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Dividing numbers by a multi-digit divisor
Example 1
A class of \latex{ 24 } students collected \latex{ 3,264\,kg } of paper for recycling purposes. To enrol in the competition between classes, they have to calculate the amount of paper per person.
Solution
Plan:
Estimation:
The amount per person is \latex{3,264\;kg\;\div\;24 =} ?
The quotient is between \latex{100\;kg} and \latex{200\;kg}, because
\latex{24 \times 100\;kg=2,400\;kg\lt} \latex{3,264\;kg} and \latex{24 \times 200\;kg=4,800\;kg \gt 3,264\;kg}.
\latex{24 \times 100\;kg=2,400\;kg\lt} \latex{3,264\;kg} and \latex{24 \times 200\;kg=4,800\;kg \gt 3,264\;kg}.
Then, perform the division using the column method.
\latex{\text{2'}}
\latex{\text{3}}
\latex{\text{6'}}
\latex{\text{4'}}
\latex{\text{÷}}
\latex{\text{2}}
\latex{\text{4}}
\latex{\text{hundreds}}
\latex{\text{3}}
\latex{\text{2}}
\latex{\text{÷}}
\latex{\text{2}}
\latex{\text{4}}
\latex{\text{=}}
\latex{\text{1}}
\latex{\text{hundreds}}
\latex{\text{−}}
\latex{\text{4}}
\latex{\text{2}}
\latex{\text{2}}
\latex{\text{4}}
\latex{\times}
\latex{\text{1}}
\latex{\text{8}}
\latex{\text{6}}
\latex{\text{tens}}
\latex{\text{2}}
\latex{\text{÷}}
\latex{\text{4}}
\latex{\text{=}}
\latex{\text{3}}
\latex{\text{tens}}
\latex{\text{7}}
\latex{\text{2}}
\latex{\text{−}}
\latex{\text{2}}
\latex{\text{4}}
\latex{\times}
\latex{\text{3}}
\latex{\text{1}}
\latex{\text{4}}
\latex{\text{4}}
\latex{\text{ones}}
\latex{\text{÷}}
\latex{\text{2}}
\latex{\text{4}}
\latex{\text{=}}
\latex{\text{6}}
\latex{\text{ones}}
\latex{\text{1}}
\latex{\text{4}}
\latex{\text{4}}
\latex{\text{−}}
\latex{\text{2}}
\latex{\text{4}}
\latex{\times}
\latex{\text{6}}
\latex{\text{0}}
\latex{\text{ones}}
\latex{\text{=}}
\latex{\text{hundreds}}
\latex{\text{1}}
\latex{\text{+}}
\latex{\text{3}}
\latex{\text{tens}}
\latex{\text{+}}
\latex{\text{6}}
\latex{\text{ones}}
\latex{\text{=}}
\latex{\text{=}}
\latex{\text{=}}
\latex{\text{1}}
\latex{\text{3}}
\latex{\text{6}}
\latex{\text{COLUMN DIVISION:}}
\latex{\text{DIVISION IN HEAD:}}
\latex{\text{CHECK:}}
\latex{\text{3}}
\latex{\text{2'}}
\latex{\text{6'}}
\latex{\text{4'}}
\latex{\text{÷}}
\latex{\text{2}}
\latex{\text{4}}
\latex{\text{=}}
\latex{\text{1}}
\latex{\text{3}}
\latex{\text{6}}
\latex{\text{−}}
\latex{\text{2}}
\latex{\text{4}}
\latex{\text{8}}
\latex{\text{6}}
\latex{\text{−}}
\latex{\text{7}}
\latex{\text{2}}
\latex{\text{1}}
\latex{\text{4}}
\latex{\text{4}}
\latex{\text{−}}
\latex{\text{1}}
\latex{\text{4}}
\latex{\text{4}}
\latex{\text{0}}
\latex{\text{3}}
\latex{\text{2'}}
\latex{\text{6'}}
\latex{\text{4'}}
\latex{\text{÷}}
\latex{\text{2}}
\latex{\text{4}}
\latex{\text{=}}
\latex{\text{1}}
\latex{\text{3}}
\latex{\text{6}}
\latex{\text{8}}
\latex{\text{6}}
\latex{\text{1}}
\latex{\text{4}}
\latex{\text{4}}
\latex{\text{0}}
\latex{\text{1}}
\latex{\text{3}}
\latex{\text{6}}
\latex{\times}
\latex{\text{2}}
\latex{\text{4}}
\latex{\text{2}}
\latex{\text{2}}
\latex{\text{7}}
\latex{\text{5}}
\latex{\text{4}}
\latex{\text{4}}
\latex{\text{3,}}
\latex{\text{2}}
\latex{\text{4}}
\latex{\text{6}}
The amount of paper per person is \latex{ 136\,kg }.
Example 2
During afforestation, \latex{ 5,179 } trees were planted in rows containing \latex{ 54 } trees. How many rows of trees are there in the forest?
Solution
- Plan: the number of rows is \latex{5,179\;\div\;54 =\;?}
Estimation:
\latex{\text{5,400} \div \text{54 = 100}}
\latex{\text{5,179} \div \text{54} \approx 100}
\latex{\text{5,400} \div \text{54 = 100}}
\latex{\text{5,179} \div \text{54} \approx 100}
Division:
\latex{517'9'\;\div\;54=95}
\latex{319}
\latex{49}
\latex{517'9'\;\div\;54=95}
\latex{319}
\latex{49}
Check:
\latex{95 \times 54 + 49 =}
\latex{=5,130 + 49 = 5,179}
\latex{95 \times 54 + 49 =}
\latex{=5,130 + 49 = 5,179}
The quotient is \latex{ 95 }, the remainder is \latex{ 49 }, so the trees are planted in \latex{ 96 } rows, and the last row only has \latex{ 49 } trees in it.
Exercises
If possible, make an estimation before solving the exercises, then check the results.
{{exercise_number}}. Perform the divisions in your head. What do you notice?
a) \latex{81\div27}
\latex{27\div9}
\latex{9\div3}
\latex{27\div9}
\latex{9\div3}
b) \latex{960\div26}
\latex{960\div48}
\latex{960\div24}
\latex{960\div48}
\latex{960\div24}
c) \latex{56\div14}
\latex{560\div14}
\latex{5,600\div14}
\latex{560\div14}
\latex{5,600\div14}
d) \latex{51\div17}
\latex{510\div170}
\latex{5,100\div1,700}
\latex{5,100\div1,700}
{{exercise_number}}. Perform the divisions using the column method.
a) \latex{660\div66}
b) \latex{6,666\div66}
c) \latex{66,066\div66}
d) \latex{660,660\div66}
e) \latex{528\div48}
f) \latex{528,528\div48}
g) \latex{5,328\div48}
h) \latex{5,808\div48}
{{exercise_number}}. Perform the divisions.
a) \latex{7,230 \; \div \;13}
b) \latex{72,300\div13}
c) \latex{72,300\div130}
d) \latex{723,000\div1,300}
e) \latex{2,289\div109}
f) \latex{6,867\div327}
g) \latex{19,789\div981}
h) \latex{20,601\div981}
{{exercise_number}}. How long did the rain last in \latex{ minutes } if it rained for \latex{ 1,080 } \latex{ seconds? }
{{exercise_number}}. Which is the greatest quotient? Arrange the operations in ascending order according to the divisors, then perform the divisions.
a) \latex{37,995\div51}
b) \latex{37,995\div745}
c) \latex{37,995\div149}
d) \latex{37,995\div255}
e) \latex{37,995\div85}
f) \latex{37,995\div447}
{{exercise_number}}. The bus costs €\latex{ 1,200 } for the trip. How much does one person have to pay if everyone pays the same amount and there are
a) \latex{ 50 } passengers;
b) \latex{ 48 } passengers?
{{exercise_number}}. Tickets for the play cost €\latex{ 25 }, €\latex{ 20 } and €\latex{ 10 }. A group of people decides to buy tickets that cost the same for €\latex{ 200 }. How many tickets of each price can they buy?
{{exercise_number}}. Agnes sells memory cards. How much does one memory card cost if she has sold \latex{ 98 } and earned €\latex{ 6,370 }?
{{exercise_number}}. \latex{ 6 } crates of juice boxes were delivered to the school cafeteria for €\latex{ 342 }. How much does a crate cost? How much does one juice box cost if one crate contains \latex{ 27 } boxes?
{{exercise_number}}. The dividend is \latex{ 5,593 }, the quotient is \latex{ 147 } and the remainder is \latex{ 7 }. What is the divisor?
{{exercise_number}}. \latex{ 170 } memory cards were sold at a supermarket. How much does a single memory card cost if the supermarket earned €\latex{ 1,190 } from the sales?
{{exercise_number}}. A spaceship travelling from Earth to Mars would have to travel \latex{ 189 } million \latex{ kilometres } and it would travel for \latex{ 259 } \latex{ days. } How many \latex{ kilometres } does the spaceship travel \latex{ each\; day? } What distance can it travel in an \latex{ hour? }
Quiz
Nine times which four-digit number is the same as
four times \latex{ 2,178 }?
\latex{\text{2,}}
\latex{\text{1}}
\latex{\text{7}}
\latex{\text{8}}
\latex{\times}
\latex{\text{4}}
\latex{\text{8,}}
\latex{\text{7}}
\latex{\text{1}}
\latex{\text{2}}
