In Classical Arithmetic, we have learned how to write a basic proof using the axioms and definitions of Arithmetic. In this lesson we will learn the formal way to solve a problem in Arithmetic. Please study this lesson carefully as it teaches a method to be used throughout the course. You will have eight problems to solve to pass this lesson.

## Problem Solving

When we studied proofs, we learned that the goal of a proof is demonstration–showing some proposition to be true in a way that leaves no room for doubt. The proof begins with a proposition that “is to be demonstrated” (*est demonstrandum*). We then go on to show that the proposition is related to other statements that are clearly true. We reason that if those statements are true, then the proposition to be demonstrated must also be true.

In this lesson, we are going to begin problem solving in a formal or orderly way. Problem solving does not begin with a proposition to be demonstrated, but with a problem to be solved (effected). It is our task to discover the solution to the problem and then provide the rules or steps one must take to solve the problem themselves.

We solve problems not by quickly answering a single example, but by writing the rules or precepts by which one may solve the given problem or any problem like it. The difficulty in this is deciding on which precepts to be followed. This requires us to use our knowledge of the axioms, rules and definitions of Arithmetic to determine the surest steps by which a problem may be solved. When we rightly discover the steps to solve the problem, and express them clearly, others can follow our instructions and solve these problems for themselves. After the problem has been solved and the steps given for its solution, we should be able to write a proof for our solution.

## Effecting a Problem

In Arithmetic, we call the solution to a problem its effection. To write out the effection of a problem, we use a method just as we used a method to write out the proof of a proposition. The method looks like this:

- State the problem.
- List each step as a Precept and number them in the order they should be followed.
- Summarize the solution and end with,
(Which was to be effected.) You can abbreviate this as*Quod Erat Efficiendum**Q.E.E.*

Let us look at an example.

First, the problem is stated:

Problem: Reduce heterogeneous fractions into homogeneous fractions of the same value.

Second, the effection, is given with rules or precepts by which the problem is to be solved. Once the effection has been completed, the effection is concluded with Q.E.E.. Notice that the problem does not consist of one specific example, but applies to any heterogeneous fractions. Your effection must then work for any such problem and any example that your effection doesn’t solve proves your effection to be false. Here is the effection for this problem, notice how each precept is listed in order:

Precept 1. Multiply all of the given denominators together for a new and common denominator.

Pre. 2. Multiply each Numerator into all the Denominators, except its own for new Numerators.

Pre. 3. Write the new and common Denominator under each of these new Numerators.

Each homogeneous fraction is equal to the given heterogeneous fraction from which its Numerator was formed, Q.E.E.

In this lesson, you are required to write the effection for eight problems that concern fractions. Therefore, you should make diligent use of all of your previous lessons and all of your axioms to help you solve these problems. You must follow the instructions and write out your effection as shown above. These are challenging problems to solve and we give you problem above to help you get started.

## Notes on Problems

**Problem 1. Reduce heterogeneous fractions into homogeneous fractions of the same value.**

This problem is quite simple because you already have the effection written above. What is important to note is that it will allow one to reduce ANY heterogeneous fractions into homogeneous fractions of the same value. Therefore the effection must supply precepts that apply to any heterogeneous fractions.

**Problem 2. Reduce a given integer b into a fraction of the same value whose denominator shall be a given quantity d.**

To solve this problem you must understand how to convert an integer into a fraction. Review memory question #140 and the section on “Adding and Subtracting Heterogeneous Fractions” in Lesson 17 for help.

**Problem 3. Reduce a mixed quantity b + x/z into an improper fraction of the same value.**

Review memory question #s 132, 134 and 140 along with the section in lesson 17 on improper fractions. It is important in this problem to understand how the integer in an improper fraction relates to the denominator in that fraction.

**Problem 4. Reduce an improper fraction ( bz + x ) / z to an integer or mixed fraction of the same value.**

This problem is asking the reverse of problem 3 above. Here, we’re starting with an improper fraction and trying to reduce it to an integer or mixed fraction. Here, the rules of division will help you. Review lesson 15 and consider how the fact that division is distributive can help you to solve this problem.

**PROBLEM 5. To add one given fraction to another given fraction.**

This may look easier than it is! Remember that these problems must be solved in a way that would apply to ANY fraction–proper, improper, mixed, heterogeneous, etc.. Therefore, make sure you include precepts for first establishing fractions that can be added easily from improper and/or heterogeneous fractions. Fractions cannot be mixed or heterogeneous when they are to be added! Review the rules for adding fractions in lesson 17.

**PROBLEM 6. To subtract a given fraction from another given fraction.**

The note for problem 5 applies to this problem as well.

**PROBLEM 7. To multiply a fraction b/c by a fraction d/p.**

Remember that the species b, c, d and p might represent any number. Explain the precepts clearly.

**PROBLEM 8. To divide a fraction bd/cp by a fraction d/p.**

Remember there are two different methods for dividing fractions and they are used in two different situations. Review the section in lesson 19 on Dividing Fractions and the section on Reciprocals.

## Summary

In this lesson, you are challenged to use what you know to discover what you don’t know. That’s problem solving! You will need to review all of your past lessons carefully and this exercise will test whether or not you have been studying carefully. In the future, we will continue to solve problems in this way, so time spent mastering the exercises is time well spent.

## Memory Work

Directions: The following questions help you to memorize the most important points of this lesson. Commit them perfectly to memory and have a parent or praeceptor quiz you to test your mastery before taking your lesson exam.

- What is an arithmetical Problem?
**An arithmetical problem is a problem to be effected by the rules, definitions and axioms of the art of arithmetic.**

- What is an effection or solution?
**An effection or solution is an explanation of precepts or steps by which a problem can be solved.**

- What is the meaning of the abbreviation Q.E.E.?
**The abbreviation Q.E.E. represents the phrase, “Quod Erat Efficiendum”, which means “Which was to be effected.” It is used to conclude the effection of a problem.**