Problem 58 - Prealgebra

Question

Add or subtract as indicated. $$3.4-1.975$$

Step-by-Step Solution

Answer

The result of 3.4 - 1.975 is 1.425.

Step 1: Understand the Problem

We need to subtract 1.975 from 3.4. This requires aligning the decimal points to correctly perform the subtraction.

Step 2: Align the Decimals

Write the numbers one below the other, ensuring that the decimal points are directly aligned:\[\begin{array}{r} 3.400 \- 1.975 \\hlined.\end{array}\]

Step 3: Subtract the Thousandths Place

Subtract the digits in the thousandths place. Since we need to subtract 5 from 0, borrow 1 from the hundredths place:\[\begin{array}{r} 3.400 \- 1.975 \\hlined.025\end{array}\]

Step 4: Subtract the Hundredths Place

With the borrowing from Step 3, we now subtract 7 from 9 (the number we borrowed from makes this 10):\[\begin{array}{r} 3.390 \- 1.975 \\hlined25\end{array}\]

Step 5: Subtract the Tenths Place

Subtract the tenths place digits, now 9 (borrowed from 3) minus 9:\[\begin{array}{r} 3.390 \- 1.975 \\hline5\end{array}\]

Step 6: Subtract the Units Place

Finally, subtract the units: 2 from 3, considering the borrowing:\[\begin{array}{r} 3 \- 1 \\hline 1\end{array}\]

Step 7: Compile the Final Answer

Compile all the differences for each column to form the final answer:\[3.4 - 1.975 = 1.425\]

Key Concepts

Aligning Decimals

When dealing with decimal subtraction,

• the first and the most important step is ensuring that the decimal points are perfectly aligned.
• This means placing one number directly under the other, so their decimal points line up vertically.

Aligning decimals is crucial because it ensures you’re subtracting from the correct place value in each column. Misalignment can result in incorrect calculations.

For example, if you write 3.4 as 3.400, it becomes clear that you have three places in the tenths, hundredths, and thousandths to work with. When you line up the numbers vertically, this visual alignment helps you see which digits to subtract in the correct order, starting from the digits farthest to the right.

Borrowing in Subtraction

Borrowing in subtraction, often referred to as 'regrouping', is a technique used when the digit being subtracted is larger than the digit above it.

• In such cases, you ‘borrow’ 1 from the next column to the left.
• This increases the value of the current column, making it possible to perform the subtraction.

For instance, when subtracting in the thousandths place from 3.400 - 1.975, you cannot subtract 5 from 0 directly.
Instead, you borrow 1 from the hundredths, turning the 0 in the thousandths place into a 10.

This borrowing process continues as needed through each subsequent place value, ensuring all digits can be properly subtracted.

Place Value

Place value is the key to understanding how numbers are structured in our decimal system. Each digit in a number has its own value depending on its position.

• The rightmost position, after the decimal, is the tenths.
• The next position is the hundredths, and the third position is the thousandths.

During subtraction, it’s essential to understand the value of the digits you are working with, as misinterpreting this can lead to errors.

When subtracting 3.4 from 1.975, consider that 3 is in the ones place, the 4 is in the tenths, whereas the 0 in 3.400 is in both hundredths and thousandths places, representing a part that gives the precise value in those places.

Using the concept of place value helps ensure you are dealing with numbers accurately, especially when borrowing is needed in the subtraction process.