Problem 96 - Fundamentals of Mathematics

Question

Compute each product using the distributive property. \(95 \cdot 11\)

Step-by-Step Solution

Answer

The product is 1045.

Step 1: Express One Number as a Sum

We can express 95 as 90 + 5. This will allow us to use the distributive property to simplify the product.

Step 2: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). Apply this to the problem: \((90 + 5) \cdot 11 = 90 \cdot 11 + 5 \cdot 11\).

Step 3: Compute Each Product Separately

First compute \(90 \cdot 11 = 990\). Next, compute \(5 \cdot 11 = 55\).

Step 4: Sum the Products

Add the results from the previous step: \(990 + 55 = 1045\).

Key Concepts

Multiplication

In mathematics, multiplication is one of the primary arithmetic operations that allow us to find the total number of items when they are grouped. It's essentially repeated addition, where we take one number and add it to itself multiple times depending on the other number in the calculation.

For example, if we want to multiply 3 by 4, we can see it as adding 3 four times:

• 3 + 3 + 3 + 3 = 12

Although it can be seen as simple repeated addition, multiplication offers a more efficient and faster way to calculate large numbers. In our example from the exercise, the distributive property is used to simplify the multiplication of two larger numbers, making them easier to calculate. This property allows us to break down a complex multiplication into several smaller pieces, like we did with 95 and 11.

Arithmetic Operations

Arithmetic operations include the basic mathematical processes such as addition, subtraction, multiplication, and division. These are the building blocks for more advanced mathematical concepts and operations.

When tackling arithmetic operations, each has its specific rules and properties that help in solving problems efficiently. Multiplication and addition are particularly important when dealing with the distributive property, as seen in our exercise.
Let's refresh our understanding of each operation:

• Addition - combining two numbers to get a sum.
• Subtraction - finding the difference by taking one number away from another.
• Multiplication - finding the product by combining groups of numbers.
• Division - splitting a number into equal parts.

Each of these plays a crucial role in mathematical problem-solving, and being comfortable with these fundamental operations helps tackle larger problems with ease.

Number Properties

Number properties are certain rules or laws that numbers follow, helping us simplify complex problems. These properties include commutative, associative, identity elements, inverse elements, and the distributive property.

In our example exercise, we utilize the distributive property, which states:

• For all numbers \(a, b,\) and \(c\), the equation \(a(b + c) = ab + ac\) holds true.

This property is extremely useful for breaking down expressions into simpler components, especially when dealing with multiplication and addition.
On the other hand, the commutative property tells us that the order of addition or multiplication does not affect the result:

• \(a + b = b + a\)
• \(a \cdot b = b \cdot a\)

Knowing these properties allows us to manipulate and solve mathematical expressions more easily, giving us control over the complexity of calculations. As students get more familiar with these properties, they can solve problems more accurately and efficiently.