The value of 95 × 105 is:
9975
Calculating the Value of 95 × 105
The problem asks us to find the value of the product of 95 and 105. We can perform this multiplication directly, or we can use a helpful algebraic identity to make the calculation simpler.
Method 1: Direct Multiplication
We can multiply 95 by 105 directly:
| 1 | 0 | 5 | |||
|---|---|---|---|---|---|
| × | 9 | 5 | |||
| 5 | 2 | 5 | (5 × 105) | ||
| + | 9 | 4 | 5 | 0 | (90 × 105) |
| 9 | 9 | 7 | 5 | ||
So, 95 × 105 = 9975.
Method 2: Using Algebraic Identity
We can notice that both 95 and 105 are close to 100. We can write 95 as \(100 - 5\) and 105 as \(100 + 5\). The multiplication becomes:
\(95 \times 105 = (100 - 5) \times (100 + 5)\)
This expression is in the form \((a - b)(a + b)\), which is equal to \(a^2 - b^2\). Here, \(a = 100\) and \(b = 5\). Applying the identity:
\((100 - 5)(100 + 5) = 100^2 - 5^2\)
Now, we calculate the squares:
- \(100^2 = 100 \times 100 = 10000\)
- \(5^2 = 5 \times 5 = 25\)
Substitute these values back into the identity:
\(100^2 - 5^2 = 10000 - 25\)
Performing the subtraction:
\(10000 - 25 = 9975\)
Thus, the value of 95 × 105 is 9975.
Conclusion on the Value of 95 x 105
Both methods yield the same result. The value of 95 × 105 is 9975.
Revision Table: Key Concepts
| Concept | Description | Example |
|---|---|---|
| Multiplication | The process of calculating the product of two or more numbers. | \(3 \times 4 = 12\) |
| Algebraic Identity: Difference of Squares | A formula used to factor or expand expressions of the form \(a^2 - b^2\). | \((a - b)(a + b) = a^2 - b^2\) |
Additional Information: Applying Identities
Algebraic identities are powerful tools that can simplify calculations and algebraic expressions. The difference of squares identity \((a - b)(a + b) = a^2 - b^2\) is particularly useful when multiplying numbers that are equidistant from a round number (like 10, 100, 1000, etc.).
For instance, to calculate \(48 \times 52\), we can write it as \((50 - 2) \times (50 + 2)\). Using the identity:
\((50 - 2)(50 + 2) = 50^2 - 2^2 = 2500 - 4 = 2496\)
This method often requires fewer steps than direct multiplication, especially for mental calculations.
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