## 10 Questions

What does the numerator represent in a fraction?

The number of equal parts being considered

Which term is used to describe fractions that may look different but express the same value?

Equivalent fractions

What must be done to find equivalent fractions?

Multiply both the numerator and denominator by the same nonzero integer

In which operation are you likely to encounter a common denominator when working with fractions?

Adding fractions

What is compared when comparing fractions?

Both numerators and denominators

When adding fractions with different denominators, what is the first step to take?

Find the least common multiple (LCM) of the denominators

In subtracting fractions with the same denominator, what operation should be applied to one of the fractions?

Subtraction

To compare fractions easily, what should be done if the denominators differ?

Find the least common multiple (LCM) of the denominators

What is the correct process for subtracting fractions when they have different denominators?

Find the least common multiple (LCM) of the denominators and adjust the numerators accordingly

When comparing fractions, why is it important to find the greatest common divisor (GCD) of the denominators?

To facilitate converting higher denominators to lower ones for accurate comparison

## Study Notes

## Exploring Fractions

Fractions are a fundamental part of mathematics that help us represent portions or parts of a whole. They allow us to deal with quantities more finely divided than just integers. Let's delve into this crucial concept by examining its key facets—introduction to fractions, equivalent fractions, adding fractions, subtracting fractions, and comparing them.

### Understanding Fractions

A fraction is made up of two numbers separated by a horizontal line called the *bar* or *vinculum*. For example ( \frac{3}{5} ) represents three fifths of something, where ( 3 ) is known as the numerator and ( 5 ) is the denominator. A fraction tells you how many equal parts out of the total number denoted by the denominator make up the given portion specified by the numerator.

### Equivalent Fractions

Two different-looking fractions can still express the same value if they have the same numerical worth. These are referred to as *equivalent fractions*, like ( \frac{1}{3} ), ( \frac{2}{6} ) (or simply ( \frac{1}{2} )), and ( \frac{3}{9} ) (which we also recognize as ( \frac{1}{3} )). To find equivalent fractions, one must multiply both the numerator and denominator by the same nonzero integer, while ensuring that both values remain positive when dividing the original numerators and denominators.

For instance, ( \frac{1}{2} = \frac{5}{10}= \frac{3}{6}= \ldots), since they all result from multiplying either the top or bottom by ( 2,\ 5,\ 3 ), respectively. In each case, these operations maintain the ratio between the numerator and denominator, yielding mathematically identical results.

### Adding Fractions

When adding fractions having the same denominator, the process is straightforward. One merely adds their corresponding numerators without altering the common denominator. If the denominators differ, first convert those fractions to a common denominator before performing addition. This step involves finding equivalent fractions.

Suppose we wish to combine ( \frac{2}{3}+\frac{1}{4}=\frac{8+3}{12} =\frac{11}{12}). First, identify the least common multiple (LCM) of the two denominators ((3\times 4=12)). Then rewrite the second fraction using equivalence: ( \frac{1}{4} = \frac{3}{12} ). Next, sum the new numerators (in this case ( \frac{2+3}{12} )). Lastly, reduce the resulting expression, if possible ((\frac{5}{12}\rightarrow\frac{11}{12})).

### Subtracting Fractions

Subtraction follows similar principles to addition, though some adjustments might need to be made based on whether the denominators are alike or unlike. When dealing with unlike denominators, proceed as described above for adding them. However, when subtracting fractions sharing the same denominator, change one of the operands to its opposite (i.e., ( -\text{operand} )) before applying the rule mentioned earlier.

To illustrate: ( \frac{4}{7}-\frac{1}{7}=\frac{4}{7}- \underbrace{\frac{-1}{7}}_{\frac{7(-1)}{7}})=(\frac{4+1}{7}=\frac{5}{7})

### Comparing Fractions

Comparisons involve determining which of two fractions has a larger or smaller value. As with addition and subtraction, fractions may either have the same or distinct denominators. By locating the greatest common divisor (GCD) among the denominators (( GCD(x,y) ) ) and converting any higher denominators to lower ones through division, we simplify comparison tasks.

If the GCD appears only once within the denominators, compare the ratios directly; otherwise, divide both the numerators and denominators by the GCD before making the comparison. With this methodology, it becomes easier to ascertain relationships such as ( \frac{3}{4}>\frac{1}{2} > \frac{1}{3} ).

Delve into the fundamental concept of fractions by exploring key aspects such as introduction to fractions, equivalent fractions, adding fractions, subtracting fractions, and comparing them. Learn how fractions are represented and how to perform operations like addition, subtraction, and comparison to gain a better understanding of this essential mathematical concept.

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