Problem 88 - Algebra for College Students

Question

Answer the question with an algebraic expression. Jesse has \(n\) nickels, \(d\) dimes, and \(q\) quarters in his bank. How much money, in cents, does he have in his bank?

Step-by-Step Solution

Answer

Jesse has \( 5n + 10d + 25q \) cents in his bank.

Step 1: Value of Nickels

A nickel is worth 5 cents. If Jesse has \( n \) nickels, he can calculate the total value of his nickels by multiplying the number of nickels by 5. Therefore, the value of the nickels is \( 5n \) cents.

Step 2: Value of Dimes

A dime is worth 10 cents. To find the total value of all dimes Jesse has, multiply the number of dimes \( d \) by 10. Thus, the value of the dimes is \( 10d \) cents.

Step 3: Value of Quarters

A quarter is worth 25 cents. Similarly, the total value of all quarters can be calculated by multiplying the number of quarters \( q \) by 25. The value of the quarters is \( 25q \) cents.

Step 4: Total Value in Cents

To find the total value in cents, sum up the values of the nickels, dimes, and quarters calculated in the previous steps. Thus, the total value in cents is: \( 5n + 10d + 25q \).

Key Concepts

Nickels: Small but Mighty

Nickels might be small in size, but they play an important role in building up the total value of coins in your pocket. Each nickel is worth 5 cents. This is important to remember because when you are figuring out how much money you have, you need to multiply the number of nickels you possess by its value.
For example:

• If you have 1 nickel, it is worth 5 cents.
• 2 nickels equal 10 cents, and 3 nickels equal 15 cents.

If Jesse has 'n' nickels, the algebraic expression to find out their total value is simple: multiply the number of nickels by 5. Thus, you have the expression: \(5n\). It’s an easy calculation, but getting into the habit of translating coin counts into their monetary value can be very helpful.

Dimes: A Decade of Value

Dimes have a unique value of 10 cents each. This means that they are double the worth of a nickel. This can make dimes quite handy when you're trying to rack up value quickly. The process of calculating the total worth of dimes is similar to that of nickels; you simply multiply the number of dimes by their value.
Instead of counting coins, think of it in terms of tens. For instance:

• 1 dime equals 10 cents.
• 2 dimes equal 20 cents, and so on.

If Jesse has 'd' dimes, the equation becomes \(10d\).
This represents the total value of his dimes when summed together. Dimes make the math straightforward due to their whole number value, which corresponds directly to the decimal base system.

Quarters: Bigger Changes

Quarters are quite valuable, each being worth 25 cents. They are the largest of the three coins mentioned here and therefore, they contribute significantly to the sum of your coins. Just like with nickels and dimes, you find the total value of quarters by multiplying the number of quarters by their value.
Here's how to look at it:

• 1 quarter equals 25 cents.
• 2 quarters are 50 cents.
• 3 quarters sum up to 75 cents.

Jesse's quarters, represented as 'q' in the equation, leads to the expression \(25q\).
This conversion helps when trying to get a quick estimate of the total using fewer coins. Quarters also make it easy to convert larger sums into dollars when paired with nickels and dimes. Altogether, even a few quarters can add up to a relatively large sum of money quickly.