Problem 50 - Algebra for College Students

Question

Solve each equation. Abby has 37 coins, consisting only of dimes and quarters, worth \(\$ 7.45\). How many dimes and how many quarters does she have?

Step-by-Step Solution

Answer

Abby has 12 dimes and 25 quarters.

Step 1: Define Variables

Let's represent the number of dimes Abby has as \( d \) and the number of quarters as \( q \). We need to find the values of \( d \) and \( q \).

Step 2: Set Up Equations

We have two conditions based on the problem statement: 1. The total number of coins is 37, so we have the equation \( d + q = 37 \).2. The total value of the coins is \\(7.45. Since dimes are worth \\)0.10 and quarters are worth \$0.25, we have the equation \( 0.10d + 0.25q = 7.45 \).

Step 3: Eliminate Decimals from the Second Equation

To eliminate decimals, multiply the second equation by 100: \[ 10d + 25q = 745 \].

Step 4: Use Substitution to Solve for One Variable

From the first equation, express \( d \) in terms of \( q \): \[ d = 37 - q \]. Substitute \( d = 37 - q \) into the equation \( 10d + 25q = 745 \).

Step 5: Solve the Equation for q

Substitute \( d = 37 - q \) into \( 10d + 25q = 745 \):\[ 10(37 - q) + 25q = 745 \]Simplify to get:\[ 370 - 10q + 25q = 745 \]\[ 370 + 15q = 745 \]Subtract 370 from both sides: \[ 15q = 375 \]Divide by 15: \[ q = 25 \].

Step 6: Substitute Back to Find d

Now that we know \( q = 25 \), substitute back into the equation \( d = 37 - q \): \[ d = 37 - 25 \]\[ d = 12 \].

Step 7: Verify the Solution

Check the solution by calculating the total value:- Dimes: \( 12 \times 0.10 = 1.20 \)- Quarters: \( 25 \times 0.25 = 6.25 \)Add the values: \( 1.20 + 6.25 = 7.45 \). The solution satisfies both conditions.

Key Concepts

Equation Solving

Equation solving is a fundamental concept in algebra, essential for finding the unknown values in mathematical problems. In our word problem, we need to discover how many dimes and quarters Abby has. To do that, we first define our unknowns: let \( d \) be the number of dimes and \( q \) the number of quarters.

Next, we set up equations based on the problem. The core idea is to formulate equations that describe:

• The total quantity of items (in this case, coins).
• The total value or relationship between those items.

For Abby's coins:

• The first equation reflects the total number of coins: \( d + q = 37 \)
• The second equation represents the total monetary value: \( 0.10d + 0.25q = 7.45 \)

Equations help organize and simplify real-world information into a manageable form, making it easier to solve for unknowns.

Substitution Method

The substitution method is an effective strategy to solve linear equations with multiple variables. It involves expressing one variable in terms of another and then substituting this expression back into the other equation.

Here’s a step-by-step of how you can apply it:

• From the equation \( d + q = 37 \), express \( d \) in terms of \( q \) by rearranging it to \( d = 37 - q \).
• Now, substitute \( d = 37 - q \) into the value equation \( 0.10d + 0.25q = 7.45 \).
• Transform the equation by replacing \( d \) to get a single variable equation.

Substitution turns complex systems into simple, single-variable equations, making it easier to find concrete solutions.

Linear Equations

Linear equations are equations where the highest power of the variable is one, making them straightforward to solve and graphically, they represent straight lines.

In our problem, both equations \( d + q = 37 \) and the altered \( 10d + 25q = 745 \) are linear. Solving linear equations often involves algebraically manipulating them to isolate one variable.

• Rewrite the equation, as shown in \( 10(37 - q) + 25q = 745 \).
• Simplify and solve using basic arithmetic to find the variable values.

Linear equations form the bedrock of algebra, serving as a building block for understanding more complex functions and systems.