Problem 61 - Algebra 1: Concepts and Skills

Question

Use the distributive property and mental math to simplify the expression. $$ -3(4.10) $$

Step-by-Step Solution

Answer

\(-12.30\)

Step 1: Identify the numbers to be multiplied

Here, the bracket has the purpose to indicate multiplication. Therefore we want to find the product of -3 and 4.10.

Step 2: Apply the distributive property

The distributive property states that a number multiplied with a sum or difference of two numbers is equal to the number multiplied with each number of the sum or difference and then summed or subtracted. In this case, -3 will be multiplied with 4.10.

Step 3: Multiply using mental math

Finally, we multiply -3 by 4.10. Counting up by 3's four times gives the result of -12.30.

Key Concepts

Multiplication

Multiplication is one of the four basic arithmetic operations and involves adding a number (called the multiplicand) to itself a certain number of times (the multiplier). In the context of our exercise, we are looking at multiplying -3 by 4.10. Here,

• -3 is the multiplicand,
• 4.10 is the multiplier.

It helps to think of multiplication as a quick way to perform repeated addition. You can visualize - multiplying -3 by 4, as adding -3 together four times: -3 + (-3) + (-3) + (-3) = -12. To complete the multiplication with 0.10, realize you are multiplying -3 by a portion off the whole, which simplifies to an additional -0.30 when you add -0.30 (since -3 * 0.10 = -0.30). This gives us the total of -12 + (-0.30) = -12.30. Understanding multiplication does not have to be intimidating, it's a scalable addition!

Mental Math

Mental math involves carrying out calculations in our minds, without the help of paper, a calculator, or other aids. This skill can be immensely useful for quick everyday calculations or for simplifying expressions when using the distributive property. For the given expression, -3 times 4.10, we break down 4.10 as 4 and 0.10. Hence,

• Multiply -3 by 4: This is typically quicker as humans are naturally adept at basic multiplication and addition.
• Next, multiply -3 by 0.10: This uses the concept of parts or fractions in multiplication. Just like with percentages, here 0.10 is 10% of -3.

Through mental math, the multiplication is -3 * 4 = -12 and -3 * 0.10 = -0.30 These results are then summed to get -12 + (-0.30) = -12.30. Practicing mental math enhances speed and confidence in solving math problems.

Simplifying Expressions

Simplifying expressions is an important step in processing mathematical problems and can often help in finding the most efficient way to solve them. The primary goal in simplification is to make the math as easy to handle as possible. When simplifying expressions like -3(4.10), we apply the distributive property to break it down into simpler components:

• Recognize that 4.10 can be split into whole numbers and fractions, namely 4 and 0.10.
• Multiply each part by -3 separately.

This turns the expression into smaller, easier calculations: -3 multiplied by (4 + 0.10) is -3 * 4 + (-3 * 0.10). By computing these separately and then combining them, the process becomes straightforward. Such an approach reduces errors and saves time, especially in more complex expressions. Simplifying expressions paves the way to better problem-solving and is a valuable technique in mathematics.