Introduction To Fractions

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Notes

Introduction to Fractions

A fraction is written as **a/b** where **a** (numerator) and **b** (denominator) are integers.
The **denominator** shows how many equal parts a whole is split into; the **numerator** shows how many parts you have.
If numerator < denominator, the fraction is less than 1; if numerator > denominator, it is greater than 1.
To shade a fraction of a diagram: divide total parts by denominator, then multiply by numerator.

Fractions of Amounts

Method 1: **Divide by denominator, multiply by numerator** – $e.g., \frac{2}{5}$ of $60: 60 \div 5 = 12, 12 \times 2 = 24$ .
Method 2: Convert fraction to **decimal** and multiply – $e.g., \frac{1}{4}$ of $16 = 0.25 \times 16 = 4$ .
Method 3: Write the amount as a fraction $(e.g., 60 = \frac{60}{1})$ and **multiply fractions** – $e.g., \frac{2}{3} \times \frac{60}{1} = \frac{120}{3} = 40$ .
For fractions of fractions, multiply the two fractions directly.

Equivalent Fractions

**Equivalent fractions** represent the same amount but are written differently $(e.g., \frac{1}{2} = \frac{6}{12})$ .
To find an equivalent fraction, **multiply numerator and denominator by the same number** $(e.g., \frac{5}{6} = \frac{10}{12} = \frac{15}{18})$ .
There are infinitely many equivalent fractions for any given fraction.

Simplifying Fractions

A **simplified fraction** uses the smallest possible integers $(e.g., \frac{12}{18} = \frac{2}{3})$ .
Simplify by **dividing numerator and denominator by a common factor** $(e.g., \frac{25}{45} \div \frac{5}{5} = \frac{5}{9})$ .
This is also called **cancelling** the fraction.

Mixed Numbers & Improper Fractions

A **mixed number** has an integer part and a fraction part $(e.g., 3 \frac{3}{4}$ means three and three quarters).
An **improper fraction** (top-heavy) has numerator > denominator $(e.g., \frac{15}{4})$ .
To convert mixed to improper: multiply whole number by denominator, add numerator, write over denominator $(e.g., 5 \frac{2}{7} = (5\times 7+2)/7 = \frac{37}{7})$ .
To convert improper to mixed: divide numerator by denominator; quotient is whole number, remainder is numerator over denominator $(e.g., \frac{22}{3} = 7 \frac{1}{3})$ .

Shading a Fraction of a Grid

Equivalent Fractions on a Number Line

Converting Mixed Number to Improper Fraction

Finding a Fraction of an Amount

Practice questions

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1.Write $\frac{60}{105}$ in its simplest form.
Easy
- A $\frac{4}{7}$
- B $\frac{12}{21}$
- C $\frac{6}{10}$
- D $\frac{3}{5}$
2.Write a fraction which is equivalent to 3/5.
Easy
- A $\frac{6}{10}$
- B $\frac{3}{10}$
- C $\frac{6}{5}$
- D $\frac{9}{10}$
3.Write $\frac{28}{140}$ as a fraction in its simplest form.
Easy
- A $\frac{1}{5}$
- B $\frac{2}{7}$
- C $\frac{7}{35}$
- D $\frac{4}{20}$
4.Shade five-eighths of this rectangle. [Diagram: a rectangle divided into 8 equal squares]
Easy
- AShade 5 squares
- BShade 3 squares
- CShade 8 squares
- DShade $\frac{5}{8}$ of a square
5.Write down a fraction that is equivalent to 7/12.
Easy
- A $\frac{14}{24}$
- B $\frac{7}{24}$
- C $\frac{14}{12}$
- D $\frac{21}{36}$
6.Find $\frac{1}{5}$ of $200.
Easy
- A$40
- B$50
- C$100
- D$20
7.There are 12 brass instruments in an orchestra. $\frac{1}{3}$ are trumpets, 3 are trombones and the remainder are horns. Find the number of horns.
Medium
- A5
- B4
- C3
- D6
8.Records show that $\frac{3}{14}$ of the population of a particular type of fish, called mullet, caught are underweight. In one day, 168 mullet are caught. Work out the expected number of mullet that are underweight.
Medium
- A36
- B24
- C42
- D48

Summary

## Introduction to Fractions - A fraction is written as $\frac{a}{b}$ where $a$ and $b$ are integers. - **Numerator** (top) tells how many parts you have; **denominator** (bottom) tells how many parts make one whole. - Fraction < 1 if numerator < denominator; > 1 if numerator > denominator. ### Shading Fractions - To shade $\frac{3}{5}$ of 20 squares: divide 20 by 5 = 4, multiply by 3 = 12, shade 12 squares. ### Fractions of Amounts - **Method 1**: Divide by denominator, multiply by numerator. E.g., $\frac{2}{5}$ of 60: $60 \div 5 = 12$, $12 \times 2 = 24$. - **Method 2**: Convert fraction to decimal and multiply. E.g., $\frac{1}{4}$ of 16: $0.25 \times 16 = 4$. - **Method 3**: Multiply fractions: $\frac{2}{3} \times \frac{60}{1} = \frac{120}{3} = 40$. ### Equivalent & Simplified Fractions - **Equivalent fractions**: Multiply numerator and denominator by the same number. E.g., $\frac{5}{6} = \frac{10}{12} = \frac{15}{18}$. - **Simplifying**: Divide numerator and denominator by a common factor. E.g., $\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$. ### Mixed Numbers & Improper Fractions - **Mixed number**: integer part + fraction part, e.g., $3\frac{3}{4}$. - **Improper fraction**: numerator > denominator, e.g., $\frac{15}{4}$. - **Convert mixed to improper**: Multiply whole number by denominator, add numerator. E.g., $5\frac{2}{7} = \frac{5 \times 7 + 2}{7} = \frac{37}{7}$. - **Convert improper to mixed**: Divide numerator by denominator; quotient is whole number, remainder over denominator. E.g., $\frac{22}{3} = 7\frac{1}{3}$. ### Key Tips - Use calculator to simplify fractions (but show division steps if required). - For "of" problems, multiply. - Always simplify final fractions where possible.

Key terms

Fraction: A number written as a/b where a and b are integers; a is the numerator, b is the denominator.
Numerator: The top number of a fraction; indicates how many parts are taken.
Denominator: The bottom number of a fraction; indicates how many equal parts the whole is divided into.
Equivalent fractions: Fractions that represent the same value, obtained by multiplying numerator and denominator by the same number.
Simplified fraction: A fraction written with the smallest possible integers by dividing numerator and denominator by a common factor.
Mixed number: A number consisting of an integer part and a fraction part, e.g., 3 3/4.
Improper fraction: A fraction where the numerator is greater than the denominator, e.g., 15/4.

Slides

What is a Fraction?

A fraction is written as a/b where a and b are integers.
The top number (a) is the numerator; the bottom number (b) is the denominator.
The denominator tells how many parts each whole is split into; the numerator tells how many parts you have.
If numerator < denominator, the fraction is less than 1. If numerator > denominator, it is greater than 1.

In the fraction $\frac{3}{5}$ , what does the 5 represent?

The number of parts you have
The number of parts each whole is split into
The total number of parts
The whole number

Shading Fractions on a Diagram

To shade a fraction of a shape, divide the total number of parts by the denominator, then multiply by the numerator.
Example: To shade $\frac{3}{5}$ of a grid of 20 squares, divide 20 by 5 to get 4, then multiply by 3 to get 12 squares shaded.
You can shade any 12 squares; grouping them neatly is often clearer.

To shade $\frac{2}{3}$ of a rectangle with 15 squares, first divide 15 $by ____ to$ get 5, then multiply $by ____ to$ $get ____$ squares.

Fractions of Amounts

Method 1: Divide by the denominator, then multiply by the numerator. $E.g., \frac{2}{5}$ of $60: 60 \div 5 = 12$ , then $12 \times 2 = 24$ .
Method 2: Convert the fraction to a decimal and multiply. $E.g., \frac{1}{4}$ of $16 = 0.25 \times 16 = 4$ .
Method 3: Write the amount as a fraction $(e.g., 60 = \frac{60}{1})$ and multiply the two fractions.

What is $\frac{3}{8}$ of 40?

Equivalent Fractions

Equivalent fractions represent the same amount but are written differently.
To find an equivalent fraction, multiply the numerator and denominator by the same number.
Example: $\frac{5}{6} = (5\times 2)/(6\times 2) = \frac{10}{12} = (5\times 3)/(6\times 3) = \frac{15}{18}$ , etc.
There are infinitely many equivalent fractions for any given fraction.

Which of these are equivalent to $\frac{2}{3}$ ?

$\frac{4}{6}$
$\frac{6}{9}$
$\frac{8}{12}$
$\frac{3}{2}$

Simplifying Fractions

A simplified fraction uses the smallest possible integers (numerator and denominator).
Divide the numerator and denominator by a common factor (a number that divides both).
Example: $\frac{12}{18} = (12\div 6)/(18\div 6) = \frac{2}{3}$ . Example: $\frac{25}{45} = (25\div 5)/(45\div 5) = \frac{5}{9}$ .
On a calculator, typing a fraction and pressing = often simplifies it automatically.

Simplify $\frac{20}{35}$ : divide numerator and denominator $by ____ to$ $get ____$ .

Mixed Numbers & Improper Fractions

A mixed number has an integer part and a fraction part, $e.g., 3 \frac{3}{4}$ means 'three and three quarters'.
An improper fraction (top-heavy) has numerator > denominator, $e.g., \frac{15}{4}$ means 'fifteen quarters'.
Any fraction $> 1$ can be written as either a mixed number or an improper fraction.

What is $\frac{7}{3}$ as a mixed number?

$2 \frac{1}{3}$
$3 \frac{1}{3}$
$2 \frac{2}{3}$
$1 \frac{4}{3}$

Converting Mixed Numbers to Improper Fractions

Step 1: Multiply the whole number by the denominator of the fraction part.
Step 2: Add the numerator of the fraction part to that product.
The result becomes the numerator; keep the same denominator.
Example: $5 \frac{2}{7} = (5\times 7 + 2)/7 = (35+2)/7 = \frac{37}{7}$ .

Convert $4 \frac{3}{5}$ to an improper fraction: $(4 \times ____) + ____ = ____, so$ the fraction $is ____$ .

Converting Improper Fractions to Mixed Numbers

Divide the numerator by the denominator.
The quotient becomes the whole number part.
The remainder becomes the numerator of the fraction part; keep the same denominator.
Example: $\frac{22}{3}: 22 \div 3 = 7$ remainder 1, so 7 1/3.

Convert $\frac{19}{4}$ to a mixed number.

$4 \frac{3}{4}$
$4 \frac{1}{4}$
$5 \frac{3}{4}$
$3 \frac{7}{4}$

Flashcards

What is a fraction? Write the general form and label the numerator and denominator.

A fraction is written as a/b where a and b are integers. The top number (a) is the numerator, the bottom number (b) is the denominator.

How do you shade $\frac{3}{5}$ of a grid of 20 squares?

Divide 20 by 5 to get 4, then multiply by 3 to get 12. Shade 12 squares.

Find $\frac{1}{5}$ of $200.

Divide $200 by 5: $200 $\div 5 =$ $40.

Write $\frac{60}{105}$ in its simplest form.

Divide numerator and denominator by $15: 60\div 15=4, 105\div 15=7$ , so 4/7.

Write a fraction equivalent to 3/5.

Multiply numerator and denominator by the same number, $e.g., \frac{3}{5} = \frac{6}{10}$ .

Write $\frac{28}{140}$ as a fraction in its simplest form.

Divide numerator and denominator by $28: 28\div 28=1, 140\div 28=5$ , so 1/5.

How do you find a fraction of an amount?

Divide by the denominator and multiply by the numerator. For example, $\frac{2}{5}$ of $60: 60\div 5=12$ , then $12\times 2=24$ .

Find $\frac{1}{3}$ of 90.

Divide 90 by $3: 90 \div 3 = 30$ .

What are equivalent fractions?

Equivalent fractions are two fractions that represent the same amount, $e.g., \frac{1}{2}$ and 6/12.

How do you find an equivalent fraction?

Multiply the numerator and denominator by the same number. For example, $\frac{5}{6} = \frac{10}{12} (multiply$ by 2).

What is a simplified fraction?

A simplified fraction uses the smallest possible integer values. It is found by dividing numerator and denominator by a common factor.

Simplify 12/18.

Divide numerator and denominator by $6: 12\div 6=2, 18\div 6=3$ , so 2/3.

What is a mixed number? Give an example.

A mixed number has an integer part and a fraction part, $e.g., 3 \frac{3}{4}$ means three and three quarters.

What is an improper fraction?

An improper fraction has a numerator larger than the denominator, e.g., 15/4.

Convert $5 \frac{2}{7}$ into an improper fraction.

5 wholes $= 5\times 7=35$ sevenths, plus 2 sevenths $= 37$ sevenths, so 37/7.

Convert $\frac{22}{3}$ into a mixed number.

$22 \div 3 = 7$ remainder 1, so 7 1/3.

Write $5 \frac{3}{4}$ as an improper fraction.

5 wholes $= 5\times 4=20$ quarters, plus 3 quarters $= 23$ quarters, so 23/4.

Write $\frac{17}{5}$ as a mixed number.

$17 \div 5 = 3$ remainder 2, so 3 2/5.

There are 12 brass instruments. $\frac{1}{3}$ are trumpets, 3 are trombones, the rest are horns. Find the number of horns.

Trumpets: $12\times \frac{1}{3}=4$ . Trombones: 3. Horns: $12-4-3=5$ .

$\frac{3}{14}$ of caught mullet are underweight. If 168 mullet are caught, how many are expected underweight?

$168 \times \frac{3}{14} = (168\div 14)\times 3 = 12\times 3 = 36$ .

Nora uses $\frac{7}{10}$ of 250g cherries. Find the mass left.

Used: $250\times \frac{7}{10} = 175g$ . Left: $250-175 = 75g$ .

A ticket costs $13.50. A student gets a one-third reduction. How much does she pay?

Reduction: $13.50\times \frac{1}{3} =$ $4.50. Pay: $13.50-4.50 =$ $9.00.

A fuel tank holds 52 litres. It is a quarter full. Jim fills it at $2.18 per litre. How much does he pay?

Amount to fill: $52\times (\frac{3}{4})=39$ litres. Cost: $39\times 2.18 =$ $85.02.

Saskia has $600. She spends $\frac{1}{5}$ on a coat and gives $\frac{1}{3}$ to her son. What fraction of $600 does she have left? (Give simplest form)

Spent: $\frac{1}{5}+\frac{1}{3} = \frac{3}{15}+\frac{5}{15}=\frac{8}{15}$ . Left: $1-\frac{8}{15}=\frac{7}{15}$ .

Jeff's share is $160, which is $\frac{5}{11}$ of the prize. Find the value of the prize.

Prize $= 160 \div (\frac{5}{11}) = 160 \times \frac{11}{5} = 160\times 11\div 5 = 1760\div 5 =$ $352.

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