Problem 24 - Fundamentals of Mathematics

Question

Write a two-digit number that has an eight in the tens position.

Step-by-Step Solution

Answer

One such number is 85.

Step 1: Understand the Exercise

The exercise asks you to find a two-digit number that has 8 as the value in the tens position. This means the number takes the form \( 8x \) where \( x \) is a digit between 0 and 9.

Step 2: Identify Possible Values

Since the number is a two-digit number of the form \( 8x \), the variable \( x \) can be any digit from 0 to 9. This is because the unit position can contain any single digit.

Step 3: Choose a Number

Choose any value for \( x \) from the identified range. For example, let \( x = 5 \). This gives us the number 85.

Key Concepts

Understanding the Tens Position

In a two-digit number, the tens position plays a crucial role in determining its value. It indicates how many tens are present in the number. For example, in the number 85, the digit '8' is in the tens position. This means there are 8 tens, or 80, as part of the number. The placement of a digit in this position significantly impacts the overall numerical value. When we say a number has '8 in the tens position', we're referring to it consistently having 80 as a component, regardless of the other digits. Understanding this concept requires observing where a digit is placed and how it contributes to the number's overall value.

Decoding the Value of Digits

Each digit within a number has its own value, which is influenced by its position. In a two-digit number like 85, each digit represents a different quantity.

• The tens digit represents multiples of ten. So, '8' in the tens position equals 80.
• The unit or ones digit represents units or ones. Thus, '5' in the units position equals 5.

Understanding this breakdown helps in comprehending how numbers increase and decrease in value based on each digit's contribution. A change in the tens digit alters the number’s base value, whereas a change in the units digit adjusts it slightly.

Identifying Values in Numbers

Identifying values within a number involves recognizing each digit's contribution based on its position. In two-digit numbers, it is essential to observe the tens and units places distinctly. To identify a number's value correctly, examine the digit in both the tens and units position. For instance, in a number like 85:

• Identify '8' in the tens place, equivalent to 80.
• Identify '5' in the units place, which adds another 5 to the total.

Such analysis helps grasp how each digit forms the complete value of a number, enabling quick recognition and comparison of different numbers.

Exploring the Digit Range

The digit range in a number refers to the scope of possible values a digit can take, particularly in specific positions. For a two-digit number, this range is crucial when selecting digits for both the tens and units position. For the tens position, when you know it has to be '8', the range is fixed. However, the unit position allows for flexibility, as it can include any digit from 0 to 9. Choosing from this range forms various numbers, such as:

• 80, with '0' in the units place.
• 89, with '9' in the units place, and so on.

This flexibility allows the formation of diverse numbers even within a seemingly fixed constraint, demonstrating the innovative use of digit ranges in number formation.