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Multi-digit multiplication
\latex{\times}
\latex{\times}
We can count in a number of ways:
\latex{ 12 } pairs of shoes cost €\latex{ 384 }.
\latex{ 2 }
\latex{ 3 }
\latex{ 1 }
\latex{ 1 }
\latex{ 2 }
\latex{ 2 }
\latex{ 2 }
\latex{ 6 }
\latex{ 4 }
\latex{ 4 }
\latex{ 8 }
\latex{32.- }
\latex{ 6 }
\latex{ 3 }
\latex{ 3 }
\latex{ 8 }
\latex{ 4 }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 3 }
Numbers with three or more digits are multiplied in a similar way to numbers with two digits. The digits of one factor are multiplied by the digits of the other factor. The place values of the sub-products also need to be taken into account.
Example
Do the following multiplications:
a) \latex{624 \times 473}
b) \latex{732 \times 461}
c) \latex{347 \times 204}
Solution
a) \latex{624 \times 473 = 624 \times \left(400 + 70 + 3\right) = 624 \times 400 + 624 \times 70 + 624 \times 3}
Start the multiplication with the hundreds.
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
BRIEFLY:
\latex{ , }
\latex{ , }
\latex{3}
\latex{3}
\latex{2}
\latex{4}
\latex{4}
\latex{7}
\latex{2}
\latex{9}
\latex{2}
\latex{2}
\latex{2}
\latex{2}
\latex{2}
\latex{6}
\latex{2}
\latex{6}
\latex{6}
\latex{6}
\latex{6}
\latex{0}
\latex{7}
\latex{2}
\latex{4}
\latex{9}
\latex{5}
\latex{1}
\latex{3}
\latex{6}
\latex{8}
\latex{1}
\latex{5}
\latex{7}
\latex{8}
\latex{0}
\latex{0}
\latex{4}
\latex{4}
\latex{4}
\latex{4}
\latex{4}
\latex{4}
\latex{4}
\latex{0}
\latex{0}
\latex{0}
\latex{3}
\latex{3}
\latex{7}
\latex{6}
\latex{6}
\latex{2}
\latex{9}
\latex{4}
\latex{7}
\latex{4}
\latex{6}
\latex{9}
\latex{5}
\latex{4}
\latex{4}
\latex{3}
\latex{6}
\latex{8}
\latex{7}
\latex{5}
\latex{2}
\latex{2}
\latex{2}
\latex{2}
\latex{8}
\latex{1}
\latex{1}
b) \latex{732 \times 461 = 732 \times \left(400 + 60 + 1\right) = 732 \times 400 + 732 \times 60 + 732 \times 1}
Now begin the multiplication with the units.
Now begin the multiplication with the units.
\latex{1}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
BRIEFLY:
\latex{ , }
\latex{ , }
\latex{\times}
\latex{7}
\latex{3}
\latex{2}
\latex{4}
\latex{6}
\latex{7}
\latex{3}
\latex{2}
\latex{2}
\latex{0}
\latex{2}
\latex{6}
\latex{3}
\latex{1}
\latex{7}
\latex{3}
\latex{2}
\latex{7}
\latex{3}
\latex{2}
\latex{2}
\latex{1}
\latex{0}
\latex{0}
\latex{4}
\latex{6}
\latex{1}
\latex{0}
\latex{4}
\latex{7}
\latex{9}
\latex{3}
\latex{8}
\latex{2}
\latex{3}
\latex{3}
\latex{7}
\latex{4}
\latex{5}
\latex{3}
\latex{2}
\latex{0}
\latex{0}
\latex{4}
\latex{9}
\latex{2}
\latex{7}
\latex{7}
\latex{7}
\latex{3}
\latex{2}
\latex{4}
\latex{6}
\latex{3}
\latex{2}
\latex{2}
\latex{2}
\latex{2}
\latex{9}
\latex{3}
\latex{4}
\latex{9}
\latex{2}
\latex{3}
\latex{3}
\latex{7}
\latex{2}
\latex{8}
\latex{4}
\latex{5}
If there is a \latex{ 1 } in any of the factors, you can write fewer sub-products.
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
BRIEFLY:
\latex{ , }
\latex{ , }
\latex{1}
\latex{7}
\latex{3}
\latex{4}
\latex{4}
\latex{6}
\latex{1}
\latex{0}
\latex{2}
\latex{5}
\latex{4}
\latex{7}
\latex{3}
\latex{3}
\latex{2}
\latex{9}
\latex{2}
\latex{8}
\latex{9}
\latex{2}
\latex{3}
\latex{2}
\latex{0}
\latex{0}
\latex{7}
\latex{7}
\latex{3}
\latex{3}
\latex{2}
\latex{2}
\latex{1}
\latex{7}
\latex{3}
\latex{2}
\latex{7}
\latex{3}
\latex{2}
\latex{7}
\latex{3}
\latex{0}
\latex{2}
\latex{4}
\latex{4}
\latex{6}
\latex{6}
\latex{0}
\latex{0}
\latex{1}
\latex{2}
\latex{4}
\latex{6}
\latex{3}
\latex{2}
\latex{9}
\latex{3}
\latex{4}
\latex{9}
\latex{2}
\latex{2}
\latex{8}
\latex{5}
\latex{4}
\latex{7}
\latex{3}
So the product we get is the same as before.
c) \latex{347 \times 204 = 347 \times \left(200 + 4\right) = 347 \times 200 + 347 \times 4}
\latex{4}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
BRIEFLY:
\latex{ , }
\latex{ , }
\latex{\times}
\latex{0}
\latex{4}
\latex{7}
\latex{6}
\latex{0}
\latex{3}
\latex{6}
\latex{9}
\latex{4}
\latex{7}
\latex{7}
\latex{0}
\latex{1}
\latex{3}
\latex{8}
\latex{8}
\latex{8}
\latex{8}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
\latex{3}
\latex{4}
\latex{4}
\latex{7}
\latex{7}
\latex{7}
\latex{7}
\latex{2}
\latex{2}
\latex{0}
\latex{0}
\latex{0}
\latex{8}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
\latex{4}
\latex{4}
\latex{2}
\latex{2}
\latex{3}
\latex{4}
\latex{4}
\latex{9}
\latex{7}
\latex{7}
\latex{7}
\latex{3}
\latex{3}
\latex{3}
\latex{4}
\latex{4}
\latex{4}
\latex{0}
\latex{1}
\latex{3}
\latex{8}
\latex{8}
\latex{8}
Do not write out sub-products that are made up only of zeros.
\latex{6}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
BRIEFLY:
\latex{ , }
\latex{ , }
\latex{\times}
\latex{3}
\latex{0}
\latex{7}
\latex{8}
\latex{3}
\latex{4}
\latex{9}
\latex{7}
\latex{7}
\latex{0}
\latex{1}
\latex{0}
\latex{8}
\latex{8}
\latex{8}
\latex{0}
\latex{4}
\latex{8}
\latex{2}
\latex{4}
\latex{4}
\latex{4}
\latex{4}
\latex{3}
\latex{4}
\latex{4}
\latex{4}
\latex{4}
\latex{0}
\latex{2}
\latex{0}
\latex{0}
\latex{1}
\latex{3}
\latex{8}
\latex{8}
\latex{8}
\latex{7}
\latex{3}
\latex{3}
\latex{4}
\latex{7}
\latex{7}
\latex{7}
\latex{7}
\latex{7}
\latex{0}
\latex{9}
\latex{2}
\latex{0}
\latex{6}
\latex{3}
\latex{2}
Did you know? We can multiply multi-digit numbers by writing the last digit of the corresponding sub-product under the digit we are multiplying it with. This way there is no confusion about the position of the sub-products, even when multiplying with the digits of a factor in any order.
\latex{3}
\latex{ , }
\latex{\times}
\latex{9}
\latex{4}
\latex{6}
\latex{1}
\latex{4}
\latex{2}
\latex{2}
\latex{2}
\latex{4}
\latex{4}
\latex{9}
\latex{6}
\latex{8}
\latex{8}
\latex{2}
\latex{2}
\latex{5}
\latex{5}
\latex{1}
\latex{6}
\latex{3}
\latex{7}
\latex{7}
Exercises
{{exercise_number}}. Perform the multiplications in writing.
a) \latex{57 \times 96}
b) \latex{957 \times 96}
c) \latex{43\times 49}
d) \latex{843 \times 49}
e) \latex{79 \times 98}
f) \latex{79\times298}
g) \latex{94\times61}
h) \latex{943 \times 618}
i) \latex{23 \times 14}
j) \latex{231 \times 314}
k) \latex{44 \times 884}
l) \latex{707 \times 608}
m) \latex{395 \times 531}
n) \latex{194 \times 685}
o) \latex{2,001 \times 376}
p) \latex{606 \times 1,001}
{{exercise_number}}.
a) There are \latex{ 37 } rows of \latex{ 24 } seats in a theatre. How many people can sit there?
b) In another theatre, there are \latex{ 28 } rows and \latex{ 17 } balcony rows with \latex{ 26 } seats in each row. How many people can watch the film when the theatre is full?
b) In another theatre, there are \latex{ 28 } rows and \latex{ 17 } balcony rows with \latex{ 26 } seats in each row. How many people can watch the film when the theatre is full?
{{exercise_number}}. There are \latex{ 3,720 } seats in each sector of the stadium. If there are \latex{ 16 } sectors, how many seats are there in total?
{{exercise_number}}. The following numbers are one hundred times and ten times what number?
a) \latex{25 \times 5 \times 34 \times 8 \times 2}
b) \latex{125 \times 32 \times 4}
c) \latex{6 \times 33 \times 421}
d) \latex{12 \times 15 \times 40 \times 4}
{{exercise_number}}. One crate can hold \latex{ 24 } boxes of chocolate. Each box contains \latex{ 36 } bars of chocolate. A bar of chocolate costs €\latex{ 2 }. How much does a box of chocolates cost?
Quiz
What number should we use to replace the letters to make the multiplication correct both horizontally and vertically?
\latex{ C}
\latex{ \times }
\latex{ \times }
\latex{ \times }
\latex{ \times }
\latex{ \times }
\latex{ A\,\,B }
\latex{ B\,\,\,A\,\,\,C }
\latex{ G\,\,\,H\,\,\,C }
\latex{ B\,\,F}
\latex{ E\,\,A}
\latex{ A}
\latex{ D}
\latex{ C}
\latex{ =}
\latex{ =}
\latex{ =}
\latex{ =}
\latex{ =}
