Question

David has two quarters for every five dimes in his change dish, with no other coins present. If he has a total of \(\$ 2\) in coins in the dish, how many total coins does he have? (A) 12 (B) 14 (C) 16 (D) 18

Short answer

David has a total of 14 coins (option B).

Step-by-step solution

Identify Variables

Let the number of dimes be represented by \( d \), and accordingly, based on the problem, the number of quarters will be \( \frac{2}{5}d \). Each quarter is worth \( \\(0.25 \) and each dime is worth \( \\)0.10 \).

Express Total Value

Set up the equation for the total value of coins: \( 0.10d + 0.25 \left( \frac{2}{5}d \right) = 2 \). This equation represents the total monetary value of all the coins in dollars.

Simplify Equation

Simplify the equation: \( 0.10d + 0.25 \times \frac{2}{5}d = 2 \). Calculate the value of the quarters' part: \( 0.10d + 0.10d = 2 \). This simplifies to \( 0.20d = 2 \).

Solve for Number of Dimes

Solve the equation \( 0.20d = 2 \) for \( d \): \( d = \frac{2}{0.20} = 10 \). Therefore, David has 10 dimes.

Calculate Number of Quarters

With \( d = 10 \), calculate the number of quarters: \( \frac{2}{5}d = \frac{2}{5} \times 10 = 4 \). David has 4 quarters.

Sum Total Coins

The total number of coins is the sum of the dimes and quarters: \( 10 + 4 = 14 \). Therefore, David has a total of 14 coins.

Key concepts

Problem-Solving Steps

When tackling math problems, especially on tests like the PSAT, it's crucial to adopt a structured approach. This ensures that you don't miss important details and arrive at the correct answer efficiently. Here are some general problem-solving steps to guide you through any math problem:

- **Identify what's being asked:** Before diving into calculations, make sure you understand the problem statement. In our original exercise, the goal was to find the total number of coins.
- **Define knowns and unknowns:** Clearly outline what information is given and what needs to be determined. Use variables to represent unknown quantities. For example, we defined the number of dimes as \(d\).
- **Set up equations:** Based on the information, formulate equations that represent the relationships in the problem. This step was crucial when we represented the total value of quarters and dimes in the original exercise.
- **Solve systematically:** Once the equations are set, solve them step by step. Simplify as needed and use algebra to find the unknowns. Always check that your solutions make sense in the context of the problem.
- **Verify your answer:** After solving the problem, it's helpful to go back and ensure that the solution meets all the conditions set in the problem. This helps catch any mistakes and confirms the answer's correctness.

Coin Math Problems

Coin math problems are a classic type of math challenge that often appears in standardized tests and textbooks. They typically involve finding the number of coins or the value of different types of coins given certain conditions. Here's an overview of how to approach these problems:

- **Identify coin types and values:** Start by listing the types of coins involved, such as quarters, dimes, nickels, and pennies, and their respective values. Quarters are worth \(0.25\), and dimes are worth \(0.10\), as seen in our example.
- **Determine relationships between coins:** Look for any ratios or proportions. In the problem we solved, the number of quarters was given as a function of the number of dimes, specifically \(\frac{2}{5}\) of the dimes.
- **Formulate the value equation:** Use the information to set up an equation based on the total value of all coins in the problem. This involves multiplying the coin quantity by its value and adding them together to equal the total amount of money specified.
- **Solve using algebraic methods:** Apply algebraic techniques to solve for the unknown number of each coin, as shown in the original exercise when calculating the number of dimes and quarters.
- **Review the solution:** Ensure that the solution logically satisfies all the given conditions and fits within the context of the problem, such as adding up to the correct total value.

Linear Equations in Algebra

Linear equations are foundational in algebra and are essential for solving a broad array of mathematical problems, including coin problems. A linear equation is an equation of the first degree, meaning it has no exponents higher than one. Here's how to master them:

- **Understanding the structure:** Linear equations often take the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and the power of \(x\) is one. In our exercise, the equation \(0.20d = 2\) is an example of a linear equation.
- **Solving techniques:** To solve linear equations, you typically isolate the variable using basic algebraic operations like addition, subtraction, multiplication, or division. For instance, we found \(d\) by dividing both sides of the equation by \(0.20\).
- **Application in real-life problems:** Linear equations are highly applicable in scenarios like the coin problem, where relationships between different quantities need to be expressed mathematically. Representing these relationships accurately allows you to use algebra to find solutions easily.
- **Checking solutions:** After obtaining a solution, plug it back into the original equation to verify that both sides are equal. This confirmation is crucial in ensuring the calculations were done correctly, as demonstrated by recalculating the total coin values in the original problem.
- **Practice and familiarity:** The best way to become comfortable with linear equations is through practice. Frequent exposure to these equations in different contexts will bolster your problem-solving skills and confidence in handling similar problems.