Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other is decreased by 4.

Question

Homi · Accepted Answer

We need to find pairs of numbers where their product remains the same after one number is increased by 2 and the other is decreased by 4.

Step 1 — Finding the relationship between the numbers

Let the first number be $a$ . Let the second number be $b$ . The original product of these two numbers is $a \times b$ . When the first number is increased by 2, it becomes $(a+2)$ . When the second number is decreased by 4, it becomes $(b-4)$ . The new product of these changed numbers is $(a+2) \times (b-4)$ . The problem states that the product remains unchanged, meaning the original product is equal to the new product.

$a \times b = (a+2)(b-4)$

Now, we will expand the right side of the equation using the distributive property. (This means we multiply each term in the first bracket by each term in the second bracket.)

$ab = a \times b + a \times (-4) + 2 \times b + 2 \times (-4)$

$ab = ab - 4a + 2b - 8$

Next, we can subtract $ab$ from both sides of the equation to simplify it.

$ab - ab = ab - ab - 4a + 2b - 8$

$0 = -4a + 2b - 8$

To find a relationship between $a$ and $b$ , let us rearrange this equation. First, we add $4a$ to both sides.

$4a = 2b - 8$

Then, we add 8 to both sides.

$4a + 8 = 2b$

Finally, we divide both sides by 2.

$\frac{4a + 8}{2} = \frac{2b}{2}$

$2a + 4 = b$

$\boxed{b = 2a + 4}$

This equation shows us the relationship that $a$ and $b$ must satisfy for their product to remain unchanged.

Step 2 — Finding three examples

We can now choose different values for $a$ and use the relationship $b = 2a + 4$ to find the corresponding value for $b$ . Then we will check the products.

Example 1: Let us choose the first number $\mathbf{a = 1}$ . Using the relationship $b = 2a + 4$ , we find $b$ :

$b = 2(1) + 4$

$b = 2 + 4$

$b = 6$

So, the pair of numbers is **(1, 6)**.
Original product: $1 \times 6 = \mathbf{6}$.
New numbers: $(1+2) = 3$ and $(6-4) = 2$.
New product: $3 \times 2 = \mathbf{6}$.
The product remains unchanged.

Example 2: Let us choose the first number $\mathbf{a = 2}$ . Using the relationship $b = 2a + 4$ , we find $b$ :

$b = 2(2) + 4$

$b = 4 + 4$

$b = 8$

So, the pair of numbers is **(2, 8)**.
Original product: $2 \times 8 = \mathbf{16}$.
New numbers: $(2+2) = 4$ and $(8-4) = 4$.
New product: $4 \times 4 = \mathbf{16}$.
The product remains unchanged.

Example 3: Let us choose the first number $\mathbf{a = 3}$ . Using the relationship $b = 2a + 4$ , we find $b$ :

$b = 2(3) + 4$

$b = 6 + 4$

$b = 10$

So, the pair of numbers is **(3, 10)**.
Original product: $3 \times 10 = \mathbf{30}$.
New numbers: $(3+2) = 5$ and $(10-4) = 6$.
New product: $5 \times 6 = \mathbf{30}$.
The product remains unchanged.

Answer

(i) The numbers are 1 and 6. (Original product: $1 \times 6 = 6$ ; New product: $(1+2)(6-4) = 3 \times 2 = 6$ ) (ii) The numbers are 2 and 8. (Original product: $2 \times 8 = 16$ ; New product: $(2+2)(8-4) = 4 \times 4 = 16$ ) (iii) The numbers are 3 and 10. (Original product: $3 \times 10 = 30$ ; New product: $(3+2)(10-4) = 5 \times 6 = 30$ )

Question 3

Answer

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