Problem 18 - Algebra and Trigonometry

Question

\(7-18 \cdot\) 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

Answer

The value is \(131p + 140\) cents.

Step 1: Identify Relationships and Variables

Let's denote the number of nickels as \( n \), the number of dimes as \( d \), and the number of quarters as \( q \). We are given that there are twice as many nickels as pennies. Therefore, \( n = 2p \).

Step 2: Establish Number of Dimes

We know there are four more dimes than nickels. So the equation for dimes is \( d = n + 4 \). Substituting \( n = 2p \) gives us \( d = 2p + 4 \).

Step 3: Determine Number of Quarters

The number of quarters is equal to the sum of the dimes and nickels. Therefore, \( q = n + d = 2p + (2p + 4) = 4p + 4 \).

Step 4: Calculate Value of Each Coin Type

Calculate the value in cents of each type of coin: \( 1p \) for pennies, \( 5n \) for nickels, \( 10d \) for dimes, and \( 25q \) for quarters. This becomes \( 1p, 5(2p), 10(2p + 4), 25(4p + 4) \).

Step 5: Combine Values for Total Value Expression

Substitute the equations for \( n \), \( d \), and \( q \) into their value equations: \( 5(2p) = 10p \), \( 10(2p + 4) = 20p + 40 \), and \( 25(4p + 4) = 100p + 100 \). Now, sum them all together with the pennies: \( 1p + 10p + 20p + 40 + 100p + 100 \).

Step 6: Simplify the Expression

Combine like terms: \( 1p + 10p + 20p + 100p + 40 + 100 = 131p + 140 \). This represents the total value in cents.

Key Concepts

Variable Relationships

In algebra, a lot of problems involve understanding how different quantities relate to each other. These relationships can often be represented by variables. Variables are simply symbols that stand in for unknown numbers or values. In our exercise, we are dealing with coins in a purse, where each type of coin can be represented by a variable amount. We introduced variables like \( p \) for pennies, \( n \) for nickels, \( d \) for dimes, and \( q \) for quarters. These variables help to simplify the process of describing the amounts of each coin in mathematical terms.

• Understanding variable relationships helps break down complex problems into manageable parts.
• By establishing a variable \( p \) as the number of pennies, we can define other amounts in terms of \( p \).
• For example, if there are twice as many nickels as pennies, then using \( n = 2p \) helps us express the number of nickels in terms of pennies.

The power of using variables lies in their ability to generalize and make calculations repeatable. So, once you define the relationships, you can apply them consistently, making the problem-solving process more straightforward.

Coin Value Calculations

Calculating the total value of coins involves both knowing the number of each type of coin and their individual values in cents. In this scenario, each type of coin has a fixed value: pennies are worth 1 cent, nickels 5 cents, dimes 10 cents, and quarters 25 cents. To find the total value, you need to multiply the number of each type of coin by its respective value, then sum these products.

• Pennies contribute \( 1p \) to the total value.
• Nickels add \( 5n \), which becomes \( 10p \) when using our variable relationship \( n = 2p \).
• Dimes add \( 10d \), substituting gives \( 20p + 40 \) based on \( d = 2p + 4 \).
• Quarters contribute \( 25q \) amounting to \( 100p + 100 \) when \( q = 4p + 4 \).

Adding all these values gives us a single expression that represents the total value. This technique is broadly applicable whenever you need to compute total values involving different unit prices.

Linear Equations

Linear equations form the backbone of solving many algebraic problems. They are equations that appear as a straight line when graphed and usually have terms that are either constants or variables only raised to the first power. In our exercise, we formed multiple linear equations based on the relationships between different types of coins. For instance, \( n = 2p \), \( d = 2p + 4 \), and \( q = 4p + 4 \) are linear equations that help us translate word problems into mathematical statements.

• Linear equations are versatile and allow us to model real-life scenarios using math.
• They can easily be solved or rearranged to express one variable in terms of another.
• In our case, each linear equation simplifies the relationship between types of coins and their numbers.

Mastering linear equations empowers you to interpret and solve vast problems just by manipulating these simple mathematical relationships. Whether you're dealing with finances, physics, or everyday decisions, linear equations provide a reliable framework for finding solutions.