Problem 24 - Algebra 1

Question

Write the verbal sentence as an equation or an inequality. Twenty-five is the quotient of a number \(y\) and 3.5

Step-by-Step Solution

Answer

The equation is \(y/3.5 = 25\).

Step 1: Identify the parts of the statement

In the statement 'Twenty-five is the quotient of a number \(y\) and 3.5', you can identify 25 as the quotient, \(y\) as the unknown number we are looking for and 3.5 as the divisor.

Step 2: Translate to mathematical operation

The operation being made in this sentence is division. The sentence can be translated to the operation as following: \(y\) divided by 3.5 equals to 25.

Step 3: Write the equation

Finally, write the operation as an equation. \(y\) divided by 3.5 equals to 25 turns to \(y/3.5 = 25\).

Key Concepts

Equations

An equation is a mathematical statement indicating that two expressions are equal. In simpler terms, think of it as a balance scale, where what is on the left side needs to equal what is on the right side. When creating or solving equations, we aim to find out what value of a variable (unknown number) will make the equation true.

Equations usually include variables such as \( y \), numbers, and mathematical operations. In the context of our exercise, we derived the equation \( y/3.5 = 25 \) from the verbal sentence. It tries to compare the division operation of \( y \) by 3.5 to the number 25.

Equations are vital in algebra for expressing relationships between numbers and for solving problems by finding unknown values:

Translation of Verbal Statements

Translating verbal statements into mathematical expressions is an important skill. It makes it easier to work with complex verbal problems by converting them into a more understandable and solvable format.

Verbal statements often contain key phrases that hint at specific operations. For example:

• "Quotient of" suggests division.
• "Sum of" would indicate addition.
• "Difference between" points to subtraction.

In our exercise, the verbal statement "Twenty-five is the quotient of a number \( y \) and 3.5" was translated into the equation \( y/3.5 = 25 \).

Here are simple steps to translate such statements:

• Identify the parts of the sentence: numbers and operations.
• Assign a variable for any unknown number.
• Turn the sentence into a mathematical equation.

Division Operations

Division is one of the basic arithmetic operations where you split a number into equal parts. When you divide a number, you want to find out how many times a number (divisor) is contained in another number (dividend). The result is called the quotient.

In mathematical language, if you have two numbers, say \( a \) (dividend) and \( b \) (divisor), division is represented as \( a/b \) or \( a \div b \).

In the exercise, we are dealing with the division of \( y \) by 3.5, and the result is 25. Here, 3.5 is the divisor, and 25 is the quotient. This means that \( y \), when divided by 3.5, gives you 25 as a result.