FRACTIONS

Fractions are used in everyday life when:

• Dividing a pizza, cake, or loaf of bread among friends.
• Measuring ingredients while cooking or baking (e.g., half a cup of sugar).
• Telling the time (e.g., a quarter past four).
• Calculating shopping discounts (e.g., half-price sale).
• Measuring fuel levels in a vehicle tank (e.g., a quarter tank left).
• Dividing land or properties into equal shares.

Example

If a circular pizza is cut into 4 equal slices and you eat 3 slices, you have eaten:

3 4 of the pizza.

This means you have taken 3 parts out of a total of 4 equal parts.

Examples of Proper Fractions:

1 2 , 3 5 , 7 10

Examples of Improper Fractions:

5 4 , 9 2 , 11 7

Examples of Mixed Numbers:

1 1 4 , 3 2 5

Equivalent fractions are used in everyday life when:

• Adjusting baking recipes to serve more or fewer people (e.g., using two 1/4 cups to make a 1/2 cup).
• Measuring lengths or tools in carpentry and construction (e.g., recognizing that 2/4 of an inch is the same as 1/2 of an inch).
• Splitting bills or sharing food equally among different sized groups of friends.

Examples of Equivalent Fractions:

Example 1

Find an equivalent fraction for 2 3 .

Solution

Step 1: Choose a whole number to multiply by.

Let us choose 4.

Step 2: Set up the multiplication for the top and bottom numbers.

The top numerator and bottom denominator are multiplied by the same number correctly.

Step 3: Simplify to find the new fraction.

Equivalent Fraction = 8 12

This means both fractions are completely equal in value.

Example 2

Find an equivalent fraction with smaller numbers for 10 15 .

Solution

Step 1: Find a number that divides evenly into both numbers.

The Greatest Common Factor for 10 and 15 is 5.

Step 2: Set up the division for the top and bottom numbers.

The top numerator and bottom denominator are divided by the same number correctly.

Step 3: Simplify to find the new fraction.

Equivalent Fraction = 2 3

This is the fraction reduced to its simplest form.

Comparing fractions is used in everyday life when:

• Comparing store prices or sales to find which deal gives you more value for your money.
• Checking recipe ingredients to see if you need more of one item than another.
• Reading measurement gauges or fuel levels on a dashboard.

Math Symbols Used for Comparing:

• > means Greater Than (e.g., 5 > 3)
• < means Less Than (e.g., 2 < 7)
• = means Equal To (e.g., 4 = 4)

Example

Compare the two fractions below by inserting the correct comparison symbol (>, <, or =):

Solution

Step 1: Identify the given values.

First Fraction = 3/4

Second Fraction = 2/3

We are required to find which fraction is larger by matching their denominators.

Step 2: Find the Least Common Multiple (LCM) of the denominators 4 and 3.

Multiples of 4: 4, 8, 12, 16

Multiples of 3: 3, 6, 9, 12, 15

The Least Common Multiple is 12. This will be our new common denominator.

Step 3: Convert both fractions into equivalent fractions with a denominator of 12.

For the first fraction, multiply the top and bottom by 3:

For the second fraction, multiply the top and bottom by 4:

Step 4: Compare the new numerators.

Now that both bottom numbers are 12, look at the top numbers. Since 9 is greater than 8, we write:

Step 5: Write the final comparison using the original fractions.

Replace the modified fractions with their original forms to complete the statement:

3 4 > 2 3

Answer = 3 4 > 2 3

Ordering from smallest to largest is used in everyday life when:

• Sorting tool drill bits or mechanical socket sets from the thinnest size to the thickest size.
• Arranging packages or ingredients by their weight fractions from lightest to heaviest.
• Organizing data shares or performance results from the lowest score to the highest score.

Example

Arrange the following list of fractions from smallest to largest (ascending order):

Solution

Step 1: Identify the given values.

Given fractions: 1/2, 2/5, and 3/10

We are required to arrange these fractions in order starting with the smallest value and ending with the largest value.

Step 2: Find the Least Common Multiple (LCM) of the denominators 2, 5, and 10.

Multiples of 2: 2, 4, 6, 8, 10, 12

Multiples of 5: 5, 10, 15, 20

Multiples of 10: 10, 20, 30

The Least Common Multiple for all three denominators is 10. This becomes our common denominator.

Step 3: Convert each fraction to have a denominator of 10.

For the first fraction (1/2), multiply the top and bottom by 5:

For the second fraction (2/5), multiply the top and bottom by 2:

For the third fraction (3/10), it already has a denominator of 10, so it remains the same:

Step 4: Compare and arrange the new numerators.

Now look at our new fractions: 5/10, 4/10, and 3/10. Comparing the top numbers from smallest to largest gives: 3, then 4, then 5. We line them up like this:

Step 5: Write the final answer using the original fractions.

Swap the modified fractions back to their original forms to complete the ascending sequence:

Order = 3 10 , 2 5 , 1 2

Final Answer = 3 10 , 2 5 , 1 2

Ordering from largest to smallest is used in everyday life when:

• Ranking students or businesses by their market or test score shares from top to bottom.
• Arranging wood cuts or plumbing pipes by thickness from the widest to the narrowest.
• Sorting remaining inventory or container fluid levels from the most full to the least full.

Example

Arrange the following list of fractions from largest to smallest (descending order):

Solution

Step 1: Identify the given values.

Given fractions: 2/3, 3/4, and 5/6

We are required to arrange these fractions in order starting with the largest value and ending with the smallest value.

Step 2: Find the Least Common Multiple (LCM) of the denominators 3, 4, and 6.

Multiples of 3: 3, 6, 9, 12, 15

Multiples of 4: 4, 8, 12, 16

Multiples of 6: 6, 12, 18, 24

The smallest number that 3, 4, and 6 can all divide into perfectly is 12. This becomes our common denominator.

Step 3: Convert each fraction to have a denominator of 12.

For the first fraction (2/3), multiply the top and bottom by 4:

For the second fraction (3/4), multiply the top and bottom by 3:

For the third fraction (5/6), multiply the top and bottom by 2:

Step 4: Compare and arrange the new numerators.

Now look at our new fractions: 8/12, 9/12, and 10/12. Comparing the top numbers from largest to smallest gives: 10, then 9, then 8. We line them up like this:

Step 5: Write the final answer using the original fractions.

Swap the modified fractions back to their original forms to complete the descending sequence:

Order = 5 6 , 3 4 , 2 3

Final Answer = 5 6 , 3 4 , 2 3

Example

Convert the fraction below into a decimal:

3 8

Solution:

Step 1: Set up long division by dividing 3 by 8.

Since 8 cannot go into 3, we write 0. and add a zero to 3 to make it 30.

Step 2: Divide 30 by 8.

8 goes into 30 a total of 3 times (because 8 × 3 = 24). The remainder is 6 (30 - 24 = 6).

Step 3: Add another zero to the remainder 6 to make it 60.

8 goes into 60 a total of 7 times (because 8 × 7 = 56). The remainder is 4 (60 - 56 = 4).

Step 4: Add a zero to the remainder 4 to make it 40.

8 goes into 40 exactly 5 times (because 8 × 5 = 40). The remainder is now 0.

Decimal Value = 0.375

Example

Convert the fraction below into a percentage value:

3 5

Solution:

Step 1: Multiply the given fraction by 100%.

Percentage = 3 5 × 100%

Step 2: Simplify by dividing 100 by the denominator 5.

100 ÷ 5 = 20

Step 3: Multiply the remaining numerator value by 20.

3 × 20% = 60%

Percentage Value = 60%

Example

Convert the given fraction into a clean ratio format:

2 5

Solution:

Step 1: Identify the numbers in your fraction.

The Numerator (top number) is 2.

The Denominator (bottom number) is 5.

Step 2: Write the numbers side-by-side separated by a colon (:). The numerator always comes first.

Numerator : Denominator → 2 : 5

Ratio Value = 2 : 5

Example

Calculate the sum of the two fractions below:

1 4 + 2 3

Solution:

Step 1: Find the Least Common Multiple (LCM) of the denominators 4 and 3.

Multiples of 4: 4, 8, 12, 16

Multiples of 3: 3, 6, 9, 12, 15

The Least Common Multiple is 12. This becomes our new common denominator.

Step 2: Convert both fractions to have the common denominator of 12.

For the first fraction, multiply the numerator and the denominator by 3:

1 × 3 4 × 3 = 3 12

For the second fraction, multiply the numerator and the denominator by 4:

2 × 4 3 × 4 = 8 12

Step 3: Add the numerators together over the common denominator.

3 12 + 8 12 = 3 + 8 12 = 11 12

Final Answer = 11 12

Example

Find the difference between the two fractions below:

4 5 - 1 2

Solution:

Step 1: Find the Least Common Multiple (LCM) of the denominators 5 and 2.

Multiples of 5: 5, 10, 15, 20

Multiples of 2: 2, 4, 6, 8, 10, 12

The Least Common Multiple is 10. This becomes our new common denominator.

Step 2: Convert both fractions to have the common denominator of 10.

For the first fraction, multiply the numerator and the denominator by 2:

4 × 2 5 × 2 = 8 10

For the second fraction, multiply the numerator and the denominator by 5:

1 × 5 2 × 5 = 5 10

Step 3: Subtract the second numerator from the first numerator over the common denominator.

8 10 - 5 10 = 8 - 5 10 = 3 10

Final Answer = 3 10

Example

Multiply the two fractions below:

2 3 × 5 7

Solution:

Step 1: Set up the multiplication across a single common fraction bar by combining the numerators and denominators.

Expression = 2 × 5 3 × 7

Step 2: Multiply the top numbers together to find the new numerator.

2 × 5 = 10

Step 3: Multiply the bottom numbers together to find the new denominator.

3 × 7 = 21

Final Answer = 10 21

Example

Divide the two fractions below:

3 4 ÷ 2 5

Solution:

Step 1: Apply the "Keep, Change, Flip" workflow adjustments.

• Keep the first fraction exactly as it is written: 3/4
• Change the division operator sign (÷) into a multiplication sign (×).
• Flip the second fraction completely upside down: 2/5 turns into 5/2.

Now rewrite your problem layout using these components:

3 4 × 5 2

Step 2: Combine the values straight across over a single mathematical dividing dash.

Expression = 3 × 5 4 × 2

Step 3: Solve the top and bottom values directly.

Top Numerator: 3 × 5 = 15

Bottom Denominator: 4 × 2 = 8

Final Answer = 15 8

Example

Simplify the fraction below to its lowest terms:

12 18

Solution:

Step 1: Find the factors of both numbers to see what can divide them evenly.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

The Greatest Common Factor (GCF) that both numbers share is 6.

Step 2: Divide both the numerator and the denominator by 6.

12 ÷ 6 18 ÷ 6 = 2 3

Simplest Form = 2 3

Example

Convert the improper fraction below into a mixed number:

14 3

Solution:

Step 1: Divide 14 by 3.

3 goes into 14 exactly 4 times (because 3 × 4 = 12).

Step 2: Find the remaining value leftover.

Remainder = 14 - 12 = 2.

Step 3: Assemble your mixed number layout components:

• Whole Number = 4
• New Numerator = 2
• Original Denominator = 3

Mixed Number = 4 2 3

Final Answer = 4 2 3

Example

Convert the mixed number below into an improper fraction:

5 1 4

Solution:

Step 1: Multiply the whole number 5 by the bottom denominator 4.

5 × 4 = 20

Step 2: Add the top numerator 1 to your result.

20 + 1 = 21 (This becomes the new top number).

Step 3: Place this total value over the original denominator 4.

Expression Layout = (5 × 4) + 1 4

Improper Fraction = 21 4

Final Answer = 21 4

Example 1

A school library received a donation of books. 4 9 of the books were Science textbooks, and 3 5 of the remaining books were Mathematics textbooks. The rest of the books were English novels. If there were 120 English novels, calculate the total number of books donated to the library.

Solution

Step 1: Identify the given values.

Science textbooks fraction = 4 9

Mathematics textbooks fraction = 3 5 of the remainder.

Total English novels = 120 books

We are required to calculate the total initial number of donated books.

Step 2: Calculate the remainder fraction after accounting for Science textbooks.

Let the total initial donation be represented as 1 whole.

Remainder Fraction = 1 − Science Fraction

Remainder Fraction = 1 − 4 9

Remainder Fraction = 5 9

This is the fraction of books left over after counting the Science textbooks.

Step 3: Calculate the fraction spent on Mathematics textbooks.

Mathematics Fraction = 3 5 × Remainder Fraction

Multiplying the numerators together and denominators together:

Step 4: Find the fraction left over for English novels.

English Novels Fraction = Remainder Fraction − Mathematics Fraction

The Least Common Multiple (LCM) of 9 and 3 is 9. Convert the second fraction:

Step 5: Calculate the initial total book donation.

Since two-ninths of the total books corresponds directly to 120 books, we divide 120 by this final fraction:

Apply the reciprocal division rule by changing division to multiplication and flipping the fraction:

Total Donated Books = 60 × 9

60 × 9 = 540

Total Donated Books = 540 books

Example 2

Evaluate the complex numerical expression shown below using standard operational hierarchies, and leave the solution in its lowest possible terms as a mixed number:

Solution

Step 1: Identify the required sequence of operations.

According to standard hierarchy operational laws, we must execute the division statement block and the multiplication statement block first, before resolving their difference via subtraction.

Step 2: Solve the division block.

Apply the Keep-Change-Flip protocol steps directly:

Step 3: Solve the multiplication block.

Multiply across top numerators and bottom denominators:

Step 4: Perform the subtraction step between the two results.

The Least Common Multiple (LCM) of denominators 2 and 5 is 10. Convert both values:

Step 5: Simplify and combine.

Converting the improper fraction 29/10 into a mixed number format:

Mixed number = 2 9 10

Final Answer = 2 9 10

Example 3

A town fuel storage tank was filled to 2 15 of its capacity. When 7,800 litres of fuel were poured into it, the fuel level reached exactly 3 5 of its total capacity. Calculate the full holding capacity of the town fuel storage tank.

Solution

Step 1: Identify the given values.

Initial storage fraction = 2/15

Final storage fraction = 3/5

Volume of fuel added = 7,800 litres

We are required to determine the absolute full scale volume capacity of the tank.

Step 2: Calculate the fractional difference matching the added volume.

Fraction Added = Final State − Initial State

The LCM of 5 and 15 is 15. Convert the first fraction to match denominators:

This means 7/15 of the total capacity matches the added 7,800 litres of fuel.

Step 3: Set up the total capacity formula based on this value.

To find the full capacity (1 whole), we divide the added volume by its matching fraction value:

Step 4: Convert division to multiplication using the reciprocal rule.

Step 5: Simplify and solve.

Full Capacity = (7,800 × 15) ÷ 7

Full Capacity = 117,000 ÷ 7

Full Capacity = 16,714.29 litres

Example 4

Convert the decimal 0.4375 into a fraction in its simplest form, and then write your final answer as a ratio.

Solution

Step 1: Identify the given values.

Decimal value = 0.4375

We are required to change this decimal into a fully simplified fraction first, and then convert it into a ratio format.

Step 2: Write the decimal as a fraction using place value.

Since there are 4 digits after the decimal point, the last digit is in the ten-thousandths place. This means we write the number over 10,000:

Step 3: Simplify the fraction using common factors.

Both numbers end in 25, so we can divide the top and the bottom by 25 to make them smaller:

4,375 ÷ 25 = 175

10,000 ÷ 25 = 400

Step 4: Reduce the fraction to its lowest terms.

We can divide by 25 one more time because 25 fits evenly into both numbers:

175 ÷ 25 = 7

400 ÷ 25 = 16

Step 5: Convert the simplest fraction into a ratio.

To write a fraction as a ratio, use the numerator (top number) as the first part and the denominator (bottom number) as the second part, separated by a colon (:).

Ratio Form = 7 : 16

Final Answer = 7 : 16

Example 5

A transport container terminal evaluated a wholesale parcel batch containing 3,600 manufacturing units. The terminal inspectors noted that 2 9 of the manufacturing units failed configuration checks and were labeled defective, 1 4 were packed directly for regional market dispatch delivery, and the remainder units were set aside for oceanic cargo export. Calculate the count of manufacturing units set aside for cargo export.

Solution

Step 1: Identify the given values.

Total processed units batch = 3,600

Defective units subset fraction = 2/9

Regional dispatch subset fraction = 1/4

We are required to isolate the total integer piece count set aside for oceanic cargo export.

Step 2: Add the defective and regional market fraction states together.

Combined Allocated Parts = Defective Fraction + Regional Fraction

The LCM of denominators 9 and 4 is 36. Scale both fraction statements to match:

Step 3: Determine the fraction balance representation left for cargo export.

Subtract the combined allocated subset fraction from 1 whole aggregate container unit value:

Export Fraction = 1 − Combined Fraction

Perform fractional conversion subtractions directly:

Step 4: Compute the absolute quantity of items matching this export state.

Multiply the final cargo export fraction by the total initial batch volume scale size of 3,600 pieces:

Step 5: Simplify terms to find the absolute solution.

Divide 3,600 by 36, which yields 100. Then multiply by 19:

Cargo Export Total Count = 100 × 19

Cargo Export Count = 1,900 units