Recall formulas connecting division and multiplication. If a number c is a sum of addends a and b, then |
a b = c c a = b c b = a |
1. Commutative property of multiplication: changing the order of factors does not change the product.
Formula: | a c = c a | |
Example: | 2 4 = 4 2 = 8; | 3 7 = 7 3 = 21 |
2. Associative property of multiplication: changing the grouping of factors does not change the product.
Formula: | (a b) c = a (b c) |
Example: | (3 6) 5 = 18 5 = 90; 3 (6 5) = 3 30 = 90 => (3 6) 5 = 3 (6 5) |
Division does not have associative or commutative properties. |
3. Distributive property of multiplication:
- over addition: sum of two numbers times a third number equals product of the first and third numbers plus product of the second and third numbers.
Formula: | (a + b) c = a c + b c |
Example: | (2 + 3) 5 = 5 5 = 25; 2 5 + 3 5 = 10 + 15 = 25 => (2 + 3) 5 = 2 5 + 3 5 |
- over subtraction: difference of two numbers times a third number equals product of the first and third numbers minus product of the second and third numbers.
Formula: | (a - b) c = a c - b c |
Example: | (6 - 3) 5 = 3 5 = 15; 6 5 - 3 5 = 30 - 15 = 15 => (6 - 3) 5 = 6 5 - 3 5 |
Be careful with distributive property. c (a+b) does not equal c a + c b ! |
However, distributive property works when the sign follows the brackets.
(a + b) c = a c + b c
(a - b) c = a c - b c
Because of the distributive property of multiplication we can collect like terms:
Example: | 2z + 7z = (2 + 7)z = 9z; 6x - x = (6 - 1)x = 5x |
4. Combined properties of multiplication and division.
- a b c = a (b c)= a c b
- a b c = (a b) c = a (b c) = b (a c)
- a (b c) = (a c) b
It is easy to illustrate combined properties of multiplication and division with fractions. As you know,
5. Multiplication or division by 1 does not change the number.
For every number a, a 1 = a and a 1= a
Example: | 7 1 = 7 and 7 1 = 7. |
6. Multiplying or dividing 0 by another number gives you 0.
For every number a, 0 a = 0 and 0 a = 0
Since multiplication is associative, for every number a, a 0 = 0
Example: | 0 7 = 7 0 = 0 and 0 7 = 0. |
7. Dividing by 0 is not defined in mathematics.
It means, for example, that 7 0 does not equal anything. There is a reason for that. Let us assume that there is such number b that 7 0 = b . Then the equation b 0 = 7 must be true. But we know, that b 0 = 0 for any b. So we got a contradiction.
Exercises
1.1. Find the value of the numerical expression:
a) (89 + 7)2 +16 x (6 -3) =
b) (63 - 8 x 2) x 2- 123 =
c) 85 - (7 x 10 + 15)5 + 12 =
1.2. Calculate:
a) 72 - 243 + 8 =
b) 7224+ 3 x 8 =
c) 72 - (243 +8) =
1.3. Find the values of these algebraic expressions:
a) 14x - 3 for: x = 12, x = 3.6 and x = $$ \frac{17}{20} $$;
b) x - 60x for: x = 12, x = 20 and x = 15;
c) 78 + 3x for: x = 8, x = 2.3 and x = $$ \frac{2}{7} $$;
d) 54(x - 7) for: x = 8.5, x = 13 and x = 11;
1.4. Word problems.
The problems below are not hard. The most important thing in this exercise is not to find the answer, but to practice translating problems from English to math language. In each problem try to the solution as one numerical expression.
Example:
A gardener planted 408 tulip bulbs. He planted 168 bulbs along the alley in the garden, and the remaining bulbs in 4 flowerbeds. He planted the same number of tulips in each flowerbed. Write a numerical expression that represents the number of tulips in each flowerbed. Find the value of this numerical expression.
Solution: (408-168) is the number of tulips in four flowerbeds together, so the number of tulips in each flowerbed is (408-168)4.
(408-168)4 = 60.
1) There are three bags with coffee beans. If 20,040 g of the beans is taken out of the first bag and put into the third one, and after that 12,004 g is taken out of the second bag and also put into the third one, then all three bags will weigh the same - 40 kg. Write a numerical expression that represents the weight of the third bag in the beginning. Find the value of the numerical expression.
2) One group of geologists collected 15 kg of mineral samples, and another group collected 2 more kg than the first group. They put all the samples into the boxes, so that each box contained 4 kg of minerals, and sent those boxes to the lab. Write a numerical expression that represents the number of the boxes they sent. Find the value of the numerical expression.
3) A train consists of 4 first class cars and 12 economy class cars. There are 16 seats in a first class car and x more in an economy class car. Write an algebraic expression that represents how many seats are there in the whole train. Find the value of the algebraic expression if x = 8.
1.5. Find the value of the numerical expression.
a) 36 + 48(6 -18(6 + 3))=
b) ((75 - 39)12 + 24)9 =
c) 52 - (357 + 15x2)5 =
1.6. Calculate:
a) 72243 x 8 =
b) 7224(3 x 8) =
c) 72(243) x 8 =
1.7. Find the values of these algebraic expressions:
a) 126 ÷ x + 10x for x = 21 and for x = $$ \frac{2}{5} $$;
b) (x - 7) ÷ 5 for x = 42 and for x = 75.5;
c) 156 - (2x + 9) for x = 17 and for x = $$ \frac{7}{8} $$;
d) 9x - 56 for x = 8.8 and for x = 23.
Puzzle of the week
1.8. You have three buckets: 10 liters, 4 liters, and 3 liters. The 10-liter bucket is full of water and you have no other water available. You have to divide the water so there are exactly 5 liters are in the 10-liter bucket, 1 liter in the 3-liter container, and 4 liters in the 4-liter bucket. You may only pour back and forth between the three given buckets. Describe how to do that using a table below. First and last columns are done for you.