Question

A boy finds $$\$ 1.05$$ in dimes, nickels, and pennies. If there are 17 coins in all, how many coins of each type can he have?

Short answer

8 dimes, 4 nickels, and 5 pennies.

Step-by-step solution

Define Variables

Let \( d \) represent the number of dimes, \( n \) the number of nickels, and \( p \) the number of pennies. We need to express everything in terms of these variables.

Set Up Equations

We have two main equations. The first is based on the total number of coins: \( d + n + p = 17 \). The second is based on the total value of the coins: \( 10d + 5n + 1p = 105 \). These are derived from the values of the dimes, nickels, and pennies in cents.

Express One Variable

From the total number of coins equation, express \( p \) in terms of \( d \) and \( n \): \( p = 17 - d - n \).

Substitute into Value Equation

Substitute \( p = 17 - d - n \) into the value equation: \( 10d + 5n + 1(17 - d - n) = 105 \). Simplify to get: \( 9d + 4n = 88 \).

Solve for Variables

We solve \( 9d + 4n = 88 \) for integer solutions:- If \( d = 8 \), then \( 72 + 4n = 88 \) \( \Rightarrow 4n = 16 \) \( \Rightarrow n = 4 \).- Substitute \( d = 8 \) and \( n = 4 \) back into \( p = 17 - d - n = 5 \).

Verify the Solution

Verify that these values satisfy both original equations: \( d + n + p = 8 + 4 + 5 = 17 \) and \( 10(8) + 5(4) + 1(5) = 80 + 20 + 5 = 105 \). Both conditions hold true.

Key concepts

Linear Equations

Linear equations form the backbone of algebraic problem solving. They involve finding the values of unknown variables that make the equation true, and they are called 'linear' because they graph as straight lines.

In our problem, the two equations that were set up were:

• The total coin count equation: \( d + n + p = 17 \)
• The total value equation: \( 10d + 5n + p = 105 \)

These equations are linear because each variable (\(d\), \(n\), and \(p\)) appears only once and is not raised to any power. To solve them, we express variables in terms of each other and substitute to reduce the equations. This process illustrates how linear equations can handle multiple constraints well, such as in word problems involving sums and values.

Understanding and setting up these linear equations is crucial as they provide the structure needed to systematically find solutions.

Variable Substitution

Variable substitution is an effective technique in solving systems of equations. It involves replacing one variable in an equation with an expression from another equation. This helps simplify the problem, allowing us to solve for other variables step by step.

In the exercise, once the equations \(d + n + p = 17\) and \(10d + 5n + p = 105\) were established, substitution was used as follows:

• From the equation for the total number of coins, solve for \(p\): \(p = 17 - d - n\).
• Substitute this expression for \(p\) into the value equation to simplify it.

This approach reduced the equations to one with two variables: \(9d + 4n = 88\). Such simplification is key in making complex problems more manageable by ensuring focus is only on the necessary parts of the problem at a time. This technique is powerful in algebra because it accelerates finding solutions by decreasing the number of variables and unknowns.

Integer Solutions

Finding integer solutions is often necessary in real-world problems where results need to be whole numbers. This is especially true in situations involving counts, like the number of coins in our problem.

After using substitution, the simplified equation was \(9d + 4n = 88\). The objective was to find integers \(d\) and \(n\) that satisfied this equation. Through checking possible values, it was found:

• If \(d = 8\), \(72 + 4n = 88\) leading to \(4n = 16\) and therefore \(n = 4\).
• With these values, \(p\) was calculated as \(17 - 8 - 4 = 5\).

Ensuring that these results also satisfy the original equations confirms their correctness. Integer solutions are feasible when dealing with practical counting scenarios, as only whole numbers make sense in these contexts. This emphasizes the importance of verifying that solutions both logically and mathematically fit the problem's requirements.