Problem 57 - Prealgebra

Question

Add or subtract as indicated. $$3.18-2.79$$

Step-by-Step Solution

Answer

The result is 0.39.

Step 1: Align the numbers

Write the numbers vertically aligning the decimal points:\[\begin{array}{c} 3.18 \- 2.79 \\hlinedots \\end{array}\]

Step 2: Subtract the hundredths

Subtract the rightmost digits (8 and 9) in the hundredth place. Since 8 is smaller than 9, we need to borrow from the left:\[18 - 9 = 9\]

Step 3: Borrow for the tenths place

Since we borrowed 1 from the tenths, the 1 in 3.18 becomes 0 and we need to adjust the subtraction in the tenths place:\[10 - 7 = 3\]

Step 4: Subtract the units

Subtract the whole numbers, 2 from 3 (accounting for the borrowing done earlier makes the 3 become 2):\[2 - 2 = 0\]

Step 5: Compile the result

Combine all parts of the subtraction to get the result:\[3.18 - 2.79 = 0.39\]

Key Concepts

Decimal Subtraction

Subtracting decimals is very similar to subtracting whole numbers, but with one important detail: the decimal points must be aligned. This is crucial because each digit must be in the correct place value column to ensure an accurate result. After aligning the decimal points, we simply subtract each column starting from the rightmost column, which is usually the hundredths or tenths, depending on the numbers you're working with.

• Write down the numbers one on top of the other, making sure to keep the decimal points aligned.
• Start from the rightmost side and work your way to the left.

By aligning decimals, you are essentially ensuring that you are subtracting the correct place values (such as tenths from tenths, hundredths from hundredths), which will help avoid confusion and errors.

Borrowing in Subtraction

Sometimes during subtraction, you might encounter a situation where the top digit is smaller than the digit below. This is common when subtracting decimals, especially in places like the hundredths or tenths. This is where 'borrowing' becomes useful.

• If the top digit in the column is smaller than the bottom digit, look to the next column to the left.
• Borrow '1' from this column, but remember this '1' is actually '10' of the next lower value column (like 10 tenths if you are borrowing from the units column).
• Adjust your numbers accordingly and continue with the subtraction.

Using borrowing correctly allows us to perform subtraction even when our top numbers are smaller. This is a useful mental math tool that makes decimal subtraction more manageable.

Aligning Decimal Points

Aligning decimal points correctly is the foundation of subtracting decimals. If the decimal points are not correctly aligned, the place values will not match, and the subtraction will be incorrect. It's a fundamental step you'll always need to consider for accurate calculations.

• Make sure each column of digits aligns perfectly on top of each other.
• Use placeholders (like zeros) if necessary, to keep your columns straight.

By ensuring your decimals are aligned, you are making sure that you're subtracting values in the same place column from each other, which ensures accuracy and reduces chances of error. This practice is not just about making numbers look neat; it is a critical part of performing accurate arithmetic operations.