Problem 37 - Intermediate Algebra

Question

A deposit slip listed $$\$ 700$$ in cash to be deposited. There were 100 bills, some of them five-dollar bills and the remainder ten-dollar bills. How many bills of each denomination were deposited?

Step-by-Step Solution

Answer

There are 60 five-dollar bills and 40 ten-dollar bills.

Step 1: Define Variables

Let \( x \) represent the number of five-dollar bills and \( y \) represent the number of ten-dollar bills. We need to find the values of \( x \) and \( y \).

Step 2: Set Up Equations

We have two pieces of information: the total number of bills is 100, and the total amount of money is $700. This leads to two equations: \( x + y = 100 \) and \( 5x + 10y = 700 \).

Step 3: Simplify Equation 2

Simplify the second equation \( 5x + 10y = 700 \) by dividing the entire equation by 5: \( x + 2y = 140 \).

Step 4: Solve the System of Equations

Now solve the system of equations: \( x + y = 100 \) and \( x + 2y = 140 \). Subtract the first equation from the second: \( (x + 2y) - (x + y) = 140 - 100 \), which simplifies to \( y = 40 \).

Step 5: Find the Value of x

Substitute \( y = 40 \) back into the first equation: \( x + 40 = 100 \). Solving this gives \( x = 60 \).

Key Concepts

Linear equations

Linear equations are a cornerstone of algebra, used to find the values of unknown variables by setting up relationships between them. In the context of our problem, linear equations help represent the total count and total value of bills, both being linear relationships.A linear equation is simply an equation of degree one, meaning the variables are not raised to any power higher than one. For example:

• In our exercise, the equation \( x + y = 100 \) is a linear equation, representing the total number of bills.
• The equation \( 5x + 10y = 700 \) represents the total amount of money in terms of the bills.

These types of equations are relatively simple to solve and can be manipulated easily, such as adding, subtracting, or substituting to find values for the unknown variables. Linear equations are foundational for solving more complex problems in algebra and everyday scenarios.

Variable substitution

Variable substitution is a powerful technique used to simplify the process of solving systems of equations. It involves replacing one variable in an equation with an expression containing another variable. This is often used to reduce the complexity of the equations involved.In our problem, variable substitution was used in the following way:

• We started with two equations: \( x + y = 100 \) and the simplified \( x + 2y = 140 \).
• To find the value of \( y \), we substituted one equation into the other. By subtracting the first equation from the second, we got an equation directly solved: \( y = 40 \).
• With the value of \( y \), we substituted it back into the first equation to find \( x \), resulting in \( x = 60 \).

This systematic substitution allows us to solve for one variable at a time efficiently, reducing potential mistakes and making complex problems more manageable.

Word problems in algebra

Word problems in algebra are sentences or paragraphs that present a math problem grounded in real-life scenarios, requiring you to translate language into mathematical equations. These problems help students connect math to everyday life, enhancing understanding and application skills. In this particular problem, we were tasked with figuring out the number of five-dollar and ten-dollar bills, given a certain total in cash and number of bills. This required:

• Identifying what the problem is asking (the number of each type of bill).
• Turning the given information into equations: one for the total number of bills and another for the financial total.
• Solving these equations step-by-step as per algebraic principles.

This approach makes algebra more relatable and shows its practical applicability in everyday situations, like budgeting or understanding monetary exchanges. It also improves problem-solving and critical thinking skills, which are invaluable in academic and real-world contexts.