Problem 54 - Fundamentals of Mathematics

Question

For the following problems, perform the additions and round to the nearest hundred. $$ \begin{array}{r} 1,468 \\ 2,183 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum rounded to the nearest hundred is 3,700.

Step 1: Add the Numbers

Begin by aligning the numbers vertically by their rightmost digits. Then add from the rightmost column to the leftmost column. First add the units place: 8 + 3 = 11. Write 1 and carry over 1 to the tens place. Next add the tens place: 6 + 8 + 1 (carry) = 15. Write 5 and carry over 1 to the hundreds place. Now add the hundreds place: 4 + 1 + 1 (carry) = 6. Write 6. Finally add the thousands place: 1 + 2 = 3. The result of the addition is 3,651.

Step 2: Round to the Nearest Hundred

Examine the tens digit of the number 3,651, which is 5. Since the digit in the tens place is 5 or more, round the hundreds place up by 1. Thus, increase the hundreds place from 6 to 7, making the rounded number 3,700.

Key Concepts

Addition

Addition is a fundamental math operation that involves combining two or more numbers to get a total sum. It is one of the essential skills you will use not just in math, but in everyday life. When adding numbers, it's important to align them based on their place value to ensure accuracy. For instance, when you are adding numbers such as 1,468 and 2,183, you should start from the rightmost digits and move left.

• Begin with the units place: Add 8 and 3 to get 11. Write down 1 and carry over the 1 to the next column.
• Move to the tens place: Add 6, 8, and the carried 1, which equals 15. Again, write down 5 and carry over the 1.
• Proceed to the hundreds place: Add 4, 1, and the carried 1, ending with 6.
• Finally, add the thousands place: 1 plus 2 equals 3.

By aligning and adding numbers systematically, you achieve the correct sum, which in this example, is 3,651.

Place value

Understanding place value is crucial in mathematics as it helps in performing various operations like addition or rounding numbers. Place value refers to the numerical value that a digit has based on its position in a number. Each position represents a power of ten. For example, in the number 1,468:

• The 8 is in the units place (10^0).
• The 6 is in the tens place (10^1).
• The 4 is in the hundreds place (10^2).
• The 1 is in the thousands place (10^3).

Being aware of where each digit sits is key when you're adding numbers together, as we just saw. It allows you to carry numbers correctly and avoid simple mistakes. Moreover, understanding place value helps when rounding numbers, as it dictates which digits will change and which will remain the same.

Estimating

Estimating is a useful skill, especially when you need to quickly assess the size of a number without calculating the exact amount. Rounding is a common method of estimation that involves altering a number to make calculations easier but still relatively precise. To round a number to the nearest hundred, like rounding 3,651:

• Look at the tens digit, which is 5 in this case.
• If the tens digit is 5 or more, you increase the hundreds digit by 1.
• In 3,651, the hundreds digit is originally 6. You then increase it by 1, getting 7.
• Thus, 3,651 rounds to 3,700.

This technique of rounding helps to simplify calculations, particularly in mental math or when you need an approximate answer quickly. Estimating is especially helpful in making sense of very large numbers or more complex calculations by reducing them to more manageable figures.