Chapter 5 — Decimal Fractions

Introduction: Decimal fractions are numbers that show parts of a whole using a point called the decimal point. They let us write values smaller than one using digits after the decimal point.

We write decimal fractions with a decimal point between the units (ones) place and the tenths place. For example, 4.2 means four and two tenths. The digits to the right of the decimal point represent parts of one: tenths, hundredths, thousandths, and so on. A decimal like 0.5 means five tenths, which is the same as 1/2. Decimal fractions are another way to show parts, similar to vulgar fractions, but they follow place-value rules like whole numbers.

Decimal fractions are useful because they make arithmetic easier in many situations, especially when adding and subtracting parts. They are used widely in money (rupees and paise), measurements (metres and centimetres), and scientific quantities. Understanding decimals helps in converting between fractional and decimal forms and in comparing sizes of fractional parts quickly.

Important points:

• The decimal point separates whole-number part and fractional part.
• Places right of the point are: tenths (1st), hundredths (2nd), thousandths (3rd).
• Zeroes are important: 2.30 is same as 2.3, but 2.03 is different.

Learning decimal fractions gives students a clear and consistent way to work with parts of a whole and prepares them for more advanced arithmetic, percentages and measurements.

Understanding place value: Place value tells us the value of each digit in a number depending on its position. For decimal numbers, places to the left of the decimal point are units (ones), tens, hundreds, etc., while places to the right are tenths, hundredths, thousandths and so on.

For example, in 45.728, the digit 4 is in the tens place (value 40), 5 is in the units place (value 5). After the decimal point, 7 is in the tenths place (value 7/10 or 0.7), 2 is in the hundredths place (2/100 or 0.02) and 8 is in the thousandths place (8/1000 or 0.008). Each step to the right divides by 10. Thus tenths = 1/10, hundredths = 1/100 and thousandths = 1/1000.

Knowing place value helps in reading numbers, comparing them, aligning digits for operations, and understanding how arithmetic affects digits. If a digit moves one place to the left its value becomes ten times larger; moving one place to the right makes its value one-tenth.

Using zeros: Zeros are used as placeholders to show that a particular place value has no units. For example, 3.05 means three and five hundredths; the zero shows there are no tenths. Also 30.5 is different from 3.05 because the positions of digits are different.

Practise by writing several decimals and labelling each digit with its place and value. This habit builds accuracy for later operations like addition or rounding.

How to read decimals: Read the part before the decimal point as a whole number. Say 'point' for the decimal point then read each digit after the point separately, or name the fractional place when convenient. For example, 7.24 can be read as 'seven point two four' or 'seven and twenty-four hundredths'.

Which form to use depends on clarity. In measurement or money, you often say 'rupees and paise' for 12.50 (twelve rupees fifty paise) or 'metres and centimetres' for 2.35 metres as 'two metres and thirty-five centimetres' if conversion is suitable.

Writing decimals in words: If the digits after the decimal form a clear fraction in hundredths or thousandths, write it that way: 0.07 is 'seven hundredths'. If the digits are single or mixed, using 'point' and saying each digit is safe: 0.304 is 'zero point three zero four' or 'three hundred four thousandths'.

Always include the word that names the place value when you use the fractional form. For instance, 0.2 is 'two tenths'; 3.02 is 'three and two hundredths'. Zeroes in front of digits after the point must be read as 'zero' when pronouncing each digit, and included in the fractional name when necessary (e.g., 0.05 is 'five hundredths', not 'zero five hundredths').

Practise by converting written descriptions into decimal notation and vice versa. This strengthens place-value understanding and helps avoid mistakes in problems involving money, lengths, and weights.

Overview: Converting fractions to decimals and decimals to fractions are linked processes. For terminating decimals, conversion is straightforward using powers of 10. For some fractions, long division gives the decimal directly. Learning both directions helps to compare values and use the right form when solving problems.

Fractions to decimals (terminating): If the denominator can be turned into 10, 100 or 1000 by multiplying numerator and denominator by the same number, the fraction converts quickly to a decimal. Example: 7/25 multiply by 4 → 28/100 = 0.28. Alternatively, divide numerator by denominator: 3 ÷ 4 = 0.75. Fractions with denominators composed only of prime factors 2 and 5 will always terminate.

Decimals to fractions: Count how many digits appear after the decimal point. Write the number without the decimal as the numerator and 10^n (where n is number of decimal places) as the denominator, then simplify. Example: 0.625 → 625/1000 = 5/8. For a mixed number like 2.5, write 25/10 and simplify to 5/2 or 2 1/2.

Using division: When the denominator is not a factor of 2 or 5, dividing numerator by denominator gives a repeating decimal (e.g., 1/3 = 0.333...). This shows when fractions do not give terminating decimals. In class 6, focus on terminating cases and simple repeating examples.

Practice tip: Always simplify fractions after conversion. Check your work by converting back (divide fraction to get decimal or write decimal over appropriate power of ten to get fraction). This habit ensures accuracy and strengthens understanding of place value.

Types of decimals: A decimal is terminating if it ends after a finite number of digits, for example 0.25 or 2.5. A decimal is non-terminating when digits continue forever; such decimals can be repeating (a pattern of digits repeats) or non-repeating (no fixed repeating pattern). In elementary arithmetic we mainly meet terminating and repeating decimals.

Why some fractions terminate: A fraction, when written in simplest form, gives a terminating decimal if and only if its denominator has no prime factors other than 2 and 5. This is because powers of 10 are made of 2s and 5s (10 = 2 × 5). For example, 1/8 = 0.125 terminates because 8 = 2^3; 3/25 = 0.12 terminates because 25 = 5^2. If the denominator contains prime factors like 3, 7, 11 etc., the decimal will repeat. For example, 1/3 = 0.333..., 2/7 = 0.285714285714... (repeating cycle).

How to see repetition: When you perform long division of numerator by denominator, if you reach a remainder of zero the decimal terminates. If the remainders begin to repeat, the digits after the decimal will repeat in a cycle. The length of the repeating cycle depends on the denominator.

Noting repeating decimals: We mark repeating digits with a bar above them: 0.333... = 0.1, where the bar indicates the repeating part (here 3). In class 6, recognise simple repeating cases such as 0.666... and practices like converting 1/6 to 0.1666... which has a repeating 6 after the first digit.

Practical use: Money and measurements are usually rounded to two or three decimal places, so terminating decimals are most common in daily life. When a repeating decimal appears in a calculation, either round it to required places or convert to a fraction for an exact answer. Practise by simplifying fractions to see if denominators are only powers of 2 and 5 and by carrying out long division to observe remainder patterns.

How to compare: To compare two decimals, first compare the digits left of the decimal point as whole numbers. If these differ, the larger whole-number part means the larger decimal. If whole parts are equal, compare digits in the tenths place; the larger tenths digit is the larger number. If tenths are equal, compare hundredths, then thousandths, and so on. If one decimal has fewer digits, add zeroes at the end to make the places equal before comparing.

For example, compare 3.45 and 3.345. Whole parts both 3, tenths both 4 → compare hundredths: 5 vs 3, so 3.45 > 3.345. Similarly, 2.5 and 2.50 are equal because 2.50 = 2.5 after adding a zero in the hundredths place.

Visual methods: Align numbers in a column with the decimal points in the same place so that digits of the same place value are under each other. Another visual method is to place numbers on a number line; the number farther to the right is larger. For lists, convert all to the same number of decimal places by adding trailing zeros which makes digit-by-digit comparison clearer.

Tips and checks: Remember that adding trailing zeros does not change the value. When comparing negative decimals, the one with greater absolute value is actually smaller (for example, -0.8 < -0.3). If confused, convert decimals to fractions with a common denominator (like 100 or 1000) and compare numerators. Use simple practice problems and encourage students to explain why one number is greater using place value language.

Comparing decimals is important in real life for comparing prices, lengths and weights. Practice with many pairs, including those with different lengths after the decimal point and with equal whole parts, to become quick and accurate.

Overview: Ordering a set of decimals means arranging them from smallest to largest or vice versa. The method builds on comparing decimals place by place. With many numbers, it is helpful to standardise the number of digits after the decimal by adding trailing zeros so comparisons are direct and error-free.

Step-by-step method: First, write all decimals in a column with decimal points aligned. Next, if decimals have different lengths, add zeroes at the end of the shorter ones to create equal places (for example, change 2.5 to 2.50 to compare with 2.48). Then compare from the leftmost digit (whole number part) to the right (tenths, hundredths, thousandths) just as you would compare two numbers. For each place, eliminate numbers that are clearly larger or smaller until you place them in order.

Use number line and conversion tricks: It helps to draw a number line for visual learners and place each decimal. Another method is to convert all decimals to fractions with a common denominator (like 100 or 1000) and compare numerators; this is especially useful when numbers have many digits. For negative decimals, remember that a more negative value is smaller on the number line and comes first when arranging from smallest to largest.

Practical checks: After ordering, you can check correctness by ensuring each successive number is greater than the previous one (for ascending order). Also test equality by removing trailing zeros to see which numbers are the same value (e.g., 2.5 and 2.50).

Practice sets should include decimals with differing whole parts, decimals that require adding zeros, and negative decimals. Clear alignment and careful digit-by-digit comparison prevent most mistakes and produce quick, accurate results.

Why round? Rounding decimals gives an approximate value that is easier to use when exact precision is not required, such as estimating money, measuring lengths, or reporting results. Rounding reduces digits while keeping the number close to the original value.

Rules of rounding: Choose the place to which you want to round (tenths, hundredths, etc.). Look at the digit immediately to the right of that place.

• If that digit is 0,1,2,3,4 → keep the chosen digit the same and replace all digits to its right by zeroes (or drop them if writing decimal form).
• If that digit is 5,6,7,8,9 → increase the chosen digit by one and drop all digits to the right.

Examples: Round 3.276 to one decimal place (tenths). The hundredths digit is 7 (≥5) so increase tenths 2 → 3.3. To round 0.864 to two decimal places, look at thousandths 4 (<5) so 0.86 stays 0.86.

Special cases: Rounding may change the whole-number part when digits carry over (e.g., 2.96 to one decimal → 3.0). When rounding negative numbers, apply the same rule but be careful: -1.24 rounded to one decimal → -1.2 because the next digit 4 is less than 5 and we keep -1.2; -1.26 → -1.3 because of 6 (increase magnitude).

Teach students to practice with money or measurement examples. Always state the place value used for rounding in the answer and show the digit checked to avoid mistakes.

Key idea: When adding decimals, align the decimal points vertically so that digits of the same place value fall in the same column. Adding decimals is like adding whole numbers when each place (tenths, hundredths, thousandths) is kept under the same column.

Step-by-step:

• Write the numbers one under another with decimal points aligned.
• If some numbers have fewer decimal places, add trailing zeros so all numbers have the same number of places (for example, write 3.4 as 3.40 if adding with a number having two decimal places).
• Add digits starting from the rightmost column (smallest place) moving leftwards, carry over when a column sum is 10 or more.
• Place the decimal point in the answer directly under the decimal points of the numbers being added.

Worked example and checks: For example, add 2.75 and 1.4. Align: 2.75 + 1.40. Add hundredths: 5 + 0 = 5. Add tenths: 7 + 4 = 11, write 1 and carry 1 to the units column. Units: 2 + 1 + carried 1 = 4. Result = 4.15. Another example: 0.875 + 1.125. Align and add gives 2.000. To check, round each number to one decimal place and add to see the approximate result; or subtract one addend from the sum to recover the other addend for verification.

Common mistakes to avoid: Do not forget to align decimal points, do not ignore trailing zeros, and do not misplace the decimal in the answer. Encourage neat columns and always verify by reverse operation (subtraction) when possible.

Practice with money (rupees and paise) and length units where decimals represent sub-units; this gives meaningful experience and builds confidence in handling carries and multiple addends.

Approach: Subtraction of decimals follows the same rules as subtraction of whole numbers: align decimal points, equalise decimal places by adding trailing zeros, and subtract digit by digit from right to left, borrowing when necessary. The decimal point in the result must be placed directly below the other decimal points.

Step-by-step method:

• Write the minuend (larger number) above the subtrahend with decimal points aligned.
• Add zeros to the shorter number so both have the same number of digits after the decimal.
• Subtract from the rightmost place. If a digit in the minuend is smaller than the corresponding digit in the subtrahend, borrow 1 from the next left place (which reduces that left digit by 1 and adds 10 to the current place).
• Continue borrowing if necessary, and place the decimal point in the answer directly below the aligned decimal points.

Worked example and checking: For example, subtract 5.000 − 2.374. Equalise places: 5.000 − 2.374. Subtract thousandths: 0 − 4, borrow from hundredths, but hundredths is 0 so borrow from tenths, etc. After borrowing across places, thousandths: 10 − 4 = 6, hundredths: 9 − 7 = 2, tenths: 9 − 3 = 6, units: 4 − 2 = 2, result = 2.626. Check by adding 2.626 + 2.374 = 5.000 to confirm correctness.

Negative results and checks: If the top number is smaller than the bottom number, the result will be negative. Always check by adding the difference to the subtrahend; you should get the minuend. Practise with money (change problems) and lengths to understand borrowing across decimal places.

Effect on decimals: Multiplying or dividing a number by 10, 100 or 1000 changes the place value of each digit. Each multiplication by 10 moves the decimal point one place to the right; each division by 10 moves it one place to the left. This rule applies to whole numbers and decimals alike and gives a fast way to calculate without formal multiplication or division steps.

Why it works: Multiplying by 10 means each unit becomes ten units, so the digit that was in the ones place moves to the tens place, and digits to the right move accordingly. Dividing by 10 makes each unit one tenth of its former value, shifting digits the other way. For example, 3.45 × 10 = 34.5 because the digits shift one place right. If shifting runs out of digits on one side, you add zeros to fill places (for example, 0.37 × 100 = 37.00 = 37).

Practical applications: Conversions often use these shifts: metres to centimetres (×100), rupees to paise (×100), kilograms to grams (×1000). When converting, keep track of units and add zeros where needed. For division, converting large numbers to smaller units may involve dividing by 10, 100 or 1000; remember to move the decimal left and, if necessary, add leading zeros (for example, 5 ÷ 100 = 0.05).

Common mistakes: Students sometimes move the decimal in the wrong direction or forget to add zeros. A quick check is to estimate: multiplying by 100 should make the result 100 times larger; if it looks too small, the decimal likely moved the wrong way. Practice with mixed examples including whole numbers, simple decimals and unit conversions to become fluent.

Approach to word problems: Read the problem carefully and underline important numbers and units. Identify what is asked and whether to add, subtract, multiply or divide. Convert all quantities to the same unit and use decimals where parts of units appear. Make neat calculations and include units in the answer.

Types of problems: Common problems use money (rupees and paise), length (metres and centimetres), weight (kg and g), and quantities like litres. For example, add lengths given in metres with decimals, or find total cost when price per kg and weight in kg (with decimals) are given. Converting mixed units to a single unit before computation avoids mistakes: convert 2 m 35 cm to 2.35 m or 235 cm depending on what is easier.

Step-by-step strategy:

• Underline numbers and required answer.
• Convert mixed units to decimals in the same unit (e.g., paise to rupees by dividing by 100 or metres and centimetres to metres by writing decimal).
• Choose the correct operation and align decimal points for addition/subtraction or shift decimal for multiplication/division by 10, 100, 1000.
• Solve, round if required, and include units in the final answer.

Examples and checks: If 1 kg rice costs Rs. 45.50, the cost of 2.5 kg = 45.50 × 2.5. Multiply 45.50 by 2.5 using column or break into 45.50 × 2 + 45.50 × 0.5 for easier work. After calculation, re-read the question to ensure you answered what was asked (total cost, remaining money, or length remaining) and that units are correct. Estimation before calculation helps check whether the final answer is reasonable.

Practice varied problems and show full working with unit conversions and alignment of decimals to build confidence and accuracy.