# Arithmetic Operations: Fractions, Decimals & Integers

Instructor: Babita Kuruvilla

Babita has an electrical engineering degree and has taught engineering students and college students preparing for medical and dental college admissions tests.

In this lesson, we'll learn how to express numbers in integer, fraction, and decimal formats and how to do the four basic arithmetic operations in each of these formats.

## Counting Apples

While at the supermarket, you buy five Gala, three Fuji, and seven green apples. How many apples did you buy in total? Because we are asked for the total number of apples, all we need to do is add up the counts of each type of apples.

Total number of apples = 5 Gala apples + 3 Fuji apples + 7 Green apples

Total number of apples = 5 + 3 + 7 = 15

## Counting Wholes and Parts

In our daily lives, we come across many scenarios where we need to quickly calculate numbers. It could be counting the number of oranges in the basket, calculating tips at a restaurant, or measuring the sides of a box and calculating its volume.

These numbers could be whole items or part of an item. We use integers to express a whole item, like an apple, and fractions or decimals to write part of an item, like half an apple.

## Integers

When we use whole apples, our counts are given in whole numbers (5 apples, 6 apples, etc). Whenever we express a number using whole numbers, we call those numbers integers.

Integers can be any real whole number, including zero. Examples of integer numbers are ...,-2, -1, 0, 1, 2,...

### Arithmetic Operations Using Integers

We count and add the number of individual items.

ex: (3 apples) + (2 apples) = 5 apples

#### Subtraction

We subtract a certain number from the total.

ex: (5 apples) - (3 apples) = 2 apples

#### Multiplication

We calculate total number based on multiples of an item.

ex: (3 apples) * (2 groups) = 6 apples

#### Division

We separate items into smaller groups.

ex: (6 apples) / (group of 3 apples) = 2 groups

## Fractions

A fraction is a number used to express parts of an item. The top number (numerator) tells us how many 'parts' we have out of a 'whole' number (denominator). So, ½ means that we have 'one' out of 'two' halves.

Just like with integers, we do arithmetic operations using fractions.

Convert the denominator of the fractions to the lowest common denominator before adding.

ex: (½ apple) + (¼ apple) = ¾ apple

#### Subtraction

Convert the denominator of the fractions to the lowest common denominator before subtracting.

ex: (1 apple) - (½ apple) = ½ apple

#### Multiplication

Multiply the numerators to get the resultant numerator and multiply the denominators to get the resultant denominator.

ex: (½ apple) * (½ apple) = ¼ apple

#### Division

Invert the bottom fraction and then multiply it with the top fraction.

ex: (½ apple) / (½ apple) = 1

### Mixed Numbers

Let's say that you had two apples and you ate one half of one of those apples. Now, you have left one full apple and one half apple. To write this in mathematical form, we cannot use just integers; we need to use fractions. You have 1½ apples.

Note that 1½ is a mixed number that has an integer (1) and a fraction (½). A mixed number gives us the full picture of the situation in one glance, but in mathematical notation.

### Improper Fractions

We can convert mixed numbers into improper fractions which have a larger numerator (the top number) than the denominator (the bottom number).

Let's see what the mixed number 1½ looks like written in improper fractional format.

#### Step 1 - Convert the integer into a fraction.

To convert the number '1' into a fraction, first we need to figure out the denominator. Since the fraction ½ has a denominator of 2, the improper fraction will also have a denominator of 2.

Now, let's setup the equation to find the numerator.

There you have it! In fractional form, '1' equals 2/2.

#### Step 2 - Add the fractional form of the integer to the fraction in the mixed number.

We find that 1½ is the same as 3/2!

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