GED Mathematical Reasoning: Understanding Fractions

There are two types of fractions: proper fractions and improper fractions. For now, we will focus on proper fractions, which represent numbers smaller than one. An example of a proper fraction is \frac{1}{2}.

The top number of a fraction is called the numerator. The numerator represents the “part”. The bottom number of a fraction is called the denominator. The denominator represents the “whole”. The whole may be one item or a group of items.

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To remember which term is which, you may find this helpful: The word “denominator” begins with the letter “d” just like the word “down” begins with the letter “d.” And the denominator is written down below the numerator.

To illustrate how a fraction represents part of a whole, think of a pizza containing 6 equal slices or parts. If I ate 1 slice of pizza for lunch, I can describe the amount of pizza I ate using the fraction \frac{1}{6} where the 1 in the numerator represents the one “part” or one slice of pizza that I ate and the 6 in the denominator represents the group of 6 slices or “parts” that made up the “whole” pizza.

To describe the amount of pizza that’s left over, we can use the fraction \frac{5}{6} where the 5 in the numerator represents the 5 “parts” or 5 slices of pizza that are left and the 6 in the denominator represents the group of 6 slices or “parts” that made up the “whole” pizza.math 132

 

Improper fraction and mixed numbers

Improper fractions represent numbers greater than or equal to one. In an improper fraction, the numerator will be greater than or equal to the denominator. Examples of improper fractions include \frac{5}{5} and \frac{7}{6}.

The fraction \frac{5}{5} represents 5 parts out of 5 total parts. The value of this fraction is 1. Thinking of it in terms of pizza, it’s as if we had a pizza containing 5 equal slices and we ate all 5 slices. In other words, we ate one whole pizza.

There’s another way of thinking of this and it’s an important note. The fraction bar is another way to denote division. So the fraction  can be thought of as 5 \div 5, which is equal to 1.

The fraction \frac{7}{6} represents one whole divided into 6 pieces. The numerator of 7 refers to 7 pieces, which is one piece more than the whole. So the fraction \frac{7}{6} represents one piece greater than one whole.

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A mixed number is another way to write an improper fraction. A mixed number contains a whole number and a proper fraction. An example of a mixed number is 1\frac{1}{6}. Here, the whole number part is 1 and the proper fraction part is \frac{1}{6}. This stands for one whole object plus \frac{1}{6} of a second object.

Check out this figure – notice how 1\frac{1}{6} and \frac{7}{6} represent the same amount:

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By the way, going between improper fractions and mixed numbers is a fairly simple process.

To go from a mixed number to an improper fraction, multiply the denominator by the whole number part and then add the numerator. That result is the numerator of the improper fraction. For the denominator, keep the same denominator. So to change 1\frac{2}{3} into an improper fraction, we would multiply 3 times 1 and add 2 for a result of 5. We would place a 5 in the numerator of the improper fraction and keep the denominator of 3 for a result of \frac{5}{3}.

To go from an improper fraction to a mixed number, divide the denominator into the numerator. That becomes the whole number part of the mixed number. For the fraction part, place the remainder in the numerator and keep the same denominator. So to change \frac{5}{3} back into a mixed number, we would divide 5 by 3, which is 1 with a remainder of 2. The 1 becomes the whole number part of our mixed number, the remainder of 2 becomes the numerator of our fraction part, and we keep the denominator of 3, giving us 1\frac{2}{3}.

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Example 1

Which fraction represents the shaded area?

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  1. \frac{7}{2}
  2. \frac{5}{7}
  3. \frac{2}{7}
  4. \frac{2}{5}

 

How many parts of the pie are shaded? The answer is 2, so we will place a 2 in the numerator of the fraction. Since there are seven parts that make up the “whole” pie, we will place a 7 in the denominator. Putting this all together, the fraction that represents the shaded region is answer (3), \frac{2}{7}.

 

Example 2

Express the fraction \frac{18}{30} in lowest terms.

  1. \frac{1}{3}
  2. \frac{18}{30}
  3. \frac{9}{15}
  4. \frac{3}{5}

 

Before we complete this example, let’s talk about what it means to write a fraction in lowest terms. Formally, a fraction is said to be written in lowest terms when the numerator and denominator have no factors in common except for 1. Recall that factors are the numbers that are multiplied together to give a product. For example, since 2 times 3 equals 6, the numbers 2 and 3 are called factors.

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Informally, to write a fraction in lowest terms – which is also known as reducing or simplifying a fraction – we’ll ask ourselves: What is the largest number that divides evenly into both the numerator and denominator? So looking at our example, what’s the largest number that divides evenly into both 18 and 30? The answer is 6. So to write \frac{18}{30} in lowest terms, we will divide both the numerator and denominator by 6. Therefore, the answer is letter (4).

\frac{18 \div 6}{30 \div 6} = \frac{3}{5}

There is another approach to simplifying fractions, which entails that we use this same general idea but reduce using more than one step.

Given our fraction \frac{18}{30}  you may have initially thought to divide the numerator and denominator by 2. Doing so results in \frac{9}{15}.

\frac{18 \div 2}{30 \div 2} = \frac{9}{15}

While \frac{9}{15} is simplified, it is not written in the lowest terms possible. So let’s continue by asking ourselves: What number divides evenly into both 9 and 15? The answer is 3. So we’ll reduce further by dividing both the numerator and denominator by 3. The result is \frac{3}{5}.

\frac{9 \div 3}{15 \div 3} = \frac{3}{5}

You can see we arrived at the same answer, we just did so in two steps – which is totally fine and the preferred method for many because it allows us to think in terms of smaller numbers. When you reduce a fraction to lowest terms and before you state your final answer, always ask yourself: Can this fraction be reduced any further?

By the way, the fractions:  \frac{18}{30} and \frac{9}{15} and \frac{3}{5} may all be referred to as “equivalent fractions”. Equivalent fractions look different but have the same value. You can arrive at equivalent fractions by either multiplying the numerator and denominator by the same number OR dividing the numerator and denominator by the same number.

Here is another example of equivalent fractions: \frac{1}{3} and \frac{2}{6}. Visually, notice in these figures representing \frac{2}{6} and \frac{1}{3}, respectively, that each shaded area corresponding to the part and the whole represent the same amount of area. So while the fractions look different, they represent the same value or amount.

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Example 3

Which improper fraction represents the shaded area in the figure?

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  1. 1\frac{5}{6}
  2. 2\frac{2}{3}
  3. 1\frac{2}{3}
  4. 2\frac{5}{6}

 

In the first figure, 3 out of 3 parts are shaded, representing one whole. In the second figure, 2 out of 3 parts are shaded, representing \frac{2}{3}. Therefore, the figure shown represents one whole object plus \frac{2}{3} of an object, giving us a final answer of 1\frac{2}{3} – answer (3).

 

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