Algebra in Words: A Guide of Hints, Strategies and Simple Explanations (2014)

FRACTIONS

It is important you know how long division translates to fraction form, since fractions are a form of division, and you probably learned long division first. Also, you will revisit long division later when doing (long) division of polynomials (not covered in this book). It is also important to get the proper terminology down, which is often overlooked. An example of a long division setup is below, with the words in the proper places.

The way you would say this is: “The dividend divided by the divisor equals the quotient.”

Sometimes people use the word “in” when describing division. In that case, it would be

“How many times does the divisor go into the dividend?”

Answer: The divisor goes into the dividend (the) quotient number of times.

Consider the example:

which is the same as:

In this case, you might ask:

“How many times does two go into six?” In this case, you are saying:

“How many times does the divisor two go into the dividend six?”

Answer: Two goes into six 3 times. Which is the same as: “Six divided by two equals 3.”

Notice how the divisor and dividend translate into a fraction and long division, and vice versa:

The dividend is the numerator. The divisor is the denominator. And the denominator goes into the numerator.

Procedure for Adding & Subtracting Fractions

1) Find the LCD.

2) Convert all fractions to like-fractions (unless a fraction is already in correct form) by multiplying the numerator and denominator of each fraction by the missing factor which will make the current denominator the LCD.

3) To the right of the equal sign, write the LCD in the denominator and perform the operations (addition and subtraction) of the newly converted numerators from the left in the new numerator, over the LCD, on the right of the equal sign.

4) Simplify the fraction completely.

·   Simplify the numerator, if possible, by combining like-terms, then

·   factor if possible, and then

·   reduce the fraction, if possible.

Multiplying Fractions

Start by attempting to cross-cancel common factors. Then, separately, multiply all the numerators together, and multiply the denominators together. Then evaluate what you have. It is often taught that fractions, once combined, should be expressed in lowest terms (also called reduced or simplified form). If you properly cross-cancelled before multiplying, your answer will come out in lowest terms. But sometimes students either forget to do this step, or just miss a set of common factors to cancel out. If you forget, that’s ok, your answer won’t come out wrong, but the numbers will be bigger and you will have to continue to factor. Evaluate your answer and look for a (greatest) common factor to cancel out at the end. To recap, you can either:

·   Look for common factors in the numerator and denominator first, prior to multiplying, and cancel them out (this is called cross cancelling) – which results in a reduced, more manageable fraction, or you can

·   Do the multiplication first, then factor the numerator and denominator, and cancel out common factors last.

The truth is, you can do it either way, and sometimes you just end up doing a mix of both to achieve the correct, reduced form of the fraction, which is fine and normal. But ideally, it is better to cross cancel first. For more on this, see: Factoring and: What is a Factor? Also, for more on cross cancelling, see: Cross Multiplication vs. Cross Cancelling.

Dividing Fractions

Dividing fractions is much different than multiplying fractions. Actually, only the first major step is different. Then it becomes multiplying fractions. In short, you multiply the first fraction times the reciprocals of the fractions being divided. But here is a more specific step-by-step procedure, followed by some comments.

1.  Simplify numerators and denominators separately, and each fraction separately.

2. Keep the first (left-most) fraction the same (meaning: do not invert it… actually, you should re-write it as is on the next line down).

3. Invert (flip upside-down) all fractions that are to be divided (they will have a division sign to the left of them at first). Once fractions are inverted (they are now called reciprocals)…

4. Change what were division signs to multiplication signs (a dot, or put each fraction in parentheses). Many students overlook this simple step.

5. Now multiply all fractions, using the procedure for multiplying fractions.

Note 1: Just to be clear, when dividing a string of fractions, keep the first one on the left the same and flip each remaining fraction upside down, before multiplying. When a fraction is flipped upside down, it is called thereciprocal. It is also referred to as the “inverse fraction” or simply the “inverse.”)

Note 2: Do not attempt to cross cancel factors before flipping the fractions and multiplying. Save cross cancelling until after the fractions are flipped and the division signs are changed to multiplication signs.

Note 3: If you attempt to divide numerators across the top and divide denominators across the bottom (in the way you would do when multiplying fractions), you will notice… it works! However… you are not encouraged to do it that way for one simple reason: it can get very complicated along the way, giving you strange fractions to manage, and many places to make a mistake. For this reason, you are highly encouraged to closely follow the procedure of flipping, then multiplying. It’s easier, and if nothing else, it is much faster.