## Key Concepts

- Fractions
- Divisions
- Relating fractions and divisions

## Division

Division is one the four basic mathematical operations, the other three being addition, subtraction, and multiplication. In simpler words, **division can be defined as the splitting of a large group into equal smaller groups or method of grouping objects equally in groups.**

**Example:** Arranging students in rows equally.

### Symbols used for division

There are two basic divide symbols that represent division. They are / or ÷.

**Example:** 4/3 or 4÷3.

### Terms involved in division

While doing divisions, we come across terms such as dividend, divisor, quotient and remainder. These terms can be understood by going through the image given below.

**Note:**

- If any number is divided by 1, gives the same answer as the dividend (the quotient equals the dividend). For example, 4÷1 = 4.
- A number cannot be divided by 0 and the result is thus undefined. For example, 15 ÷ 0 = undefined.
- When the dividend equals the divisor, then the answer is always 1. For example, 15 ÷ 15 = 1.
- Anytime you are converting a smaller unit of measure to a larger unit of measure we need to divide by a conversion factor.

### How is division done?

Let us understand the process of the long division with the help of an example. For example, we are to divide 435 by 4. Hence, we need 435 ÷ 4.

Here, the first digit is 4 and it is equal to the divisor. So, 4 ÷ 4 =1; 1 is written on top. The result 4 × 1 = 4 is subtracted from the digit and 0 is written below.

- Next drop the second digit or the digit in the ten’s place beside 0. Since, 03 is less than 4, we cannot divide this number. Hence, we write a 0 on the top and drop the digit on the unit place beside 3.
- Now, we have 35. As 35 > 4, we can divide this number and to divide 35 ÷ 4, write 8 on top.
- Subtract the result 4 × 8 = 32 from 35 and write 3.
- 3 is known as the
**remainder**and 108 is called the**quotient**.

## Fractions

The word fraction is derived from the latin word “fractio” which means “to break”. In India, fractions were first written with one number over the other (numerator and denominator), without a line. It was the Arabs who introduced the line which seperates the numerator from the denominator.

### What are fractions?

In mathematics, fractions are represented as a numerical value, which can be defined as the parts of a whole. Let us understand this concept using an example.

Example: There is pizza which is made into 8 equal parts. Suppose you have taken a piece, what is the part chosen by you?

**Solution:**** **It means 1 in 8 equal parts. It can be also read as:

- One-eighth
- 1 by 8.

### Parts of a fraction

All the fractions consist of a numerator and a denominator.

- The denominator of the fraction tells us, how many parts is the whole been divided into ( 8 parts in the above example)
- The numerator of the fractions tell about how many parts are selected or represented. ( 1 part is taken)

Representation of fraction for 1 in 8 equal parts is 1/8.

### Types of fractions

There are different types of fractions that can be differentiated by using the numerator and the denominator.

**Proper fractions:** If the numerator of the fraction is less than the denominator.

**Example:** 5/7, 3/8, 2/5 etc.

**Improper fractions:** If the numerator of the fraction is greater than the denominator.

**Example: **8/4, 10/8, 12/5 etc.

**Unit fractions:** Fractions with numerators as 1 are known as unit fractions.

**Example:** 1/3, 1/5 etc.

**Mixed fractions:** A mixed fraction is a mixture of whole and proper fractions.

**Example:**_{ }2 1/5, 4 2/3, 3 5/6, etc.

**Like fractions:** Fractions with the same denominators are defined as like fractions.

**Example:** 6/5, 8/5 etc.

**Unlike fractions:** Fractions with different denominators are defined as unlike fractions.

**Example: **2/3, 4/5 etc.

### How are fractions related to divisions

Writing a division expression for fractions:

**Example 1:** Express a fraction a/b as division.

**Solution:**** **Numerator of the fraction ‘a’ is considered as dividend, and the denominator of the fraction **‘b’ **is taken as the divisor.

**Step 1**: Dividend a is written as the first term followed by the symbol of division and then divisor b is written after the symbol. ** a ÷ b**

Writing a fraction expression for division:

**Example 1:** Express **c ÷ d** as a fraction.

**Solution:**** **The first term **‘c’** before the division is considered as numerator and the second term **‘d’ **after the division symbol is taken as the denominator.

**Step 1: **The two terms c and d are separated by using a ‘/ ‘symbol horizontally between them. 𝒄/𝒅

**9.1.1 Relating fractions to divisions using word problems **

**Example 1:**

How can 3 bars of chocolates be shared among 4 people equally?

**Solution:** Partition each bar into 4 equal parts. Each part is 1/4 of one bar.

**Step 1:**** **Each of the four persons picked up one color from each of the bars.

**Step 2**: So, 3 ÷ 4 = 3 ×1/4= 3/4

Therefore, each of the friend receives 3/4 of the one bar.

**Example 2:**** **How can we equally divide 5 apples among 8 kids?

**Solution:** Partition each apple into 8 equal parts. Each part is1/8 of one apple.

**Step 1:**** **Each of the 8 kids picked up one part from each of the apples.

All the 8 kids will be receiving the below 5 parts each.

**Step 2: **So, 5÷8 = 5 ×1/8= 5/8

Therefore, each of the kid receives 5/8 of one apple.

## Exercise

- Help Justin to divide 5 packs of milk into 7 cups.
- Three friends have decided to complete a marathon of 2 km equally. Find the distance covered by each of them.
- Share $7 among 10 friends equally.
- Divide 5 oranges among six soccer players equally.
- A team of 7 workers have constructed 3 sheds in 10 days. How much of a shed each of them built?
- John used 3 packages of spaghetti for 10 guests. How much did each guest have?
- Explain if 5/6 is equal to 6÷5 or not.
- If 5 burgers cost $3. What would be the cost of each burger?
- A 3-liter pitcher of juice was poured into 7 cups. How much juice was in each cup?
- If the weight of 4 sandwiches is 9 pounds. Find the weight of each sandwich.

### What have we learned

- Fractions
- Divisions
- Relate fractions and divisions