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Chapter 1: Problem 779

Simplify. $$ 15 \cdot \frac{3}{5}(4 d+10) $$

Short Answer

Expert verified
The simplified form is \( 36d + 90 \).

Step by step solution

01

- Distribute the Fraction

First, distribute the fraction \(\frac{3}{5}\) to both terms inside the parentheses: \( 4d \) and \( 10 \). This results in \[ 15 \times \frac{3}{5}(4d) + 15 \times \frac{3}{5}(10) \].
02

- Simplify Each Term

Simplify each term separately. For \( 15 \times \frac{3}{5}(4d) \), multiply the numbers first: \(\frac{3}{5} \times 4d = \frac{12d}{5} \). Now for \( 15 \times \frac{3}{5}(10) \), multiply: \(\frac{3}{5} \times 10 = 6 \). So, we get \(\frac{12d}{5} \times 15 + 6 \times 15 \).
03

- Distribute 15

Now, distribute the 15 to both simplified terms: \(\frac{12d}{5} \times 15 + 6 \times 15 \). Calculate each distribution separately: \( \frac{12d}{5} \times 15 = 36d \), and \( 6 \times 15 = 90 \). Thus, \ (36d + 90) \ is obtained.

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Key Concepts

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

Distributive property
The distributive property is a crucial concept in algebra that helps us simplify expressions. It allows us to multiply a single term by each term within parentheses and then add the results together. For example, if you have an expression like \(a(b + c)\), you apply the distributive property to get \(ab + ac\).

This applies to our original exercise, where we need to distribute the fraction \(\frac{3}{5}\) to both terms inside the parentheses \(4d + 10\). Hence, we multiply \(\frac{3}{5}\) by \(4d\) and \(\frac{3}{5}\) by \(10\). This step-by-step breakdown ensures we cover all parts of the expression evenly.
Fraction multiplication
Fraction multiplication is another essential algebra concept. When multiplying fractions, you multiply the numerators together and the denominators together. For example, multiplying \(\frac{a}{b} \) by \(c\) results in \(\frac{ac}{b}\).

To illustrate this with our exercise, let's break it down:
1. We need to multiply \(\frac{3}{5} \) by \(4d\) and \(10\).
2. Initially, multiply \(\frac{3}{5} \) by each term separately:
\(\frac{3}{5} \times 4d = \frac{12d}{5}\)
\(\frac{3}{5} \times 10 = 6\)
3. Now, we multiply these results by \(15\). For instance, \(\frac{12d}{5} \times 15 = 36d\) and \(6 \times 15 = 90\). This step ensures that the math is accurate and organized.
Simplifying algebraic expressions
Simplifying algebraic expressions is about making an expression easier to understand and work with. It involves combining like terms and performing any necessary calculations.

Returning to our exercise, we start by distributing the fraction \(\frac{3}{5}\). After handling fraction multiplication, we get \(36d + 90\). All the multiplications have been done, and the expression is at its simplest form. Finally, we verify each step to ensure the solution is correct and no terms are left to simplify.

This approach ensures that every term is accounted for and the expression is as simplified as possible.

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