Problem 59 - Prealgebra

Question

A person buys \(\$ 4.57\) worth of candy. If he pays for the candy with a \(\$ 10\) bill, how much change should he receive?

Step-by-Step Solution

Answer

The person should receive $5.43 in change.

Step 1: Understand the Problem

The person pays for candy worth $4.57 with a $10 bill. To find out how much change they receive, we need to subtract the cost of the candy from the amount they paid with.

Step 2: Set Up the Subtraction

We will perform the subtraction using the equation: \[ \text{Change} = \text{Amount Paid} - \text{Cost of Candy} \] Substituting the values we have:\[ \text{Change} = 10.00 - 4.57 \]

Step 3: Perform the Subtraction

Carry out the subtraction for the given numbers:Align the decimal points:\[ \begin{array}{c} 10.00 \- \ 4.57 \\hline\end{array}\]Subtract seven from zero in the hundredths column; borrow as needed through the entire columns:\[ \begin{array}{c} 10.00 \- \ 4.57 \\hline \ 5.43 \\end{array}\]Thus, the change amount is $5.43.

Step 4: Verify the Calculation

To verify, add the change amount back to the price of the candy:\[ 4.57 + 5.43 = 10.00 \]Since the result is the same as the amount they paid initially, the subtraction is verified to be correct.

Key Concepts

Decimal Subtraction

Decimal subtraction involves subtracting numbers that have decimal points.
To solve such problems, begin by aligning the numbers' decimal points vertically. It ensures each digit is in the correct place value column.
For example, to subtract \(4.57 from \)10.00, line up the decimal points as follows:

• \[ \begin{array}{c} 10.00 \ - 4.57 \hline \end{array}\]

By aligning decimals, it becomes easier to perform each subtraction step column by column.

Borrowing in Subtraction

Borrowing is a key concept to understand when larger numbers are subtracted from smaller numbers in any column.
When subtracting in our example, subtract 7 from 0, and it is clear that borrowing will be needed.
Here's how you handle borrowing for our problem:

• Start from the rightmost digit (hundredths in this example), where subtraction is not possible since 0 is smaller than 7.
• Borrow 1 from the next column (tenths) which makes it a 9 (since one whole is equivalent to 10 tenths).
• Carry this 10 over to the hundredths column, so it becomes 10 - 7 = 3.
• Continue this method if needed leftwards through all columns, ending with:

\[ \begin{array}{c} 10.00 \ - 4.57 \hline 5.43 \end{array} \]

Verification of Answers

Verifying your answer ensures the subtraction was carried out correctly.
The best way to do this is using the method of reverse operation, which involves adding the result to the subtracted value.
If the addition gives you back the original minuend, the subtraction was likely performed correctly.
For our problem, add the change back to the cost of the candy:

• \[ 4.57 + 5.43 = 10.00 \]
• Compare the sum (10.00) to the amount paid initially.
• If they match, the subtraction is verified.

This checks whether the original equation holds true, adding confidence in the solution's accuracy.