Question 9

The distributive property works for different types of numbers.

Step 1 — Understanding the Patterns

We will consider two common algebraic patterns. These are known as the distributive properties.

Pattern 1 is the Distributive Property of Multiplication over Addition: $a \times (b + c) = a \times b + a \times c$

Pattern 2 is the Distributive Property of Multiplication over Subtraction: $a \times (b - c) = a \times b - a \times c$

We will test if these patterns hold for different types of numbers.

Step 2 — Checking Counting Numbers

Counting numbers are positive whole numbers like $1, 2, 3, \ldots$. Let us choose $a=2, b=3, c=4$ for our example.

First, let's check Pattern 1: $a \times (b + c) = a \times b + a \times c$. Left Hand Side (LHS): $a \times (b + c) = 2 \times (3 + 4)$ $= 2 \times 7$

$\boxed{14}$

Right Hand Side (RHS): $a \times b + a \times c = 2 \times 3 + 2 \times 4$ $= 6 + 8$

$\boxed{14}$ Since LHS = RHS, Pattern 1 holds for counting numbers.

Next, let's check Pattern 2: $a \times (b - c) = a \times b - a \times c$. Let us choose $a=5, b=6, c=2$. Left Hand Side (LHS): $a \times (b - c) = 5 \times (6 - 2)$ $= 5 \times 4$

$\boxed{20}$

Right Hand Side (RHS): $a \times b - a \times c = 5 \times 6 - 5 \times 2$ $= 30 - 10$

$\boxed{20}$ Since LHS = RHS, Pattern 2 also holds for counting numbers.

Step 3 — Checking Negative Integers

Negative integers are whole numbers less than zero, like $-1, -2, -3, \ldots$. Let us choose $a=-2, b=5, c=-3$ for our example.

First, let's check Pattern 1: $a \times (b + c) = a \times b + a \times c$. Left Hand Side (LHS): $a \times (b + c) = -2 \times (5 + (-3))$ $= -2 \times (5 - 3)$ $= -2 \times 2$

$\boxed{-4}$

Right Hand Side (RHS): $a \times b + a \times c = (-2) \times 5 + (-2) \times (-3)$ $= -10 + 6$

$\boxed{-4}$ Since LHS = RHS, Pattern 1 holds for negative integers.

Next, let's check Pattern 2: $a \times (b - c) = a \times b - a \times c$. Let us choose $a=-5, b=-6, c=2$. Left Hand Side (LHS): $a \times (b - c) = -5 \times (-6 - 2)$ $= -5 \times (-8)$

$\boxed{40}$

Right Hand Side (RHS): $a \times b - a \times c = (-5) \times (-6) - (-5) \times 2$ $= 30 - (-10)$ $= 30 + 10$

$\boxed{40}$ Since LHS = RHS, Pattern 2 also holds for negative integers.

Step 4 — Checking Fractions

Fractions are numbers like $1/2, 3/4, -5/7$, etc. Let us choose $a=1/2, b=2/3, c=1/4$ for our example.

First, let's check Pattern 1: $a \times (b + c) = a \times b + a \times c$. Left Hand Side (LHS): $a \times (b + c) = \frac{1}{2} \times \left(\frac{2}{3} + \frac{1}{4}\right)$ $= \frac{1}{2} \times \left(\frac{8}{12} + \frac{3}{12}\right)$ $= \frac{1}{2} \times \frac{11}{12}$

$\boxed{\frac{11}{24}}$

Right Hand Side (RHS): $a \times b + a \times c = \frac{1}{2} \times \frac{2}{3} + \frac{1}{2} \times \frac{1}{4}$ $= \frac{2}{6} + \frac{1}{8}$ $= \frac{1}{3} + \frac{1}{8}$ $= \frac{8}{24} + \frac{3}{24}$

$\boxed{\frac{11}{24}}$ Since LHS = RHS, Pattern 1 holds for fractions.

Next, let's check Pattern 2: $a \times (b - c) = a \times b - a \times c$. Let us choose $a=2/3, b=1/2, c=1/4$. Left Hand Side (LHS): $a \times (b - c) = \frac{2}{3} \times \left(\frac{1}{2} - \frac{1}{4}\right)$ $= \frac{2}{3} \times \left(\frac{2}{4} - \frac{1}{4}\right)$ $= \frac{2}{3} \times \frac{1}{4}$

$\boxed{\frac{2}{12} = \frac{1}{6}}$

Right Hand Side (RHS): $a \times b - a \times c = \frac{2}{3} \times \frac{1}{2} - \frac{2}{3} \times \frac{1}{4}$ $= \frac{2}{6} - \frac{2}{12}$ $= \frac{1}{3} - \frac{1}{6}$ $= \frac{2}{6} - \frac{1}{6}$

$\boxed{\frac{1}{6}}$ Since LHS = RHS, Pattern 2 also holds for fractions.

Step 5 — Justification

The distributive properties are fundamental rules of arithmetic. They apply to all real numbers. Counting numbers, negative integers, and fractions are all types of real numbers. So, these patterns hold true for all of them.

Answer

(i) No, Patterns 1 and 2 do not hold only for counting numbers. (ii) Yes, Patterns 1 and 2 hold for negative integers as well. (iii) Yes, Patterns 1 and 2 hold for fractions.