Problem 36 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$\frac{3}{4}(1.8+7.6)$$

Step-by-Step Solution

Answer

The simplified decimal value is 7.05.

Step 1: Simplify Inside the Parentheses

First, calculate the sum inside the parentheses: \(1.8 + 7.6\). Add these two numbers together:\[ 1.8 + 7.6 = 9.4 \]

Step 2: Multiply by the Fraction

Next, take the result from the previous step, which is 9.4, and multiply it by \( \frac{3}{4} \). Perform the multiplication:\[ \frac{3}{4} \times 9.4 = 3 \times \frac{9.4}{4}\] Calculate \(\frac{9.4}{4}\):\[ \frac{9.4}{4} = 2.35 \]Then multiply by 3:\[ 3 \times 2.35 = 7.05 \]

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. To do this, you divide both the top number (numerator) and the bottom number (denominator) by their greatest common divisor (GCD). In some math problems, like the exercise above, you might need to deal with fractions and decimals together. Here's a short guide to simplify fractions effectively:

• Find the GCD: Determine the greatest common factor for the numerator and the denominator. This is the largest number that divides them both without leaving a remainder.
• Divide: Once the GCD is found, divide the numerator and the denominator by this number to get the simplified fraction.
• Rewriting Decimals as Fractions: Sometimes, you'll encounter decimals. Convert these to fractions (if needed) to make operations easier. For example, 0.75 can be rewritten as \( \frac{3}{4} \).

Simplifying fractions helps to understand and solve problems with a clearer view. It aligns numbers in a form that is easier to work with, particularly in arithmetic operations.

Multiplication of Decimals

When multiplying decimals, understanding the placement of decimal points is crucial. This process is slightly different from multiplying whole numbers, primarily due to the need to correctly place the decimal point in the final result. Follow these simple steps:

• Ignore the Decimals Initially: Temporarily treat the decimal numbers as whole numbers, ignoring the decimal points. Multiply them as you normally would with integers.
• Count Total Decimal Places: Count the number of decimal places in each of the original numbers. This is the amount of decimal places you will "create" in the final answer.
• Position the Decimal: After multiplying, place the decimal point in your result. Start from the right, and move left the total number of decimal places counted before.

For instance, in the exercise above when multiplying \( \frac{3}{4} \) by 9.4, calculate \( 3 \times 2.35 = 7.05 \). This means placing the decimal correctly is just as vital as the multiplication itself.

Order of Operations

The order of operations is a fundamental concept in mathematics, guiding the sequence in which different operations should be performed to correctly solve expressions. Just like following a recipe, missing a step or doing things out of order can lead to errors in calculations. The acronym PEMDAS helps in remembering the order:

• Parentheses: Always start by solving anything inside parentheses or brackets. This ensures that parts of an expression intended to be grouped together are solved first.
• Exponents: After parentheses, calculate exponents or powers if they are present in your problem.
• Multiplication and Division: These are on the same level and should be done from left to right, whichever comes first.
• Addition and Subtraction: Lastly, handle addition and subtraction from left to right, just like multiplication and division.

Applying the order of operations correctly is crucial to obtaining a correct result. In the exercise given, first solve inside the parentheses (1.8 + 7.6), then multiply the result by the fraction. Approaching problems in this order avoids confusion and ensures precision in calculation.