Problem 36 - Beginning and Intermediate Algebra

Question

Find each quotient. \(\frac{35}{5}\)

Step-by-Step Solution

Answer

7

Step 1: Understand the Problem

The given problem is to find the quotient of the division of 35 by 5. In other words, you need to determine how many times 5 goes into 35.

Step 2: Set Up the Division

Set up the division expression as \(\frac{35}{5}\). This means dividing 35 by 5.

Step 3: Perform the Division

Divide the numerator (35) by the denominator (5). Calculate \(\frac{35}{5} = 7\).

Step 4: Write the Quotient

The quotient of 35 divided by 5 is 7.

Key Concepts

Quotient

In basic arithmetic, a quotient represents the result of dividing one number by another. When you divide a number by another number, the quotient tells you how many times the divisor fits into the dividend. For instance, when we divide 35 by 5, we are looking for how many times 5 fits into 35. The quotient in this case is 7, because 5 can fit into 35 exactly 7 times.

The concept of quotient is fundamental in arithmetic operations like division. It allows us to distribute a quantity into equal parts.

Here’s how you can visualize it:

• If you have 35 apples and you want to divide them equally into groups of 5, each group will have 7 apples.
• This makes 7 the quotient of 35 divided by 5.

Numerator

The numerator is the top number in a fraction, and it represents the number of parts we have. In the division problem \(\frac{35}{5}\), the numerator is 35. This is the number we want to divide, also known as the dividend.

In any division expression, the numerator plays a key role as it indicates the total amount to be distributed. For example:

• In \(\frac{35}{5}\), 35 is the total number of apples we want to divide equally into 5 groups.
• This numerator tells us what we are starting with before we apply any division operations.

If you consider the fraction as a part-whole relationship, the numerator is the part we are focusing on.

Denominator

The denominator is the bottom number in a fraction. It represents the number of equal parts into which the whole is divided. In \(\frac{35}{5}\), the denominator is 5. This number is also known as the divisor in a division problem.

The denominator gives us context about how we are splitting the quantity represented by the numerator. Here’s an example to visualize this concept:

• In the problem \(\frac{35}{5}\), the 5 tells us that we are dividing 35 apples into 5 equal groups.
• Each group will have an equal number of apples, which is determined by the quotient.

Understanding the role of the denominator helps in accurately dividing and achieving equal parts in division.

Basic Arithmetic

Basic arithmetic includes the fundamental operations of addition, subtraction, multiplication, and division. These operations are the building blocks for most mathematical concepts.

Division, particularly, is one of the core arithmetic operations, and it involves splitting a number into equal parts, as shown in the problem \(\frac{35}{5}\). Here’s a brief overview of how basic arithmetic principles apply:

• Addition: Combining quantities. E.g., 3 + 4 = 7.
• Subtraction: Finding the difference. E.g., 7 - 4 = 3.
• Multiplication: Repeated addition. E.g., 3 * 4 = 12.
• Division: Splitting into equal parts. E.g., 35 ÷ 5 = 7.

By mastering these basic operations, you can solve a wide range of mathematical problems and understand more advanced concepts.