Problem 54 - Intermediate Algebra

Question

Jeannette has \(\$ 5\) and \(\$ 10\) bills in her wallet. The number of fives is three more than six times the number of tens. Let \(t\) represent the number of tens. Write an expression for the number of fives.

Step-by-Step Solution

Answer

Number of fives: \(6t + 3\)

Step 1: Identify Variables

Let the number of \(\bold{tens}\) be represented by the variable \(t\). We need to express the number of \(\bold{fives}\) in terms of \(t\).

Step 2: Analyze the Relationship

The problem states that the number of fives is three more than six times the number of tens.

Step 3: Form the Expression

According to the given relationship, the number of \(\bold{fives}\) can be represented as:$$6t + 3.$$Thus, the expression for the number of fives in terms of \(t\) is \(\bold{6t + 3}\).

Key Concepts

Variables in Algebra

A variable in algebra is a symbol, usually a letter, that represents a number. It can take different values based on the given conditions, making them useful for solving equations and understanding relationships between numbers. For instance, in the exercise, 't' is the variable representing the number of tens in Jeannette's wallet.
By using 't', we can form equations to describe relationships and solve problems dynamically. Variables are the backbone of algebra, turning complex word problems into manageable mathematical expressions.

Linear Expressions

Linear expressions are algebraic expressions that represent straight lines when plotted on a graph. They are usually in the form of y = mx + by = mx + by=mx+b, where 'm' is the slope and 'b' is the y-intercept. In our exercise, the expression for the number of fives, 6t + 3, 6t + 3,is a linear expression. Here, '6t' represents the term dependent on the variable t, and '+3' is the constant.

• This linear relationship helps us quickly understand how changes in the number of tens (t) affect the number of fives.
• Since the number of fives is 'three more than six times the number of tens', the linearity makes it simple to compute and predict values.

Linear expressions make it easier to grasp and visualize direct relationships.

Word Problems in Algebra

Word problems in algebra involve translating real-world situations into mathematical statements.
They require careful reading and an understanding of the problem context to identify variables and relationships. In our exercise:

• Identify key details: Jeannette has both \(5 and \)10 bills.
• Assign variables: Let 't' represent the number of \(10 bills.
• Analyze relationships: The problem states that the number of \)5 bills is 'three more than six times the number of \(10 bills'.

By clearly defining the relationships using algebraic expressions, we solve the problem systematically.
Writing the expression [6t + 3]6t + 3, for the number of \)5 bills, directly addresses the problem requirements. Word problems enhance critical thinking by challenging students to bridge the gap between textual descriptions and mathematical formulations.