Problem 66 - Prealgebra

Question

Suppose you buy some tools at the hardware store and the bill comes to \(\$ 37.87\). If you give the cashier two \(\$ 20\) bills and 2 pennies, how much change should you receive? List the bills and coins you should receive for change.

Step-by-Step Solution

Answer

$2.15 change: Two $1 bills, one dime, one nickel.

Step 1: Calculate Total Payment

To find out how much you have paid, add the amount of money given to the cashier: two \(20 bills and 2 pennies. Two \)20 bills is \[ 2 \times 20 = 40 \text{ dollars} \]Two pennies is \[ 0.02 \text{ dollars} \]So, the total amount paid is \[ 40 + 0.02 = 40.02 \text{ dollars} \].

Step 2: Calculate the Change Due

Subtract the cost of the tools from the total payment to find the change due:\[ 40.02 - 37.87 = 2.15 \text{ dollars} \].

Step 3: Determine the Bills and Coins for Change

To make $2.15 in change, the ideal denominations would be: - One $2 bill (or two $1 bills if $2 bills are unavailable). - One 10-cent coin (dime). - One 5-cent coin (nickel). If using $1 bills instead of a $2 bill: - Two $1 bills. - One 10-cent coin (dime). - One 5-cent coin (nickel).

Key Concepts

Addition and Subtraction of Decimals

Understanding how to add and subtract decimals is essential, especially when dealing with money. Decimals represent fractions of whole numbers and are crucial for precision in monetary transactions. Let's see how it applies in real life, such as when calculating change.

When adding decimals, align the decimal points vertically. This ensures that digits of the same place value line up correctly. For example, to find out your total payment, if you add two \(20 bills and 2 pennies, it means combining a whole number with a decimal:

• Two \)20 bills: \(40 is a whole number, which we write as 40.00 for clarity.
• Two pennies: \)0.02 is just 0.02 in decimal form.

Combine these by adding: \[40.00 + 0.02 = 40.02.\]

Subtraction of decimals is similar. Align the numbers by their decimal points and subtract as you would with integers. To find out the change, subtract the total amount spent (\(37.87) from your payment (\)40.02):\[40.02 - 37.87 = 2.15.\]

This calculation reveals how much change you should receive. Practicing these simple operations with decimals can help in many daily financial situations.

Money Denominations

When receiving change, it's physically distributed in specific money denominations. Understanding these denominations helps you recognize the correct change you should receive. U.S. currency is a good example with various bills and coins.

In this task, you received $2.15 as change.

• The dollar bills available include $1, $5, $10, $20, $50, and $100. Sometimes, $2 bills are available.
• Coins are often in denominations like 1 cent (penny), 5 cents (nickel), 10 cents (dime), 25 cents (quarter), and so on.

To make $2.15:

• A $2 bill or two $1 bills cover $2.
• A 10-cent coin (dime) covers 10 cents.
• A 5-cent coin (nickel) covers 5 cents.

This combination ensures you receive the right amount in the fewest pieces possible.

Problem-Solving Steps

Breaking down a problem into simple, manageable steps can greatly ease the process of finding a solution. In the context of making change, following structured problem-solving steps can help keep things clear and organized.

Step 1: Calculate Total Payment
First, determine the total amount paid. This includes counting all bills and coins given to the cashier. In the exercise, this was counted as $40.02.

Step 2: Calculate Change Due
Subtract the total cost from what you paid. This requires subtraction of decimals, ensuring accuracy with a calculator or careful manual calculation. The result here was $2.15.

Step 3: Determine Bills and Coins for Change
Evaluate the denominations available to make up the change amount. Choose the fewest number of coins and bills to make the exact amount, minimizing the pieces you handle. Here, either a $2 bill, a 10-cent dime, and a 5-cent nickel were ideal choices.

Following these steps can simplify what initially seems a complex task. It makes for efficient problem solving as well as a useful everyday life skill.