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Changes of the quotient
\latex{18\div3=6},
that is, each child
gets \latex{ 6 } cookies.
gets \latex{ 6 } cookies.
Observe how the quotient changes in the following examples.
The dividend changes
(the divisor is unchanged)
\latex{36\;\div\;3 \;=\;12}
\latex{18\;\div\;3 \;=\;6}
multiplied
by \latex{ 2 }
by \latex{ 2 }
multiplied
by \latex{ 2 }
by \latex{ 2 }
Each person gets twice as many cookies from two sheet pans; that is, everyone can eat \latex{ 12 } cookies.
\latex{6\;\;\div\;3 \;=\;2}
\latex{18\;\div\;3 \;=\;6}
divided
by \latex{ 3 }
by \latex{ 3 }
divided
by \latex{ 3 }
by \latex{ 3 }
When the amount of cookies is decreased to one-third of the original amount, each person can eat \latex{ 2 } cookies.
The divisor changes
(the dividend is unchanged)
\latex{18\;\div\;6 \;=\;3}
\latex{18\;\div\;3 \;=\;6}
divided
by \latex{ 2 }
by \latex{ 2 }
multiplied
by \latex{ 2 }
by \latex{ 2 }
If the cookies are divided among twice as many children, then each child gets half as many cookies, that is, \latex{ 3 }.
\latex{18\;\div\;1 \;=18}
\latex{18\;\div\;3 \;=\;6}
multiplied
by \latex{ 3 }
by \latex{ 3 }
divided
by \latex{ 3 }
by \latex{ 3 }
If the cookies are divided among one-third of the children, then one child gets three times more.
The dividend and the divisor change by the same amount
\latex{36\;\div\;6 \;=\;6}
\latex{18\;\div\;3 \;=\;6}
unchanged
multiplied
by \latex{ 2 }
by \latex{ 2 }
multiplied
by \latex{ 2 }
by \latex{ 2 }
If two sheet pans of cookies are divided among two times as many children, then the number of cookies a child can eat remains unchanged.
\latex{6\;\;\div\;\;1 \;=\;6}
\latex{18\;\div\;3 \;=\;6}
divided
by \latex{ 3 }
by \latex{ 3 }
divided
by \latex{ 3 }
by \latex{ 3 }
unchanged
If one-third of the cookies are divided among one-third of the children, then the number of cookies a child can eat remains unchanged.
The quotient does not change if the dividend and the divisor are multiplied or divided by the same number
(which is not zero).
(which is not zero).
Exercises
{{exercise_number}}. Perform the divisions. Calculate in the simplest possible way.
a) \latex{6,000 \div 30}
b) \latex{8,000 \div 40}
c) \latex{100,000 \div 100}
d) \latex{400,000 \div 200}
e) \latex{400,000 \div 800}
f) \latex{150,000 \div 15}
{{exercise_number}}. Compare the quotients. Why did they change?
A) \latex{72 \div 4}
B) \latex{24 \div 8}
C) \latex{48 \div 4}
D) \latex{36 \div 12}
E) \latex{144 \div 16}
F) \latex{96 \div 12}
{{exercise_number}}. \latex{ 240 } books will be given as a prize for the winners of a crossword competition. How many books does a winner get if
a) there are \latex{ 120 } winners;
b) the number of winners is halved;
c) the number of winners is one-third of the winners in point a)?
{{exercise_number}}. What is the rule for each sequence? Write down the next \latex{ 3 } members.
a) \latex{240 \div 96;\; 240 \div 48;\; 240 \div 24; ... }
b) \latex{15,625 \div 5;\; 3,125 \div 5;\; 625 \div 5; ...}
c) \latex{700,000;\; 70,000;\; 7,000; ...}
d) \latex{960 \div 480;\; 480 \div 240;\; 240 \div 120; ...}
{{exercise_number}}. How many \latex{ hours } and \latex{ days } are
a) \latex{ 2,880 } \latex{ minutes; }
b) \latex{ 7,200 } \latex{ minutes; }
c) \latex{ 14,400 } \latex{ minutes; }
d) \latex{ 86,200 } \latex{ seconds? }
Quiz
Which equality is incorrect if, \latex{ a }, \latex{ b } and \latex{ c } are natural numbers?
a) \latex{(a \div b) \div c = a \div c \div b}; \latex{b \neq 0, c \neq 0}
b) \latex{ (a \div c) \div (b \div c) = a \div b}; \latex{b \neq 0, c \neq 0}
c) \latex{(a \times c) \div (b \div c) = a \div b}; \latex{b \neq 0, c \neq 0}
d) \latex{a \div (b \div c) = (a \div b) \times c}; \latex{b \neq 0, c \neq 0}
