In a collection of nickels, dimes, and quarters, there are...

In a collection of nickels, dimes, and quarters, there are twice as many dimes as nickels and 3 fewer quarters than dimes. If the total value of the coins is $\$ 4.50$ how many of each type of coin are there?

Key Concepts

System of Linear Equations

This concept involves creating and solving a set of equations that collectively represent the relationships described in a problem. When dealing with coin problems, each coin type contributes both to the count and to the total value, and setting up these equations is crucial to find the unknown counts.

Variable Representation

Assigning variables to unknown quantities allows the translation of a word problem into mathematical language. In coin problems, each type of coin is typically represented by a distinct variable, which forms the basis for constructing equations that model the relationships and constraints given in the problem.

Translating Word Problems to Equations

This key concept is about interpreting descriptive statements, such as proportional relationships or differences between quantities, and converting them into algebraic expressions. Recognizing phrases like 'twice as many' or 'fewer than' is essential for building a system of equations that accurately represents the problem scenario.

Coin Value Calculations

In coin problems, understanding that each coin type has a specific monetary value is essential. The total value of a collection of coins is determined by multiplying the number of each coin type by its corresponding value and summing these products, thus forming part of the equation system to be solved.

Substitution Method

A common algebraic technique used to solve systems of equations, the substitution method involves solving one of the equations for one variable and substituting that expression into the other equations. This method simplifies the system and makes it easier to find the values of the unknown variables.

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