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Fractions 1: Adding and Subtracting
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Simplify all fractions: 3/6 becomes 1/2
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Fractions illustrate an important concept, and as such are typically taught fairly early. But the process of actually using them in mathematic operations is relatively difficult. It strikes me as being a little reminiscent of adding numbers from arbitrary bases together.

This section focuses on addition and subtraction. There are two ways of going about performing operations with fractions. The first way, which is taught in school, is to get matching denominators (the number at the bottom of a fraction), multiplying the numerators (the top number) by the opposing denominator. The section below will show this process in detail. The other way is to convert the problems into decimals and figure out how to arrive back at a fraction after the fact. This is how I've often done it, and in fact, how this computer program generates the correct solutions for these problems.

Adding Fractions

First, find a common denominator. This is a number that each number can be multiplied to make. This can always be done by multiplying the two numbers together, but there may be other options that are smaller. In this case, 6 and 8 can both be multiplied to make 24.

We multiply 5 by 3, because it takes multiplying 8 by 3 to make 24. A learner might suppose that this process is changing the numbers. They should always remember that a fraction represents the value that is made by division, dividing the top by the bottom . 1 ÷ 2, 3 ÷ 6, and 100 ÷ 200 all equal 0.5. (This is why I feel it is helpful to discuss decimals before fractions.)

With the numbers adjusted, we add the numerators together.

When the numerator exceeds the denominator, it becomes a whole number.

Finally, simplify the fraction if possible, making it the smallest equivalent fraction. In some cases, this can be obtained by dividing the denominator by the numerator. The new denominator is the result of that division; the new numerator is 1. However, there are some situations where this trick doesn't work, yet the fraction can still be simplified. Some recommend breaking each number down into its component 'factors', a topic I haven't discussed here yet, and cancelling the overlapping factors out. In any case, this process requires experience, intuition, and sometimes trial and error.

For the record, this program finds simplified fractions by brute force: it converts the fraction to a decimal, then increments an integer, starting at 2, and multiplies it by the decimal, until the product of the two is a whole number. Then the integer is the denominator and the product is the numerator. This method will always eventually find the lowest simplified fraction, but it is too laborious for human use.

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