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Chapter 1: Problem 788

Simplify using the distributive property. $$ 10\left(\frac{3}{10} x-\frac{2}{5}\right) $$

Short Answer

Expert verified
3x - 4

Step by step solution

01

- Identify the distributive property

The distributive property states that for any numbers a, b, and c: \[ a(b + c) = ab + ac \] In this exercise, apply the distributive property to the expression.
02

- Apply the distributive property

Distribute the 10 to both terms inside the parentheses: \[ 10 \times \frac{3}{10}x + 10 \times \frac{2}{5} \]
03

- Simplify each term separately

Calculate each term: \ \[ 10 \times \frac{3}{10} x = 3x \] \[ 10 \times \frac{2}{5} = 4 \]
04

- Combine the simplified terms

Combine the simplified terms from step 3 to get the final simplified expression: \[ 3x - 4 \]

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Key Concepts

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

Simplifying Expressions
When you simplify expressions, you make them shorter or easier to work with. This often means reducing fractions, combining like terms, or performing basic arithmetic.

Here, we have the expression: \( \ 10\bigg(\frac{3}{10} x-\frac{2}{5}\bigg) \).
Notice all the steps to simplify, like distributing and calculating each term separately.
The goal is to turn a more complex expression into a simpler one while keeping the same value. The result in this exercise is shorter and easier to see what's happening with the variable. Simplifying expressions is a core skill in algebra and is used frequently in solving equations and real-life problems.
Algebraic Properties
Algebraic properties help us manipulate and solve equations or expressions. One of the most important properties is the distributive property.

The distributive property states: \(a(b + c) = ab + ac\).
It lets you distribute or multiply a single term across terms inside parentheses.
Here's why it's useful:
  • It helps in simplifying expressions, just like in this exercise.
  • It assists in solving equations where we need to isolate variables.
  • It breaks down more complex problems into simpler parts.


In the given problem, using the distributive property: \(10 \times\big( \frac{3}{10} x-\frac{2}{5}\big)= 10 \times \frac{3}{10} x + 10 \times \frac{2}{5}\).
It's all about applying properties to solve algebra problems in a methodical way.
Fraction Multiplication
Understanding fraction multiplication is crucial in algebra and arithmetic. Let's take a closer look:

To multiply fractions, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example: \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \).

In this exercise, we see fraction multiplication when distributing:

\( 10 \times \frac{3}{10} x \).
Here, it's helpful to simplify before multiplying:
  • \(10 = \frac{10}{1}\), making it \(\frac{10}{1} \times \frac{3}{10}\).
  • This simplifies to: \(\frac{10 \times 3}{1 \times 10} x \) = 3x.

Also, \( 10 \times \frac{2}{5}\):
  • \(10 = \frac{10}{1}\), making it \(\frac{10}{1} \times \frac{2}{5}\).
  • This simplifies to: \(\frac{20}{5}\) = 4.


Breaking down fraction multiplication step-by-step helps in understanding how to tackle similar problems.

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Most popular questions from this chapter

Simplify using the distributive property. $$ 14(c-1)-8(c-6) $$

Solve. Fawzi's SUV is 6 feet 4 inches tall. If he puts a 2 feet 10 inch box on top of his SUV, what is the total height of the SUV and the box?

Reflectors The reflectors in each lane-marking stripe on a highway are spaced 16 yards apart. How many reflectors are needed for a one mile long lanemarking stripe?

Adele's shampoo sells for $$\$ 3.99\( per bottle at the grocery store. At the warehouse store, the same shampoo is sold as a 3 pack for \)\$ 10.49 .\( To find the cost of 3 bottles at \)\$ 3.99,\( notice that 3.99 is \)4-0.01\( (a) Multiply 3(3.99) by using the distributive property to multiply \)3(4-0.01)$$. (b) How much would Adele save by buying 3 bottles at the warehouse store instead of at the grocery store?

Convert the units. A football field is 160 feet wide. Convert the width to yards.
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