Problem 10 - Fundamentals of Mathematics

Question

Perform the following subtractions. Show the expanded form for the first three problems. $$ \begin{array}{r} 76 \\ -28 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The subtraction gives 48.

Step 1: Write the Problem in Expanded Form

First, let's express the problem in an expanded form. We have the numbers 76 and 28. - 76 can be expressed as 70 + 6. - 28 can be expressed as 20 + 8. So the subtraction becomes (70 + 6) - (20 + 8).

Step 2: Subtract the Tens

Separate the numbers into their tens components: 70 and 20. Subtract 20 from 70:\[70 - 20 = 50\]

Step 3: Subtract the Units

Now, subtract the units components: 6 and 8. Since 6 is smaller than 8, we need to borrow from the tens place. Borrow 10 from the 50:\[16 - 8 = 8\]

Step 4: Combine the Results

Add the results from the previous steps to find the subtraction result:- Tens subtraction: 50 (after borrowing 10)- Units subtraction: 8Combine these results:\[50 + 8 = 58\]

Step 5: Conclusion

The result of the subtraction is the combination of the tens and the units differences, giving us a final answer of 48.

Key Concepts

Expanded Form

Understanding the expanded form is key to breaking down numbers into more digestible parts. In mathematics, expanded form refers to writing a number as the sum of its parts, usually focused on place value. For example, the number 76 can be represented as 70 + 6. This form enables us to see the separate contributions of each digit to the overall number. By expressing 76 as 70 + 6 and 28 as 20 + 8, subtraction becomes easier to handle.

• It splits bigger numbers into manageable pieces, like tens and units.
• Helps visualize the subtraction by separating tens and units.

When tackling subtraction problems, using expanded form makes complex numbers less intimidating. It serves as the foundation for performing arithmetic operations more logically, step by step.

Borrowing

Borrowing is a technique used during subtraction when a smaller number needs to be subtracted from a larger place value. This usually happens in the context of subtracting units that require borrowing from the tens place. For instance, when subtracting 6 from 8, we encounter a challenge because 6 is less than 8. In such cases, we 'borrow' from the nearest higher place value—in our example's case, the tens place—in order to perform the subtraction accurately.

• Borrowing allows us to convert a problem that appears unsolvable into an easily manageable one.
• It involves adjusting place values without changing the overall value of the number.

Effectively practicing borrowing improves overall subtraction skills and helps with understanding more advanced arithmetic concepts.

Tens and Units

Understanding the components of tens and units is crucial in arithmetic operations, especially in subtraction. Every digit in a number represents a different place value; for two-digit numbers, this includes the tens and units. In the number 76, '7' represents 70, which means 7 tens, and '6' represents the units. Similarly, in 28, '2' stands for 20, or 2 tens, and '8' is in the units place. Separating these parts:

• Helps perform arithmetic operations separately on tens and units.
• Makes borrowing more intuitive when needed.

By paying attention to tens and units, mathematical operations become simpler, providing a solid basis for understanding more complicated calculations.

Arithmetic Operations

Arithmetic operations form the core of basic mathematics, including addition, subtraction, multiplication, and division. Each operation follows specific rules to maintain logical consistency. Subtraction, a fundamental arithmetic operation, involves taking one number away from another. Successful subtraction requires:

• Understanding place value to handle numbers correctly.
• Using borrowing for accurate calculation.
• Separating processes into manageable steps using methods such as expanded form.

Mastering these operations provides students with the confidence to handle more complex mathematical tasks. Practicing arithmetic operations sharpens problem-solving skills and enhances one's ability to perform everyday mathematical tasks efficiently.