Problem 59 - Algebra 1

Question

Write the verbal sentence as an equation. A number multiplied by \(\frac{2}{3}\) is 8

Step-by-Step Solution

Answer

Therefore, the mathematical equation that represents the verbal sentence is \( \frac{2}{3} * x = 8 \).

Step 1: Identify the Unknown

The verbal sentence indicates an unknown number which will be translated as a variable, usually \( x \).

Step 2: Translate Multiplication

The phrase 'multiplied by' suggests a multiplication operation. We can represent this in our equation as \( \frac{2}{3} * x \).

Step 3: Translate the Equality

The word 'is' in a verbal sentence usually represents the equal sign in an equation. We can now extend our equation to \( \frac{2}{3} * x = 8 \).

Key Concepts

Algebraic Expressions

When it comes to turning words into numbers and symbols, algebraic expressions are the tools we need. They are combinations of numbers, variables (such as x), and operation symbols that represent a specific quantity without an equality sign. For instance, if you hear the term 'a number multiplied by three', in algebra, you would write this as 3x, where x is the unknown number.

Understanding how to form algebraic expressions is fundamental to solving problems in algebra. Translating phrases into expressions enables us to perform calculations and solve problems. For the expression in this exercise, \( \frac{2}{3} \times x \), we're dealing with a fraction multiplied by an unknown variable, which is a habitual operation in algebra. It's crucial to recognize keywords and phrases such as 'multiplied by' as they direct which algebraic operations to use.

Equations and Inequalities

Equations are statements that assert the equality of two expressions. They are composed of two algebraic expressions separated by an equals (=) sign, for example, \( x + 2 = 5 \). An equation says that both sides are identical in value. Inequalities, on the other hand, suggest that the expressions are not necessarily equal but can be less than (<), greater than (>), less than or equal to (\leq), or greater than or equal to (\geq).

Translating verbal sentences into equations or inequalities requires an understanding of certain words and phrases. In our exercise, the word 'is' signals that we should use the equals sign, showing us that our algebraic expression equals a certain number, which in this case is 8. Thus, the complete equation is \( \frac{2}{3} \times x = 8 \).

Multiplication of Fractions

Fraction multiplication is a fundamental concept in algebra where two fractions are multiplied together. The process involves multiplying the numerators (top numbers) together and the denominators (bottom numbers) together. For example, to multiply \( \frac{1}{4} \) by \( \frac{3}{5} \), we multiply the numerators 1 and 3, and the denominators 4 and 5, giving us \( \frac{1 \times 3}{4 \times 5} = \frac{3}{20} \).

When multiplying a fraction by a whole number or a variable, the same concept applies but with a slight tweak: the whole number or variable is considered to have a denominator of 1, making the calculation similar to the previous example. In the exercise, we multiply the fraction \( \frac{2}{3} \) by the variable x (which can be thought of as \( \frac{x}{1} \)), resulting in the expression \( \frac{2}{3} \times x \) or \( \frac{2x}{3} \), which is a form you'll often see in algebraic equations.