Learn Before
Solving a Coin Mixture Problem: Pennies and Nickels
Apply the seven-step problem-solving strategy and the total value of coins model to a coin mixture problem in which the relationship between the two coin counts involves both multiplication and addition.
Problem: Danny has $2.14 worth of pennies and nickels in his piggy bank. The number of nickels is two more than ten times the number of pennies. How many nickels and how many pennies does Danny have?
- Read the problem and identify the coin types involved: pennies (worth $0.01 each) and nickels (worth $0.05 each). The total value of all coins is $2.14.
- Identify what to find: the number of pennies and the number of nickels.
- Name the unknowns using a single variable. Since the number of nickels is defined in terms of the number of pennies, start with pennies. Let = the number of pennies. The phrase "two more than ten times" combines multiplication by with adding , so the number of nickels is . Organize everything in a table:
| Type | Number | Value ($) | Total Value ($) |
|---|---|---|---|
| Pennies | 0.01 | ||
| Nickels | 0.05 | ||
| 2.14 |
- Translate into an equation by adding the total values and setting the sum equal to the overall total:
- Solve the equation:
- Distribute :
- Combine like terms:
- Subtract from both sides:
- Divide both sides by :
Find the number of nickels: .
- Check: →
- Answer: Danny has four pennies and 42 nickels.
This example extends the coin mixture technique to a case where the relationship between the two coin counts involves both multiplication and addition — "two more than ten times" translates to — rather than a simple additive relationship like "nine more than." The expression introduces a larger coefficient () inside the parentheses, so distributing across produces rather than a small decimal term. After distribution, combining yields the non-obvious coefficient , making the final division step () more challenging than in simpler coin problems.
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In a professional setting such as a bank or retail store, which formula is used to calculate the total value of a collection of identical coins?
Match each component of the coin value model to its correct description as used in professional cash handling and accounting.
When a retail associate is performing a drawer count at the end of a shift, they calculate the total value of a specific coin type by multiplying the number of coins by the ____ of a single coin.
In a retail cash-handling procedure, the total value of a specific denomination of coins in a register drawer is determined by multiplying the total number of those coins by the monetary value of a single coin.
A retail associate is performing an end-of-shift cash count and needs to determine the total value of the nickels in their drawer. Arrange the steps of the 'Total Value of Coins Model' in the correct logical sequence to complete this task.
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In a professional cash-handling environment, what is the standard procedure for determining the overall value of a collection that contains several different types of coins, such as quarters, dimes, and nickels?
A banking associate is reviewing the 'Total Value of Coins Model' training manual. According to the specific example provided in the model, if an associate counts 17 dimes and each dime is worth $0.10, what is the resulting 'total value' of that collection?
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Learn After
A retail manager is performing a cash audit and needs to document the relationship between the coins found in a register. The manager determines that the number of nickels is 'two more than ten times' the number of pennies. If 'p' represents the number of pennies, which algebraic expression should the manager use to represent the number of nickels in the audit report?
A retail manager is auditing a cash drawer containing pennies and nickels. Match each algebraic component to its correct definition based on the coin mixture model used to calculate the total value.
A retail manager is documenting a cash audit procedure for a drawer containing pennies and nickels. The audit must follow a standardized seven-step strategy to determine the number of each coin when the total value is $2.14 and the number of nickels is 'two more than ten times' the number of pennies. Arrange the following steps in the correct order to model and solve this problem.
A retail manager is performing a cash audit where the number of nickels is 'two more than ten times' the number of pennies (). True or False: When setting up the total value equation, distributing the nickel's unit value (0.05) over the expression for the quantity of nickels ($10p + 2$) results in a variable term of 0.50p.
Combining Coefficients in Cash Audits
A retail manager has finished calculating the number of pennies and nickels in a register drawer. Before recording the final counts in the ledger, the manager performs a verification to ensure the total value of these coins equals the expected sum. In the seven-step problem-solving strategy, this verification process is known as the ____ step.
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Standardizing Algebraic Models for Cash Audits
A retail manager is simplifying the equation $0.01p + 0.05(10p + 2) = 2.14 to reconcile a cash drawer. To remove the parentheses by multiplying the unit value (0.05) by both terms inside the expression ($10p and 2), which algebraic property is the manager applying?
A retail manager is using a standardized seven-step strategy to reconcile a cash till containing pennies and nickels. According to this model, which step is dedicated to converting the verbal relationship between the coin counts and their unit values into a formal algebraic equation?