Problem 207 - Elementary Algebra

Question

Suppose you have six quarters, nine dimes, and four pennies. Explain how you find the total value of all the coins.

Step-by-Step Solution

Answer

The total value of the coins is 244 cents.

Step 1: Convert Quarters to Cents

Each quarter is worth 25 cents. Multiply the number of quarters (6) by 25.\[\text{Value of quarters} = 6 \times 25 = 150 \text{ cents}\]

Step 2: Convert Dimes to Cents

Each dime is worth 10 cents. Multiply the number of dimes (9) by 10.\[\text{Value of dimes} = 9 \times 10 = 90 \text{ cents}\]

Step 3: Convert Pennies to Cents

Each penny is worth 1 cent. Multiply the number of pennies (4) by 1.\[\text{Value of pennies} = 4 \times 1 = 4 \text{ cents}\]

Step 4: Sum the Total Value

Add the values of quarters, dimes, and pennies to find the total value.\[\text{Total value} = 150 + 90 + 4 = 244 \text{ cents}\]

Key Concepts

Basic Arithmetic

In this exercise, basic arithmetic plays a crucial role. To find out the value of each type of coin, we use multiplication and addition. Multiplication helps us determine the total value of a group of coins. Add up these values to get the final sum.
For example, multiplying the number of quarters (6) by 25 gives us 150 cents. Next, multiplying the number of dimes (9) by 10 gives us 90 cents. And multiplying the number of pennies (4) by 1 gives us 4 cents.
Once these values are calculated, we use addition to find the total value. Adding 150, 90, and 4 together gives us 244 cents.

Conversion of Units

Understanding the value of different coins requires converting each type of coin into the same unit, which is cents in this exercise.

• Quarters need to be converted into cents.
• Dimes also need to be converted into cents.
• Pennies are already in cents.

For quarters, since each is worth 25 cents, six quarters are 6 times 25, which is 150 cents.
For dimes, each worth 10 cents, nine dimes are 9 times 10, which is 90 cents.
Pennies remain as they are, with four pennies giving 4 times 1 cent, resulting in 4 cents.
By converting all values to the same unit (cents), we can simplify the final calculation of their total value.

Sum Calculation

To find the total value of the coins, we need to sum up the results from our conversions:

• Value of quarters: 150 cents
• Value of dimes: 90 cents
• Value of pennies: 4 cents

Adding these values together is simple:
\[150 + 90 + 4 = 244 \text{ cents} \] Breaking down the process like this ensures that we don't miss any steps and that our calculations are accurate. It's also a good practice to double-check your additions, especially in more complex problems.

Elementary Algebra

Algebra helps to generalize and solve problems like these efficiently. Let's represent the number of quarters, dimes, and pennies as variables:

• Let \( q = 6 \) (number of quarters)
• Let \( d = 9 \) (number of dimes)
• Let \( p = 4 \) (number of pennies)

Now, we use the known values for each type of coin: Quarters are worth 25 cents each, dimes 10 cents each, and pennies 1 cent each:
\[ \text{Value of quarters} = q \times 25 = 6 \times 25 = 150 \text{ cents} \] \[ \text{Value of dimes} = d \times 10 = 9 \times 10 = 90 \text{ cents} \] \[ \text{Value of pennies} = p \times 1 = 4 \times 1 = 4 \text{ cents} \] Substituting these back, we can set up our total value calculation:
\[ \text{Total value} = 150 + 90 + 4 = 244 \text{ cents} \] Using algebra in this way streamlines the problem-solving process, making it easier to see relationships and patterns.