Why are they Called Rational Numbers?

The numbers we usually call fractions should be called rational numbers.

Look at the first five letters in the word rational. They spell ratio and that's exactly what rational numbers are. A rational number is a fraction. It is composed of one WHOLE NUMBER divided by (over) another WHOLE NUMBER.

We represent fractions by where a, the numerator
and b, the denominator are integers and b ! 0.

In our number system, division by zero is undefined so the definition of a rational number must state that the denominator is not equal to zero. And since fractions are made up of integers, they can be either positive or negative just like integers can.

Integers are Fractions Too!

Any integer -- positive or negative whole number such as 5 -- is a fraction too because it
means 5 ÷ 1 or . Remember this when doing a question such as: .
It says to multiply by 2, and since 2 is really , we multiply the numerator by 2 and the denominator by 1 like this:

.

The denominator of an integer is 1.

Where Does the Negative Sign Belong?

QUESTION:

ANSWER: Yes it is.

It doesn't matter where we place the negative sign -- as long as there is only one of them. Usually, in an answer, if it is a negative fraction, we put the negative sign in the numerator.

If we have because a minus times a minus = a plus.

.

Multiplying Fractions

When we multiply fractions, the result is a fraction with the product of the numerators over the product of the denominators. We just multiply the tops and multiply the bottoms like this:

Using Equivalent Fractions to Reduce or Cancel

Reducing, often called canceling is one of the least understood operations in math.

The most common error is reducing across addition or subtraction.

For example, many students will incorrectly do this:

because they reduce or cancel the 3 with the 6.

But it's obvious that the result is not right since
.

So what is reducing or canceling?

Reducing or canceling, is a technique we use to express fractions in lowest terms We rewrite a fraction as an equivalent value because we divide by 1.

In the example above, we eliminated , because it equals 1.

In this example, we eliminate

If we "cancel" before we multiply, we'll get our answer in lowest terms. Also, the numbers we have to multiply will be smaller because we will eliminate common factors. This example shows that we can simplify and reduce fractions easily, when both numerator and denominator have been factored. Remember to reduce by dividing numerator and denominator by their common factors.

Examples:

1) , we divide top and bottom by the common factor 2.

2)

,

we divide 2 (top) and 4 (bottom) by their common factor 2.
we divide 15 (top) and 5 (bottom) by their common factor 5.

3) Now a mixed number multiplied by a fraction.
First, we change the mixed number into an improper fraction. Then we cancel and multiply.

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

Practice Exercise 1: Multiplying Fractions

1) Multiply

a)

b)

c)

2) Factor the numbers, reduce (cancel), then multiply.

a)

b)

c)

( solutions )

Dividing Fractions

When we divide a number by 3, it's exactly the same as taking 1/3 of it.

Since we know that division is the inverse operation of multiplication, to divide by any number is exactly the same as multiplying by its reciprocal (the flip). That's how we do division by all numbers so why not fractions?

To divide by a fraction, invert it, then multiply.

Examples

1)

b)

If there are mixed numbers in the question, we turn them into improper fractions, invert the divisor, reduce where we can, then multiply as usual. We must be careful to reduce or cancel after we invert the divisor.

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

Practice Exercise 2: Dividing Fractions

1) Divide

a)

b)

c)

2) Change mixed numbers into improper fractions, then divide

a)

b)

c)

( solutions )

Solutions

Practice Exercise 1: Multiplying Fractions

1) Multiply

a)

b)

c)

2) Factor the numbers, reduce (cancel), then multiply.

a)

b)

c)

.

Practice Exercise 2: Dividing Fractions

1) Divide

a)

b)

c)

2) Change mixed numbers into improper fractions, then divide

a)	b)
c)

( Primary MathRoom Index )