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Multiplying natural numbers
\latex{5 + 5 + 5 + 5 = 4 \times 5 = 20}
\latex{4 + 4 + 4 + 4 + 4 = 5 \times 4 = 20}
There are \latex{ 5 } balconies in each column: \latex{ 4 \times 5 = 20} balconies.
There are \latex{ 4 } balconies on each floor: \latex{ 5 \times 4 = 20} balconies.
Two kids have counted the number of balconies in their house. One counted by columns, the other by floors. Both children counted \latex{ 20 } balconies.
The sum of an addition involving identical numbers can also be seen as the product of a multiplication. In this case, the two sides of the multiplication are called the repeating terms and the number of terms.
factors
\latex{5}
\latex{4}
x
product
\latex{\text{=}}
\latex{20}
product
The properties of multiplication
\latex{5 \times 4 = 4 \times 5 = 20}
Changing the order of the factors does not change the product.
\latex{a \times b = b \times a}
This is true for all two-factor multipilications.
To multiply three numbers, first multiply any two of them and then multiply their product by the third. If there are parentheses, start by multiplying the numbers inside them.
\latex{(5 \times 4) \times 2=20\times2 = 40}
\latex{\text{5} \times (4 \times 2) = 5\times8 = 40}
\latex{(5 \times 4) \times 2 = 5 \times (4 \times 2) = 5 \times 4 \times 2 = 40}
When multiplying several factors, the product does not change even if the factors are grouped differently. This means that the brackets can be omitted.
\latex{a \times (b \times c) =}
\latex{= (a \times b) \times c = a \times b \times c}
Zero in Multiplication
\latex{0 + 0 + 0 + 0 + 0 + 0 = 6 \times 0 = 0 \times 6 = 0}
If one of the terms in a two-factor multiplication is \latex{ 0 }, then the product is \latex{ 0 }.
\latex{5 \times 3 \times 0 \times 2 = 0}
If any of the factors is \latex{ 0 }, then the product is \latex{ 0 }.
The product of a multiplication cannot be \latex{ 0 } if none of its factors is \latex{ 0 }.
If the product is \latex{ 0 }, then one of its factors must also be \latex{ 0 }.
Multiplying a sum or a difference
There are \latex{3 \times 2} white and \latex{3 \times 5} yellow
flowers, so the number of flowers is
\latex{3 \times 2 + 3 \times 5 = 6 +15 = 21.}
flowers, so the number of flowers is
\latex{3 \times 2 + 3 \times 5 = 6 +15 = 21.}
There are \latex{ 2 } white and \latex{ 5 } yellow
flowers in all \latex{ 3 } rows, so the number
of flowers is \latex{3 \times (2 + 5) = 3 \times 7 = 21.}
flowers in all \latex{ 3 } rows, so the number
of flowers is \latex{3 \times (2 + 5) = 3 \times 7 = 21.}
\latex{3 \times 2 + 3 \times 5 = 6 + 15 = 21 }
\latex{3 \times (2 + 5) = 3 \times 7 = 21 }
How many flowers are there?
\latex{3 \times (2 + 5) = 3 \times 2 + 3 \times 5}
You can also multiply a product and any given number by multiplying the factors of the product one by one and then adding them up.
\latex{a \times (b + c) =}
\latex{= a \times b + a \times c}
There were \latex{ 7 } flowers in all \latex{ 3 } rows. \latex{ 2 } have wilted in each of the \latex{ 3 } rows, so there are \latex{7 \times 3 - 3 \times 2 = 21- 6 =15} flowers left.
There were \latex{ 7 } flowers in all \latex{ 3 } rows. \latex{ 2 } of the \latex{ 7 } flowers in each row have wilted, so there are \latex{3 \times (7 - 2) = 3 \times 5 = 15} flowers left.
\latex{3 \times 7 - 3 \times 2 = 21 - 6 = 15}
\latex{3 \times (7 - 2) = 3 \times 5 = 15}
How many flowers have not wilted?
\latex{3 \times (7 - 2) = 3 \times 7 - 3 \times 2}
To multiply a difference by any given number, first multiply the minuend and the subtrahend separately, then subtract the two products.
\latex{a \times (b - c) =}
\latex{= a \times b - a \times c}
Example
Perform the calculations in your head.
a) \latex{4 \times 525}
b) \latex{2 \times 1,999}
c) \latex{102 \times 16}
d) \latex{980 \times 54}
b) \latex{2 \times 1,999}
c) \latex{102 \times 16}
d) \latex{980 \times 54}
Solution
a) \latex{4 \times 525 = 4 \times (500 + 25) = 4 \times 500 + 4 \times 25 = 2,000 + 100 = 2,100}
b) \latex{2 \times 1,999 = 2 \times (2,000 - 1) = 2 \times 2,000 - 2 \times 1 = 4,000 - 2 = 3,998}
c) \latex{102 \times 16 = (100 + 2)\times 16 = 100 \times 16 + 2 \times 16 = 1,600 + 32 = 1,632}
d) \latex{980 \times 54 = (1,000 - 20) \times 54 = 54,000 - 1,080 = 52,920}
\latex{(a + b) \times c =}
\latex{= a \times c + b \times c}
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Exercises
{{exercise_number}}. Perform the calculations in your head. Use the previous examples for help.
a) \latex{ 7 \times 1,995}
b) \latex{2,125 \times 8}
c) \latex{108 \times 820 - 8 \times 820}
d) \latex{58 \times 37 + 42 \times 37}
{{exercise_number}}. Choose the expressions with the same value.
a) \latex{5 \times (23 + 47)}
\latex{5 \times 23 + 47}
\latex{23 + 5 \times 47}
\latex{5 \times 23 + 5 \times 47}
b) \latex{145 - 55 \times 2}
\latex{145 \times 2 - 55}
\latex{145 \times 2 - 55 \times 2}
\latex{(145 - 55) \times 2}
{{exercise_number}}. Kate and Pam bought \latex{ 3 } bottles of paint each. The paint that Kate bought cost €\latex{ 13 } per bottle, while the paint Pam bought cost €\latex{ 27 } per bottle. How much more did Kate pay than Pam?
{{exercise_number}}. A family goes on holiday for seven \latex{ days. } The hotel costs €\latex{ 130 }, the food €\latex{ 78 } \latex{ per } \latex{ day. } Was €\latex{ 1,500 } enough for the hotel and the food? Think of several ways to solve the problem. Write them down.
{{exercise_number}}. There are a dozen (\latex{ 12 }) roses in a bunch. Eight of these bunches have been put in a box. How many roses are there in five such boxes? Write down your calculations. In which order should you calculate if you want the quickest answer?
{{exercise_number}}. The flower clock on the right is made from two kinds of flowers. Each section contains \latex{ 20 } yellow pansies or \latex{ 14 } red begonias. How many flowers have been planted in total?
{{exercise_number}}. A world champion tap dancer can tap his shoes \latex{ 35 } times a \latex{ second. } How many times can he tap his shoes at most in
a) \latex{ 15 } \latex{ seconds; }
b) half a \latex{ minute; }
c) \latex{ one\; minute; }
d) \latex{ 75 } \latex{ seconds? }
{{exercise_number}}. How many times are the product of the first six odd natural numbers greater than the product of the first five odd natural numbers?
Quiz
The product of five natural numbers is \latex{ 0 }. The largest one is \latex{ 1,024 }. What is the smallest number?
