Problem 102 - Algebra for College Students

Question

How would you help someone who is having difficulty expressing \(n\) nickels and \(d\) dimes in terms of cents?

Step-by-Step Solution

Answer

The total value in cents is \(5n + 10d\).

Step 1: Understanding Coin Values

First, we need to understand the value of each coin. In the United States, a nickel is worth 5 cents and a dime is worth 10 cents.

Step 2: Expressing Nickels in Cents

Each nickel is worth 5 cents, so if you have \(n\) nickels, the total value in cents can be expressed as \(5n\).

Step 3: Expressing Dimes in Cents

Each dime is worth 10 cents, so if you have \(d\) dimes, the total value in cents can be expressed as \(10d\).

Step 4: Combine Values of Both Coins

To find the total amount in cents from \(n\) nickels and \(d\) dimes, simply add the values we calculated: \(5n + 10d\).

Step 5: Final Expression

The final expression for the total value in cents when you have \(n\) nickels and \(d\) dimes is \(5n + 10d\).

Key Concepts

Coin Values

Coins are a common part of everyday transactions. In the United States currency system, coins such as nickels and dimes have specific values. A nickel is worth 5 cents and a dime is worth 10 cents.

Knowing these values is crucial for calculations involving cents or when converting coin amounts into larger currency sums. Memorizing these values can help in various practical scenarios, making it easier to manage cash transactions or calculate totals when handling multiple coins.

Nickels and Dimes

Nickels and dimes are small-denomination coins in the U.S., each with its own monetary value that can be easily used in various calculations.

Understanding these coins involves simple multiplication:

• Each nickel equals 5 cents. So, if you have a certain number of nickels, multiplying the number of nickels () by 5 gives the total value in cents: \( 5n \).


• Each dime equals 10 cents. If you have a certain number of dimes (d), multiplying the number of dimes by 10 gives the total value in cents: \( 10d \).

These basic multiplications form the foundation for combining the values of nickels and dimes into a single expression.

Cents Conversion

Converting nickels and dimes to cents involves understanding and applying a simple mathematical concept. When we talk about cents conversion, we're essentially expressing the total value of our coins in terms of cents.

This involves combining the value from nickels and dimes. If you have both coins, you sum up their individual values:

• The expression \( 5n \) from nickels, where each nickel contributes 5 cents.


• The expression \( 10d \) from dimes, where each dime contributes 10 cents.

To find the total value in cents, you add these two numbers together: \( 5n + 10d \). This final expression gives the comprehensive amount in cents for any given pair of nickels and dimes.