A decimal is a number that uses a decimal point followed by digits to show a value smaller than a whole number. It is a way of expressing parts of a whole based on groups of tens, hundreds, thousands, and so on.
A decimal number is split into two major regions by a small dot called the decimal point:
Whole Number Part . Fractional (Decimal) Part
Decimals are used in everyday life when:
Example
If you split a single note of currency into 10 equal smaller parts and you keep 3 parts, your financial value is written as:
0 . 3
This layout shows you hold 0 whole units and 3 tenths of a unit.
Each position after the decimal point splits the value down by a multiple of 10. Below is a fully responsive representation of how positions change value across the line:
| Whole Numbers | • | Decimal Parts (Fractional) | ||||
|---|---|---|---|---|---|---|
| Hundreds | Tens | Ones | Point | Tenths | Hundredths | Thousandths |
| 100 | 10 | 1 | . | 1/10 (0.1) | 1/100 (0.01) | 1/1000 (0.001) |
| 2 | 4 | 5 | . | 3 | 8 | 1 |
* Note: Scroll horizontally to view the full chart if you are viewing on a small smartphone display.
Decimals can be classified into distinct categories depending on how the numbers behave after the decimal point:
A terminating decimal is a number that contains a finite number of digits after the decimal point. The sequence of numbers stops completely.
Examples of Terminating Decimals:
0.5 , 2.75 , 14.125
A non-terminating decimal is a number that contains an infinite sequence of digits after the decimal point. The digits continue forward without ever coming to an end.
These are split further into two major categories:
The digits after the decimal point repeat themselves in a regular, predictable cycle forever.
Examples of Recurring Decimals:
0.3333... (or 0.3) , 1.2727... (or 1.27)
The digits after the decimal point continue forever but do not form a repeating pattern.
Examples of Non-Recurring Decimals:
3.14159265... (Pi) , 1.41421356... (Square root of 2)
Decimals can also be grouped together by comparing the total count of places occupied to the right of the point.
Visual Examples:
Like Decimals (Each has exactly 2 decimal places): 4.12, 11.85, 0.09
Unlike Decimals (Varying counts of decimal places): 5.4, 2.185, 96.34
Comparing decimals means looking at two decimal numbers to find out which one is larger, smaller, or if they are completely equal in value.
To compare decimals, you look at their digits from left to right. First, look at the whole number parts. If they are the same, look at the tenths place, then the hundredths place, and so on. You can also add extra zeros to the end of a decimal to make them have the same length, which makes comparing them very easy.
Comparing decimals is used in everyday life when:
Math Symbols Used for Comparing:
Compare the two decimals below by inserting the correct comparison symbol (>, <, or =):
Step 1: Identify the given values.
First Decimal = 0.42
Second Decimal = 0.418
We need to find out which decimal is larger.
Step 2: Make the number of decimal places equal.
0.42 has two decimal places, and 0.418 has three decimal places. We add a zero to the end of 0.42 so it has three places too. This does not change its value.
First Decimal becomes: 0.420
Second Decimal stays: 0.418
Step 3: Compare the numbers from left to right.
Step 4: Write the final comparison using the original decimals.
Since 2 is greater than 1, 0.420 is greater than 0.418. We write:
0.42 > 0.418
Answer = 0.42 > 0.418
Ordering decimals from smallest to largest means arranging a list of three or more numbers in ascending order.
When decimals have different lengths, it can be confusing. To order them easily, add extra trailing zeros to the end of each number until all numbers have the same amount of digits after the decimal point. Once they match, you can read them like normal numbers and rank them from lowest to highest.
Ordering from smallest to largest is used in everyday life when:
Arrange the following list of decimals from smallest to largest (ascending order):
Step 1: Count the decimal places.
The highest number of decimal places is 3.
Step 2: Add zeros to make all lengths equal to 3 places.
Step 3: Compare the new numbers.
Now look at the numbers after the decimal point as whole numbers: 500, 250, and 375.
So, the order from smallest to largest is: 0.250, 0.375, 0.500
Step 4: Write the final answer using the original numbers.
Answer = 0.25 , 0.375 , 0.5
Ordering decimals from largest to smallest means arranging a list of three or more numbers in descending order.
When decimals have different lengths, it can be confusing. To order them easily, add extra trailing zeros to the end of each number until all numbers have the same amount of digits after the decimal point. Once they match, you can read them like normal numbers and rank them from highest to lowest.
Ordering from largest to smallest is used in everyday life when:
Arrange the following list of decimals from largest to smallest (descending order):
Step 1: Count the decimal places.
The highest number of decimal places is 3.
Step 2: Add zeros to make all lengths equal to 3 places.
Step 3: Compare the new numbers.
Now look at the numbers after the decimal point as whole numbers: 620, 800, and 125.
So, the order from largest to smallest is: 0.800, 0.620, 0.125
Step 4: Write the final answer using the original numbers.
Answer = 0.8 , 0.62 , 0.125
To convert a decimal number into a fraction, you write the numbers after the decimal point as the numerator (top number). For the denominator (bottom number), you write a multiple of 10 (like 10, 100, or 1000) based on the number of decimal places, and then simplify the fraction.
Example
Convert the decimal below into a fraction in its simplest form:
0.375
Solution:
Step 1: Find the numerator (the number after the decimal point).
In this case, the numbers after the decimal point is 375
Therefore :
Numerator (Top number) = 375Step 2: Find the denominator by Counting the number of decimal places after the dot.
The number 0.375 has 3 decimal places (tenths, hundredths, and thousandths).
Step 3: Write the decimal as a fraction with a numerator of 375(The number after the decimal point) and denominator of 1000 (since there are 3 decimal places).
375 1000
Step 3: Simplify the fraction by dividing both top and bottom numbers by their highest common factor (125).
375 ÷ 125 = 3
1000 ÷ 125 = 8
Fraction Value = 3 8
A recurring decimal is a decimal number where a digit or a block of digits repeats itself forever in a regular pattern. To convert a recurring decimal into a simplified fraction, we use algebraic equations to cancel out the infinite repeating digits.
Example
Convert the recurring decimal number below into a simplified fraction in its lowest terms:
0.727272... (or 0.72)
Solution:
Step 1: Set up the first algebraic equation.
Let a variable like x represent our original recurring decimal number:
x = 0.727272... — (Equation 1)
Step 2: Create a second equation by multiplying by a multiple of 10.
Count the number of repeating digits in the repeating pattern cycle. Since there are exactly 2 repeating digits (7 and 2), we multiply both sides of Equation 1 by 100. This shifts the decimal point two places to the right:
100x = 72.727272... — (Equation 2)
* Note: If only 1 digit repeats (like 0.333...), multiply by 10. If 3 digits repeat (like 0.123123...), multiply by 1000.
Step 3: Subtract Equation 1 from Equation 2.
Subtracting the smaller equation from the larger equation completely wipes out the infinite repeating digits behind the decimal dot:
Step 4: Solve for x by rewriting the numbers as a fraction.
Divide both sides of the expression by 99 to isolate our fraction layout:
x = 72 99
Step 5: Reduce the fraction to its lowest terms.
Simplify by dividing both the numerator and the denominator by their highest common factor, which is 9.
Final Fraction Value = 8 11
To convert any decimal number into a percentage value, you simply multiply the decimal number by 100%. This is the same as moving the decimal point exactly two positions to the right.
Example
Convert the decimal below into a percentage value:
0.6
Solution:
Step 1: Set up the multiplication by 100%.
Percentage = 0.6 × 100%
Step 2: Move the decimal point two places to the right to complete the multiplication.
Moving the dot once makes it 6. Moving it a second time gives 60.
Percentage Value = 60%
To convert a decimal number into a ratio, you first turn it into a simplified fraction. After getting the fraction, you write the numerator as the first number and the denominator as the second number, separated by a colon (:).
Example
Convert the given decimal into a clean ratio format:
0.4
Solution:
Step 1: Change the decimal into a fraction based on its place value.
0.4 has 1 decimal place, so it is written as 4 out of 10:
4 10
Step 2: Simplify the fraction by dividing the top and bottom by 2.
2 5
Step 3: Write the simplified fraction numbers side-by-side separated by a colon (:). The numerator always comes first.
Numerator : Denominator → 2 : 5
Ratio Value = 2 : 5
Just like regular whole numbers, decimals can be added, subtracted, multiplied and divided. Following the correct rules for each operation ensures accurate calculations.
To add decimals, the most important rule is to line up the decimal points vertically. If the numbers have different lengths, you add trailing zeros to fill any empty spaces before adding normally from right to left.
Example
Calculate the sum of the two decimals below:
4.25 + 3.8
Solution:
Step 1: Line up the decimal points vertically and use a trailing zero to fill the empty place value.
The number 3.8 becomes 3.80 so it matches the length of 4.25.
Step 2: Arrange the numbers in a vertical stack to add them clearly.
Step 3: Add the digits from right to left.
Drop the decimal point straight down into the answer line.
Final Answer = 8.05
Subtracting decimals follows the same stack alignment rules as addition. Line up the decimal dots vertically, fill out any shorter rows with trailing zeros, and subtract normally from right to left while borrowing when necessary.
Example
Find the difference between the two decimals below:
7.4 - 2.65
Solution:
Step 1: Line up the decimal points and add a trailing zero to fill out empty spaces.
The number 7.4 becomes 7.40 so you can subtract 2.65 from it properly.
Step 2: Arrange the numbers in a vertical stack to subtract them clearly.
Step 3: Subtract the digits from right to left by borrowing where needed.
Drop the decimal point straight down into the answer line.
Final Answer = 4.75
To multiply decimals, you do not need to line up the decimal points. Simply multiply the numbers as if they were regular whole numbers. Once you find the product, count the total number of decimal places in both original numbers and place the decimal point that same number of positions from the right side of the answer.
Example
Multiply the two decimals below:
0.12 × 0.4
Solution:
Step 1: Count the total number of decimal places in both numbers.
Step 2: Multiply the numbers as whole numbers by ignoring the decimal points completely.
Expression = 12 × 4 = 48
Step 3: Place the decimal point into the whole number answer. Start from the right side and move 3 positions to the left.
Since 48 only has two digits, we add a placeholder zero to the left of the numbers to make it 3 places: .048
Step 4: Add a zero before the decimal point to complete the format.
The layout turns into: 0.048
Final Answer = 0.048
To divide decimals, we turn the divisor (the second number) into a whole number by moving its decimal point to the right end. You must then move the decimal point in the dividend (the first number) by the exact same number of positions before running standard long division.
Example
Divide the two decimals below:
12.5 ÷ 0.5
Solution:
Step 1: Make the divisor (0.5) a whole number by moving its decimal point 1 place to the right.
Step 2: Move the decimal point in the first number (12.5) by the same 1 place to the right.
Now rewrite your problem layout using these adjusted numbers:
125 ÷ 5
Step 3: Run long division. Since 125 is a whole number, the decimal point sits invisibly at the very end.
Step 4: Check each calculation step directly.
Final Answer = 25
Simplifying a decimal means removing extra zeros from the very end of the number to make it shorter while keeping the total value exactly the same. A decimal is in its simplest form when it does not have any unnecessary zeros tracking at the right end of the fractional part.
Example
Simplify the decimal number below to its simplest terms:
0.4500
Solution:
Step 1: Identify the extra placeholder digits at the very end of the fractional part.
In 0.4500, the last two digits are zeros sitting in the thousandths and ten-thousandths place positions.
These trailing zeros add no extra weight or actual numeric value to the original number.
Step 2: Drop the unnecessary trailing zeros directly from the end.
0.4500 → 0.45
Simplest Form = 0.45
We can rewrite long decimals to change their length by Rounding them to a specific place value (like the nearest whole number, tenths, or hundredths).
To round a decimal to the nearest tenth, look at the digit in the hundredths place (the second number after the dot). If that digit is 5 or more, add 1 to the tenths place. If it is less than 5, keep the tenths digit the same and drop all digits to the right.
Example
Round the decimal number below to the nearest tenth:
3.782
Solution:
Step 1: Find the target rounding digit and the decision digit next to it.
The digit in the tenths place is 7. The decision digit directly to its right is 8.
Step 2: Check if the decision digit is 5 or greater.
Since 8 is greater than or equal to 5, we must round up by adding 1 to the tenths digit.
Step 3: Assemble your rounded decimal layout components:
3.782 → 3.8
Final Answer = 3.8
To round a decimal to the nearest hundredth, look at the digit in the thousandths place (the third number after the dot). If it is 5 or more, round up by adding 1 to your hundredths digit. If it is less than 5, leave it unchanged.
Example
Round the decimal number below to the nearest hundredth:
6.243
Solution:
Step 1: Find the target hundredths digit and the decision digit next to it.
The digit in the hundredths place is 4. The decision digit directly to its right is 3.
Step 2: Check if the decision digit is 5 or greater.
Since 3 is less than 5, we round down by keeping the hundredths digit exactly the same.
Step 3: Drop all the remaining numbers to the right side of your target place.
6.243 → 6.24
Final Answer = 6.24
A farmer harvested a crop of maize. He sold 0.4 of his total harvest to a local grain mill, and then sold 0.35 of the remaining harvest to a wholesale merchant. The rest of the maize was stored for family consumption. If he stored exactly 780 kilograms of maize for his family, calculate the total weight of the maize harvested in kilograms.
Step 1: Identify the given values.
Fraction sold to local mill = 0.4
Fraction sold to merchant = 0.35 of the remainder
Maize kept for family = 780 kg
We are required to find the total initial weight of the harvested maize.
Step 2: Calculate the remainder after the local mill sale.
Let the total initial harvest be represented as 1 whole unit.
First Remainder = 1 − 0.4 = 0.6
This means 0.6 of the total harvest is left over.
Step 3: Calculate the fraction sold to the wholesale merchant.
Merchant Sale = 0.35 × First Remainder
Merchant Sale = 0.35 × 0.6
Multiply as whole numbers: 35 × 6 = 210. Count 3 decimal places from the right:
Merchant Sale = 0.210 = 0.21
Step 4: Find the fraction left over for the family.
Family Fraction = First Remainder − Merchant Sale
Family Fraction = 0.6 − 0.21
Line up place values (0.60 − 0.21):
Family Fraction = 0.39
Step 5: Calculate the initial total weight of harvested maize.
Since 0.39 of the total harvest matches 780 kg, we divide 780 by 0.39:
Total Harvest = 780 ÷ 0.39
Move the decimal point 2 places to the right to make the divisor a whole number (39), and add two zeros to the dividend (78,000):
Total Harvest = 78,000 ÷ 39
Since 78 ÷ 39 = 2, we append the remaining zeros:
Total Harvest = 2,000 kg
Evaluate the numerical expression shown below using standard mathematical order of operations:
4.5 + 2.5 × (0.8 − 0.2) ÷ 0.3
Step 1: Solve the operation inside the parentheses (brackets) first.
Expression = 0.8 − 0.2
0.8 − 0.2 = 0.6
Now substitute this back into the full layout: 4.5 + 2.5 × 0.6 ÷ 0.3
Step 2: Perform division and multiplication from left to right.
First, calculate the multiplication part: 2.5 × 0.6
Multiply as whole numbers: 25 × 6 = 150. Count 2 decimal places from the right:
2.5 × 0.6 = 1.50 = 1.5
Now substitute this back into the updated layout: 4.5 + 1.5 ÷ 0.3
Step 3: Perform the remaining division step.
Expression = 1.5 ÷ 0.3
Move the decimal point 1 place to the right for both numbers to make them whole numbers:
15 ÷ 3 = 5
Now substitute this back into the calculation layout: 4.5 + 5
Step 4: Perform the final addition step.
Line up place values (4.5 + 5.0):
Answer = 9.5
A specific industrial metal alloy is created by combining copper, zinc, and tin. The alloy mixture contains 0.55 copper and 0.3 zinc by total weight, with the remaining weight consisting entirely of tin. If a structural piece of this alloy contains exactly 2.7 kilograms of tin, find the total mass of the entire alloy piece.
Step 1: Identify the given decimal values.
Copper content = 0.55
Zinc content = 0.3
Tin weight = 2.7 kg
Step 2: Find the combined decimal value of copper and zinc.
Combined Value = 0.55 + 0.3
Line up place values (0.55 + 0.30):
Combined Value = 0.85
Step 3: Calculate the decimal share left over for tin.
Let the complete alloy block equal 1 whole unit.
Tin Share = 1 − 0.85
Line up place values (1.00 − 0.85):
Tin Share = 0.15
Step 4: Calculate the total mass of the alloy block.
Since 0.15 of the total mass is equal to 2.7 kg, we divide 2.7 by 0.15:
Total Mass = 2.7 ÷ 0.15
Move the decimal point 2 places to the right to make the divisor a whole number (15). Move the dividend by 2 places as well, adding a placeholder zero (270):
Total Mass = 270 ÷ 15
Run long division: 15 goes into 27 once with a remainder of 12. Bring down the 0 to make 120. 15 goes into 120 exactly 8 times.
Total Mass = 18 kg
Convert the recurring (repeating) decimal 0.363636... into a simplified fraction, and then calculate its final value when multiplied by the terminating decimal 5.5.
Step 1: Set up an algebraic equation to convert the recurring decimal.
Let x = 0.363636... (Equation 1)
Since there are 2 repeating digits (3 and 6), multiply both sides of the equation by 100:
100x = 36.363636... (Equation 2)
Step 2: Subtract Equation 1 from Equation 2 to eliminate the repeating parts.
100x − x = (36.363636...) − (0.363636...)
99x = 36
Divide both sides by 99 to isolate x:
x = 36 ÷ 99
Step 3: Simplify the resulting fraction to its lowest terms.
Divide the top and bottom numbers by their greatest common factor, which is 9:
36 ÷ 9 = 4
99 ÷ 9 = 11
Converted Fraction = 4/11
Step 4: Convert the second number (5.5) into a fraction and multiply.
5.5 has 1 decimal place, so it can be written as 55/10, which simplifies to 11/2 by dividing by 5.
Now, multiply the two fractions together:
Cancel out the 11 from the top and bottom. This leaves 4/2, which equals 2.
Answer = 2
Three business partners split a shared office operation bill of KSh 8,450.50. Alice pays 0.45 of the total bill, and Ben pays 0.35 of the total bill. Charles pays the remaining financial balance. Calculate exactly how much money Charles pays, and round his final payment to the nearest whole shilling.
Step 1: Identify the shared cost values.
Total Office Bill = KSh 8,450.50
Alice Share = 0.45
Ben Share = 0.35
Step 2: Find the combined decimal share paid by Alice and Ben.
Combined Share = 0.45 + 0.35
Line up place values:
Combined Share = 0.80 = 0.8
Step 3: Calculate the decimal share left for Charles.
Charles Share = 1 − 0.8
Charles Share = 0.2
Step 4: Compute the exact cash payment amount for Charles.
Charles Payment = 0.2 × KSh 8,450.50
Multiply ignoring the decimal points (845050 × 2 = 1690100). Then count a total of 3 decimal places from the right side (1 from 0.2 and 2 from 8,450.50):
Exact Payment = KSh 1,690.100 = KSh 1,690.10
Step 5: Round the exact cash payment to the nearest whole shilling.
Look at the digit in the tenths place, which is 1.
Since 1 is less than 5, we round down by keeping the whole number exactly as it is and dropping the decimal parts.
Charles Final Payment = KSh 1,690
Attempt the following questions on decimals.
(i) Convert the recurring decimal 0.4444... into a simplified fraction in its lowest terms.
(ii) Arrange the following decimal numbers in descending order (largest to smallest): 0.7, 0.75, 0.079, 0.702.
(iii) Round the decimal number 14.826 to the nearest tenth (1 decimal place).
(iv) Calculate the exact value of the expression: 5.6 × 0.04.
(v) Convert the decimal number 0.65 into both a simplified fraction and a percentage.
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