500 } \latex{ litres } of milk each year. How many jugs can be filled with this amount if the volume of a jug is
a) \latex{ 2 } \latex{ litres; }
b) \latex{ 5 } \latex{ litres; }
c) \latex{ 10 } \latex{ litres; }
d) \latex{ 20 } \latex{ litres; }
e) \latex{ 25 } \latex{ litres? }
{{exercise_number}}. The students bought concert tickets for €\latex{ 132 }. The prices of the tickets started at €\latex{ 22 } and increased by €\latex{ 22 }. None of the tickets cost more than €\latex{ 66 }.
a) How many students went to the concert if their tickets were all the same price?
b) How much did the tickets cost if three students went to the concert and each ticket had a different price?
{{exercise_number}}. Perform the calculations in the simplest possible way.
a) \latex{465\div5+535\div5}
b) \latex{162\div3+138\div3}
c) \latex{434\div7-217\div7}
d) \latex{6,372\div9+3,528\div9}
e) \latex{473\div2+527\div2}
f) \latex{6,952\div11-2,541\div11}
{{exercise_number}}. Write down the division and calculate the quotient if
a) the dividend is \latex{ 120 } and the factor is the smallest two-digit number;
b) the dividend is \latex{ 120 }, the factor is one-tenth of the dividend;
c) the factor is \latex{ 20 }, and the dividend is \latex{ 120 }.
{{exercise_number}}. Cheeses sold at a store:
Gouda Cheese (\latex{ 200\,g }) €\latex{ 2 }
Mild Cheddar (\latex{ 200\,g }) €\latex{ 8 }
Natural Edam Cheese (\latex{ 300\,g }) €\latex{ 6 }
Mild Cheddar (\latex{ 1\,kg }) €\latex{ 30 }
Salami cheese (\latex{ 600\,g }) €\latex{ 6 }
How can you compare the prices of the cheeses?
Why would you recommend customers to buy each cheese?
Quiz
Think of a three-digit number. Multiply it by \latex{ 13 }, then by \latex{ 7 } and finally by \latex{ 11 }. Write down the product. If your calculations are correct, it is possible to tell which number you thought of simply by looking at the product. Why?
