Problem 18

Question

Express the given quantity in terms of the indicated variable. The value (in cents) of the change in a purse that contains twice as many nickels as pennies, four more dimes than nickels, and as many quarters as dimes and nickels combined; $p=$ number of pennies.

Step-by-Step Solution

Verified

Answer

The value in cents is $ 131p + 140 $.

1Step 1: Define Variables for Coins

Let $ p $ represent the number of pennies. The number of nickels is twice the number of pennies, which can be expressed as $ 2p $.

2Step 2: Express the Number of Dimes

According to the problem, there are four more dimes than nickels. Since the number of nickels is $ 2p $, the number of dimes can be expressed as $ 2p + 4 $.

3Step 3: Express the Number of Quarters

The number of quarters is equal to the total number of dimes and nickels combined, which is $ 2p + (2p + 4) = 4p + 4 $.

4Step 4: Calculate Total Value of Pennies

Since each penny is worth 1 cent, the value of the pennies is $ p $ cents.

5Step 5: Calculate Total Value of Nickels

Each nickel is worth 5 cents. With $ 2p $ nickels, they have a total value of $ 5 \times 2p = 10p $ cents.

6Step 6: Calculate Total Value of Dimes

Each dime is worth 10 cents. With $ 2p + 4 $ dimes, they are valued at $ 10(2p + 4) = 20p + 40 $ cents.

7Step 7: Calculate Total Value of Quarters

Each quarter is worth 25 cents. With $ 4p + 4 $ quarters, they have a value of $ 25(4p + 4) = 100p + 100 $ cents.

8Step 8: Sum Total Values to Find Equation

Add the values of all coins together: pennies ($ p $ cents) + nickels ($ 10p $ cents) + dimes ($ 20p + 40 $ cents) + quarters ($ 100p + 100 $ cents). The total value in cents is: \[ p + 10p + 20p + 40 + 100p + 100 = 131p + 140 \].

Key Concepts

Variable SubstitutionCoin Value CalculationSystem of Equations

Variable Substitution

Variable substitution is a powerful concept used in algebra to simplify complex problems by replacing variables with algebraic expressions. In this exercise, we were tasked with expressing the value of various coins in terms of one main variable, which is the number of pennies, represented by $ p $.

First, we define the number of nickels as $ 2p $, meaning twice the number of pennies.
Next, for dimes, we express it as four more than nickels, $ 2p + 4 $.
For quarters, we sum the number of dimes and nickels, resulting in $ 4p + 4 $.

These substitutions allow us to easily find the total value of coins by creating a uniform variable reference for better clarity and simpler calculations.

Coin Value Calculation

Once we've established the number of each type of coin using variable substitution, the next step is to calculate the total value of each type of coin. This forms the basis of evaluating the total monetary value.

Pennies: Each penny is worth 1 cent, so if you have $ p $ pennies, it's worth $ p $ cents.
Nickels: Worth 5 cents each, and with $ 2p $ nickels, the value is $ 10p $ cents.
Dimes: Worth 10 cents each, so $ 2p + 4 $ dimes are valued at $ 20p + 40 $ cents.
Quarters: Each being 25 cents, gives $ 4p + 4 $ quarters a total value of $ 100p + 100 $ cents.

By calculating the monetary value of each type of coin, we prepare to add them all up for a complete evaluation.

System of Equations

In algebra, a system of equations consists of multiple equations that are solved together. Here, through variable substitution, we tactically created expressions for nickels, dimes, and quarters, all based on $ p $. The problem doesn't direct us to solve these equations as simultaneous equations in the traditional sense, but rather it's about substituting and summing the equations.

We identify equations like: $ \,2p$, $ \,2p + 4$, and $ \,4p + 4$.
These expressions then contribute to an overall equation for total value.
Combining the values: $ p + 10p + 20p + 40 + 100p + 100 $.
This calculation results in a simplified total value expression of $ 131p + 140 $.

This is a system as the detailed expressions for different coin types allow us to comprehensively evaluate the coin collection's total value.

Problem 18

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