Question

A toy savings bank contains \(\$ 17.30\) consisting of nickels, dimes, and quarters. The number of dimes exceeds twice the number of nickels by 3 and the number of quarters is 4 less than 5 times the number of nickels. How many of each coin are in the bank?

Short answer

There are 11 nickels, 25 dimes, and 51 quarters in the toy savings bank.

Step-by-step solution

Write the System of Equations

Our system of equations based on the given information is: 1. \(0.05n + 0.10d + 0.25q = 17.30\) 2. \(d = 2n + 3\) 3. \(q = 5n - 4\)

Substitute the Values of d and q in Equation 1

Since we have expression for d and q in terms of n, we can substitute them into equation 1. 0.05n + 0.10(2n + 3) + 0.25(5n - 4) = 17.30

Solve the Equation for n

Now we will solve the equation for n: 0.05n + 0.20n + 0.30 + 1.25n - 1.00 = 17.30 Combine like terms: 1.50n + 0.30 = 17.30 Subtract 0.30 from both sides: 1.50n = 17.00 Now, divide both sides by 1.50: n = 11.33 (approximately) Since the number of nickels must be a whole number, round it to the nearest integer: n = 11

Find the Values of d and q Using n

Plug the value of n into equations 2 and 3 to find the values of d and q: d = 2(11) + 3 d = 22 + 3 d = 25 q = 5(11) - 4 q = 55 - 4 q = 51

Check the Solution

We will check if our solution satisfies the original equation 1: 0.05(11) + 0.10(25) + 0.25(51) = 0.55 + 2.50 + 12.75 = 17.30 Our solution satisfies the equation, so we have found the correct number of nickels, dimes, and quarters.

Answer

There are 11 nickels, 25 dimes, and 51 quarters in the toy savings bank.

Key concepts

Pre-Calculus

Pre-calculus is a course designed to prepare students for the study of calculus. It lays the foundation by introducing fundamental concepts like functions, complex numbers, and polynomials. But its emphasis on algebraic processes is crucial when dealing with word problems that involve setting up and solving equations.

In the context of the given problem, the pre-calculus skills employed involve understanding the relationship between numbers and variables. These include the ability to translate written information into mathematical expressions and handle coin denominations as linear coefficients. Recognizing how to set up a system of equations from written statements is an essential pre-calculus skill that is at play here.

Solving Systems of Equations

Process of Elimination and Substitution

When working with systems of equations, there are multiple methods to find a solution, including graphing, substitution, and elimination. The coin problem presented uses the substitution method, which is a powerful pre-algebra skill that involves solving one equation for a variable and then substituting that solution into another equation.

This method is particularly helpful when the equations are not readily suited for elimination. Substitution often simplifies the problem to one variable, which is easier to solve and then proceed with finding the values of other variables.

Coin Word Problems

Coin word problems are a practical application of linear equations and are commonly used to teach students how to translate real-life situations into mathematical models. These problems typically include descriptions of different coin values and their quantities.

Effective strategies for tackling coin word problems include:

• Identifying the coins and assigning a variable to each type.
• Translating the relationships and total value into equations.
• Solving the system of equations usually involving linear functions.

Students must pay attention to the units used, i.e., converting coin values to dollars if necessary, as was done in the exercise where nickels, dimes, and quarters were converted into their dollar equivalents before being used in the equations.

Linear Equations

Role in Word Problems

Linear equations represent straight-line relationships between variables. They're fundamental in constructing and solving mathematical models of real-world problems. In the context of our coin word problem, the linear equations are made up of variables representing the number of coins and constants representing their values.

Understanding how to manipulate these equations, combining like terms, and isolating variables is crucial to finding a solution. The exercise demonstrates this process step-by-step, illustrating how the linear system corresponds to the quantities and values of the coins in the problem.