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Chapter 2: Problem 47

Simplify each of the numerical expressions. $$0.6(4.1)+0.7(3.2)$$

Short Answer

Expert verified
The simplified expression is 4.7.

Step by step solution

01

Distribute the first term

We start by distributing the first multiplication in the expression. Multiply 0.6 by 4.1: \(0.6 \times 4.1 = 2.46\).
02

Distribute the second term

Next, distribute the second multiplication in the expression. Multiply 0.7 by 3.2: \(0.7 \times 3.2 = 2.24\).
03

Add the results

Finally, add the results of the individual multiplications together: \(2.46 + 2.24 = 4.7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distribution in Algebra
Distribution, in algebra, is a fundamental concept that allows us to simplify expressions involving multiplication. When we see an expression like 鈥媆(a(b + c)\)鈥�, we use the distributive property to expand it to 鈥媆(ab + ac\)鈥�. Here, each term inside the parenthesis is multiplied separately by the value outside.

  • In this exercise, the expression is 鈥媆(0.6(4.1) + 0.7(3.2)\)鈥�. This requires distribution in two parts.
  • First, multiply 鈥媆(0.6\)鈥� with 鈥媆(4.1\)鈥�, and second, 鈥媆(0.7\)鈥� with 鈥媆(3.2\)鈥�.
Breaking down the problem into these smaller parts, and handling each part separately, helps avoid mistakes and ensures the computation remains simple and manageable.
By distributing terms, you're actively engaging with the expression to break it down into more manageable steps.
Step-by-Step Problem Solving
Approaching problems step-by-step is crucial for clear understanding and accuracy.

  • First, identify parts of the expression that need attention鈥攊n our exercise, these are the multiplications.
  • Second, handle each component one at a time. Calculate 鈥媆(0.6 \times 4.1\)鈥� in Step 1, and then 鈥媆(0.7 \times 3.2\)鈥� in Step 2.
  • Finally, in Step 3, add the outcomes of these individual calculations together to reach the final simplified expression.
This method also enhances problem-solving skills, ensuring each calculation is done correctly and facilitating thorough verification of your work. Practicing this strategy in various scenarios enables students to be confident when faced with similar mathematical challenges.
Basic Arithmetic Operations
Basic arithmetic operations, such as multiplication and addition, are the building blocks of many mathematical tasks.

  • Multiplication involves combining equal groups, and is here used in distributing terms in both parts of the expression.
  • Addition is used at the end to combine the simplified results from the multiplication.
In our example, first perform the multiplications: 鈥媆(0.6 \times 4.1 = 2.46\)鈥� and 鈥媆(0.7 \times 3.2 = 2.24\)鈥�. Then, add these two results: 鈥媆(2.46 + 2.24 = 4.7\)鈥�.

Understanding these basic operations thoroughly ensures one can handle complexity as mathematical expressions become more intricate. Mastery of these operations supports not only algebra but math in general, providing a strong foundation for continued learning.

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Most popular questions from this chapter

When the market opened on Monday morning, Garth bought some shares of a stock at \(\$ 13.25\) per share. The daily changes in the market for that stock for the week were \(0.75,-1.50,2.25,-0.25\), and \(-0.50\). What was the value of one share of that stock when the market closed on Friday afternoon?

Andrea's automobile averages \(25.4\) miles per gallon. With this average rate of fuel consumption, what distance should she be able to travel on a 12.7-gallon tank of gasoline?

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