Andrei can pack 2 sandwiches in one minute. He will be selling it in a country fair. How many sandwiches can he pack in 5 minutes? In an hour?
We can solve the first problem using repeated addition.
2 + 2 + 2 + 2 + 2 =10
You can also multiply:
$\frac{2 \ sandwiches}{1 \ minute} \ x \ 5 \ minutes $
where you cancel out the minutes and so Andre can pack 2 x 5 = 10 sandwiches in 5 minutes.
Let us review the algorithm for the multiplication of whole numbers.
23 x 4 = (20 + 3) x 4
= (20 x 4) + (3 x 4)
= 80 + 12 = 92
Short method:
23
x 4
12 <-- 4 x 3 (first partial product)
+ 8 <-- 4 x 2 (second partial product; 8 and 1 are aligned)
92 <-- complete product (sum of the partial products)
Let us answer the second question "how many sandwiches can Andrei pack in an hour?".
Note that there are 60 minutes in an hour, so we multiply 2 by 60.
To multiply numbers that end in zeroes, first multiply without considering ending zeroes. Then write the number of zeroes in the product that are present in all the factors.
2
x 6 [0]
120 <-- one zero is annexed to the complete product.
Andrei can pack 120 sandwiches in 1 hour.
Properties of Multiplication
Commutative Property
Is 2 x 60 = 60 x 2? or 23 x 4 = 4 x 23?
If yes, then you are right! This is the commutative property of multiplication. If a and b are numbers, then a x b = b x a.
Associative Property
Now, 8 x (4 x 3) = 96. But do you get the same product if it is (8 x 4) x 3?
If yes, then you're right again. If a, b, and c are any numbers, then (a x b) x c = a x (b x c). This is the associative property of multiplication.
Distributive Property
The last property of multiplication is the distributive property of multiplication. The result is the same when we multiply a sum by a number and multiplying each addend first then adding the products. It is written as a(b + c) = ab + ac.
Let us apply these properties below
3 x 41 x 90 = 3 x 90 x 41
= (3 x 90) x (40 + 1)
= 270 x (40 + 1)
= (270 x 40) + (270 x 1)
= [(200 + 70) x 40] + 270
= (200 x 40) + (70 x 40) + 270
= 8000 + 2800 + 270
= 8000 + 2000 + 800 + 200 + 70
= 10 000 + 1 000 + 70
= 11,070
Whew! that was a long one. Can you guess which properties were used in every line?
Zero Property
The last property to remember is the zero property of multiplication. Any number times zero equals zero. Example, 5 x 0 = 0.
If you multiply any number by 1 then you will get the same number. 1 is the identity element for multiplication. For example, 1 x 6 = 6.
Try the following:
Multiply
a. 12 x 23
b. 456 x 341
c. 11 021 x 35
d. 12 x 3 x 55
e. 240 x 2 x 63
f.
40,306
x 607
