Amber Has $ 1.50 \$1.50 $1.50 In Pennies And Dimes. If There Are Five Times As Many Pennies As Dimes, How Many Pennies And Dimes Does Amber Have?First, Complete The Equations Below, Where P P P Stands For Pennies And D D D Stands For

Mar 9, 2025 by ADMIN 234 views

**Solving the Penny and Dime Problem**

Introduction

In this problem, we are given that Amber has a total of $\$1.50$ in pennies and dimes. We also know that there are five times as many pennies as dimes. Our goal is to find out how many pennies and dimes Amber has.

Setting Up the Equations

Let's denote the number of pennies as $p$ and the number of dimes as $d$ . Since there are five times as many pennies as dimes, we can write the equation:

p = 5d

We also know that the total value of the coins is $\$1.50$ . Since each penny is worth $\$0.01$ and each dime is worth $\$0.10$ , we can write the equation:

0.01p + 0.10d = 1.50

Solving the System of Equations

We can substitute the first equation into the second equation to get:

0.01(5d) + 0.10d = 1.50

Simplifying the equation, we get:

0.05d + 0.10d = 1.50

Combine like terms:

0.15d = 1.50

Divide both sides by $0.15$ :

d = 10

Now that we have found the value of $d$ , we can find the value of $p$ by substituting $d$ into the first equation:

p = 5d

p = 5(10)

p = 50

Conclusion

Therefore, Amber has $\boxed{50}$ pennies and $\boxed{10}$ dimes.

Discussion

This problem is a classic example of a system of linear equations. We were given two equations and two variables, and we were able to solve for both variables by using substitution and elimination methods. This type of problem is commonly seen in algebra and mathematics.

Real-World Applications

This problem may seem simple, but it has real-world applications. For example, if you are a store owner and you want to know how many coins you have in your register, you can use this type of problem to solve it. You can also use this type of problem to solve other types of problems, such as finding the number of items in a basket or the number of people in a room.

Tips and Tricks

When solving this type of problem, make sure to read the problem carefully and understand what is being asked. Also, make sure to use the correct equations and variables. Finally, make sure to check your work by plugging in the values you found into the original equations.

Practice Problems

Here are a few practice problems to help you practice solving systems of linear equations:

A bakery has a total of $\$2.50$ in cookies and cakes. If there are three times as many cookies as cakes, how many cookies and cakes does the bakery have?
A store has a total of $\$1.25$ in pencils and pens. If there are four times as many pencils as pens, how many pencils and pens does the store have?
A restaurant has a total of $\$3.00$ in burgers and fries. If there are two times as many burgers as fries, how many burgers and fries does the restaurant have?

Answer Key

The bakery has $\boxed{15}$ cookies and $\boxed{5}$ cakes.
The store has $\boxed{20}$ pencils and $\boxed{5}$ pens.
The restaurant has $\boxed{12}$ burgers and $\boxed{6}$ fries.
Frequently Asked Questions (FAQs) =====================================

Q: What is the difference between a penny and a dime?

A: A penny is a coin worth $\$0.01$ , while a dime is a coin worth $\$0.10$ .

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use the substitution method or the elimination method. The substitution method involves substituting one equation into the other, while the elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: What is the purpose of the variable $p$ in the problem?

A: The variable $p$ represents the number of pennies in the problem.

Q: What is the purpose of the variable $d$ in the problem?

A: The variable $d$ represents the number of dimes in the problem.

Q: How do I know which equation to use first?

A: You can choose which equation to use first based on which variable you want to solve for. In this problem, we chose to use the equation $p = 5d$ first because it was easier to substitute into the other equation.

Q: What if I get stuck on a problem?

A: If you get stuck on a problem, try to break it down into smaller steps. Read the problem carefully and make sure you understand what is being asked. You can also try to draw a diagram or use a different method to solve the problem.

Q: Can I use this method to solve other types of problems?

A: Yes, this method can be used to solve other types of problems that involve systems of linear equations. You can also use this method to solve problems that involve quadratic equations or other types of equations.

Q: How do I check my work?

A: To check your work, plug in the values you found into the original equations. If the equations are true, then your solution is correct.

Q: What if I make a mistake?

A: If you make a mistake, don't worry! Just go back and re-read the problem and try again. You can also ask for help from a teacher or tutor.

Q: Can I use a calculator to solve this problem?

A: Yes, you can use a calculator to solve this problem. However, make sure to check your work by plugging in the values you found into the original equations.

Q: How do I know if my solution is correct?

A: To know if your solution is correct, plug in the values you found into the original equations. If the equations are true, then your solution is correct.

Q: Can I use this method to solve problems with more than two variables?

A: Yes, you can use this method to solve problems with more than two variables. However, the method may become more complex and require more steps.

Q: How do I know which method to use?

A: You can choose which method to use based on which variables you want to solve for and which equations are easier to work with. In this problem, we chose to use the substitution method because it was easier to substitute into the other equation.

Q: Can I use this method to solve problems with fractions or decimals?

A: Yes, you can use this method to solve problems with fractions or decimals. However, make sure to simplify the fractions or decimals before solving the problem.

Q: How do I know if my solution is reasonable?

A: To know if your solution is reasonable, check if the values you found make sense in the context of the problem. For example, if you are solving a problem about the number of people in a room, make sure the number of people is reasonable.