Solving Number Problems Involving Quarters and Dimes: A Comprehensive Guide

Quarters and dimes are common coins in many currencies, and understanding their values and relationships can be a valuable skill in various real-life scenarios. This guide will walk you through solving a series of problems involving quarters and dimes using algebraic equations. We will also look at how these solutions can be applied in different contexts.

Example 1: Triple the Number of Quarters

The first example involves a more complex scenario where the number of quarters is three times the number of dimes, and the total value is given. This problem requires us to set up and solve a system of equations.

Problem Statement

Suppose you have three times as many quarters as dimes, and the total amount of money is $6.80. How many quarters and dimes do you have?

Solution

Let d be the number of dimes and q be the number of quarters.

According to the problem, we have the following relationships:

The number of quarters is three times the number of dimes: q 3d The total value of the dimes and quarters is $6.80. Since dimes are worth $0.10 and quarters are worth $0.25, we can express this as: 0.10d 0.25q 6.80

Now, we substitute the expression for q from the first equation into the second equation:

0.10d 0.25(3d) 6.80

Simplifying this, we get:

0.10d 0.75d 6.80

0.85d 6.80

Solving for d:

d 6.80 / 0.85 8

Now, we can find the number of quarters:

q 3d 3(8) 24

To verify, we check the total value:

Value of dimes: 8 * 0.10 0.80 Value of quarters: 24 * 0.25 6.00 Total: 0.80 6.00 6.80

Thus, the solution is correct: 8 dimes and 24 quarters.

Example 2: Six Times as Many Quarters

The second example involves a simpler setup where the number of quarters is six times the number of dimes, and the total amount of money is $8.

Problem Statement

If you have six times as many quarters as dimes, and the total amount of money is $8, how many quarters and dimes do you have?

Solution

Let d be the number of dimes and q be the number of quarters.

According to the problem, we have the following relationships:

The number of quarters is six times the number of dimes: q 6d The total value of the dimes and quarters is $8. Since dimes are worth $0.10 and quarters are worth $0.25, we can express this as: 0.10d 0.25q 8.00

Now, we substitute the expression for q from the first equation into the second equation:

0.10d 0.25(6d) 8.00

Simplifying this, we get:

0.10d 1.5d 8.00

1.60d 8.00

Solving for d:

d 8.00 / 1.60 5

Now, we can find the number of quarters:

q 6d 6(5) 30

To verify, we check the total value:

Value of dimes: 5 * 0.10 0.50 Value of quarters: 30 * 0.25 7.50 Total: 0.50 7.50 8.00

Thus, the solution is correct: 5 dimes and 30 quarters.

Real-Life Applications

Understanding how to solve number problems involving quarters and dimes can be useful in various scenarios, such as balancing a piggy bank, calculating change, or solving puzzles. For instance, Parker's scenario involves balancing a piggy bank with a specific total value and a given relationship between dimes and quarters.

Parker's Scenario

Parker has 19 quarters and 23 dimes. Here's how I approached this problem:

First, take out the 4 extra dimes, which leaves a total of $6.65. Knowing there are the same number of dimes and quarters in the remaining $6.65, we divide 6.65 by 0.35 (a dime plus a quarter) to get 19 more dimes and quarters.

Therefore, Parker has 19 quarters for $4.75 and 23 dimes for $2.30, totaling $7.05.

Original Answer Correction

The original answer with the problem statement "four more dimes than quarters" would be different. The corrected solution, using the correct algebraic method, is as follows:

Definition of Variables

Note: I am counting in cents to avoid using decimals. So, the goal is to find how the piggy bank contains 705 cents.

D number of dimes Q number of quarters 10D value of dimes (in cents) 25Q value of quarters (in cents)

Equations from Stated Problem

25Q 10D 705 (total value in cents) Q D - 4 (four more dimes than quarters)

Calculation

Substitute the expression for Q from the second equation into the first equation:

25(D - 4) 10D 705

25D - 100 10D 705

35D - 100 705

35D 805

D 805 / 35 23

Calculate Q:

Q D - 4 23 - 4 19

To verify, we check the total value:

25Q 10D 705

25(19) 10(23) 705

475 230 705

19 23 - 4

Therefore, the correct solution is: 19 quarters and 23 dimes, with the total value of $7.05.

These examples demonstrate the importance of carefully reading the problem and setting up the equations correctly. Whether it's balancing a piggy bank or solving algebraic problems, understanding how to handle quarters and dimes effectively can be a valuable skill.