What are the different distributive laws, and when should we use them?
I. Distributive laws
In the following two paragraphs, a, b, and k designate any numbers.A. Distribution of multiplication with respect to addition
This gives us the rule: k × (a + b) = k × a + k × bMore simply, we can write: k(a + b) = ka + kb
Example: 2 × (3 + 4) = 2 × 3 + 2 × 4
We can check that: 2 × (3 + 4) = 2 × 7 = 14
and that: 2 × 3 + 2 × 4 = 6 + 8 = 14.
Note: We also have: (a + b) × k = a × k + b × k and therefore:(a + b)k = ak + bk = ka + kb = k(a + b).
B. Distribution of multiplication with respect to subtraction
This gives us the rule: k × (a – b) = k × a – k × bMore simply, we can write: k(a – b) = ka – kb
Example: 3 × (5 – 2) = 3 × 5 - 3 × 2
Note: We also have: (a – b) × k = a × k – b × k and therefore:(a – b)k = ak – bk = ka – kb = k(a – b).
C. Generalization
The preceding formulas can be generalized to any number of terms inside the parentheses.Example: 2 × (3 + 4 – 5) = 2 × 3 + 2 × 4 – 2 × 5
II. Examples of application
A. Mental calculation
Example 1: We want to calculate 25 × 11; 25 × 21; and 25 × 31.We can proceed like this:
25 × 11 = 25 × (10 + 1) = 25 × 10 + 25 × 1 = 250 + 25 = 275
In the same way:
25 × 21 = 25 × 20 + 25 = 500 + 25 = 525
25 × 31 = 25 × 30 + 25 = 750 + 25 = 775
Example 2: We want to calculate 24 × 9; 24 × 19; and 24 × 29.
We can proceed like this:
24 × 9 = 24 × (10 – 1) = 24 × 10 – 24 × 1 = 240 – 24 = 216
In the same way:
24 × 19 = 24 × (20 – 1) = 24 × 20 – 24 = 480 – 24 = 456
24 × 29 = 24 × (30 – 1) = 24 × 30 – 24 = 720 – 24 = 696
B. Calculating the lateral surface area of a right prism
Figure 1 shows the net of the lateral surface of a right prism. The unit of length is centimeters.
To calculate the area of this surface, we can carry out the following calculation:
24 × 14 + 24 × 19 + 24 × 12
This would amount to adding together the areas of the three rectangles.
It is, however, simpler to carry out this calculation:
24 × (14 + 19 + 12) = 24 × 45 = 1,080
So we find that the area of this surface is equal to 1,080 cm2.
Note: (14 + 19 + 12) is the size in centimeters of the perimeter of a base of the prism.
3. Calculating the area of a circular ring
We want to calculate the area of the circular ring shaded in figure 2. The unit of length is centimeters. The unit of area will be centimeters squared.
The area of this ring will be equal to the difference between the area of the large circle with a radius of 3 cm and the area of the small circle with a radius of 2 cm. We have:
area of the large circle: π × 32 = π × 3 × 3 = 9 π;
area of the small circle: π × 22 = π × 2 × 2 = 4 π;
area of the ring: 9 π - 4 π = (9 – 4) π = 5 π.
The area of the ring is therefore equal to 5 π cm2.
By taking π ≈ 3.14, we find the area is around 15.7 cm2.
Note: Generally, the area of a circular ring bordered by two concentric circles with respective radii of R and r is equal to πR² – πr², which is π(R² – r²).
Read more:
Calculating a Numerical Expression (1)
Calculating a Numerical Expression (2)
Determining the Common Factors of Two Integers
Recognizing a Proportional Relationship
Rounding to the Nearest Unit
Writing a Numerical Expression That Corresponds to a Sequence of Operations