Overview
Chapter: Decimals (Mathematics – VI) — Introduction, Importance, Key Themes and Learning Outcomes. Introduction: This chapter introduces the idea of decimals as an extension of the place value system to the right of the decimal point. Using familiar contexts (money, lengths, and measurement), students learn to represent parts of a whole in tenths, hundredths and thousandths, read and write decimal numbers, and place them on the number line. Importance: Decimals provide a flexible way to work with quantities smaller than one and are essential for everyday calculations (money, measurement, weights) and later topics in mathematics (fractions, percent, algebra). A solid foundation in decimals helps students perform accurate computations and compare precise values. Key Themes: place value and the decimal point; tenths, hundredths and thousandths; conversion between fractions (with denominators 10, 100, 1000) and decimals; representing decimals on the number line; comparing and ordering decimals; basic decimal arithmetic (addition and subtraction by aligning decimal points; multiplying and dividing by 10, 100, 1000); rounding and estimation; and practical problem solving using…
Learning Objectives
- Define decimal numbers and identify place values of digits (tenths, hundredths, thousandths).
- Explain the relationship between fractions and decimals with examples.
- Read and write decimals (up to three decimal places) in words and numerals.
- Represent decimals on a number line and locate their relative positions.
- Compare and order decimals up to three decimal places using place-value reasoning.
- Convert decimals to fractions and fractions (with denominators 10, 100, 1000) to decimals and simplify where possible.
- Round decimals to the nearest whole number, tenth, or hundredth and justify the result.
- Add and subtract decimals (aligned by place value) and solve related word problems.
Topics in this chapter
9 topics · tap a topic title to jump straight to it.
Introduction / Need for Decimals
What are decimals?
Decimals are a way to write numbers that are not whole by using a decimal point. The digits to the left of the decimal point show whole units and the digits to the right show parts (fractions) of a unit in base 10 (tenths, hundredths, thousandths, ...).
Why do we need decimals?
- Many real-life quantities are not whole: money, lengths, weights, time and temperature often come in parts of a unit. Decimals give a clear, compact way to write these parts using the base-10 system.
- Decimals align with the place-value system used for whole numbers, so operations (addition, subtraction, multiplication, division) are easier and consistent.
- Decimals are often simpler than fractions for measurement, calculation and comparison because they use the same positional notation as whole numbers.
How decimals relate to fractions
Each decimal place corresponds to a power of 10. For example, 0.3 is 3 tenths = 3/10; 0.25 is 25 hundredths = 25/100. You can convert a terminating decimal to a fraction by placing the decimal digits over 10, 100, 1000, ... and simplifying.
Reading a decimal
Read 12.47 as “twelve point four seven,” or interpret the part after the decimal as tenths and hundredths: 12.47 = 12 + 4/10 + 7/100.
Key ideas to remember
- The place immediately right of the decimal point is the tenths place, then hundredths, thousandths, etc.
- Decimals are simply another way to represent parts of a whole consistent with base 10.
- To compare decimals, line up the decimal points and compare digits from left to right (or convert to equal number of decimal places).
- Money: ₹1.25 means 1 rupee and 25 paise. As a decimal: 1.25 = 1 + 25/100.
- Length: 1.5 m means 1 metre and 50 centimetres (because 0.5 m = 50 cm). So 1.5 m = 1 + 5/10 metres.
- Weight: 0.75 kg = 75 hundred grams = 3/4 kg (0.75 = 75/100 = 3/4).
- Temperature: 36.6°C means thirty-six point six degrees; the 0.6 is six tenths of a degree.
- Petrol gauge or fuel quantity: 12.8 L means 12 litres and 800 millilitres (0.8 L = 800 mL).
- Fraction to decimal example: 3/10 = 0.3; 7/100 = 0.07; 1/4 = 0.25 (by dividing 1 by 4).
- Place values: 10 tenths = 1 unit, 10 hundredths = 1 tenth, etc. So 1 = 10 × 0.1 = 100 × 0.01 = 1000 × 0.001.
- Decimal to fraction (terminating): If a decimal has n digits after point, decimal = integer / (10^n). Example: 0.456 = 456/1000.
- Fraction to decimal: Divide numerator by denominator. If denominator is a factor of a power of 10 (2 and/or 5 only), you get a terminating decimal.
- Comparing decimals: Align decimal points and compare digits from left to right. Or make equal decimal places by adding zeros.
- Converting repeating decimals (introduction level): repeating forms (like 0.333...) represent fractions (0.333... = 1/3) — detailed conversion is taught later.
Tenths and Hundredths
What are tenths and hundredths?
When we write numbers with a decimal point, digits to the right of the decimal show parts less than one. The first place right of the decimal is the tenths place (1/10), and the second place is the hundredths place (1/100).
Place-value positions around the decimal point (from left to right): … | tens | ones | decimal point | tenths | hundredths | thousandths | …
Value of a digit = (digit) × (place value). For example, in 4.37:
- 4 is in the ones place = 4 × 1 = 4
- 3 is in the tenths place = 3 × 1/10 = 3 × 0.1 = 0.3
- 7 is in the hundredths place = 7 × 1/100 = 7 × 0.01 = 0.07
Reading Decimals: 0.1 is read as "one tenth"; 0.01 is read as "one hundredth". A number like 2.05 is read as "two and five hundredths".
Converting between fractions and decimals: To convert a fraction whose denominator is 10 or 100, write the numerator in the tenths or hundredths place. Example: 3/10 = 0.3 and 7/100 = 0.07. To convert a decimal to a fraction, write the decimal digits as the numerator and the place value as denominator then simplify: 0.6 = 6/10 = 3/5; 0.06 = 6/100 = 3/50.
Comparing and operating with tenths and hundredths: To compare decimals with tenths and hundredths, compare digits from left to right (ones, then tenths, then hundredths). For addition and subtraction, align decimal points and, if needed, convert tenths to hundredths by adding a zero: e.g., 0.4 = 0.40 to add to 0.27.
Why hundredths matter: Some quantities need finer measurement than tenths (for example money, where 1 rupee = 100 paise), so we use hundredths to express such small parts precisely.
- 3/10 = 0.3 (three tenths).
- 7/100 = 0.07 (seven hundredths).
- Write 5.28: 5 is ones, 2 is tenths (0.2), 8 is hundredths (0.08). So 5.28 = 5 + 0.2 + 0.08.
- Convert 0.6 to a fraction: 0.6 = 6/10 = 3/5 (simplified).
- Convert 0.06 to a fraction: 0.06 = 6/100 = 3/50 (simplified).
- Adding 0.4 and 0.27: write 0.4 as 0.40, then 0.40 + 0.27 = 0.67.
- Value of digit = (digit) × (place value). Example: in 4.37, 3 = 3 × 1/10 = 0.3; 7 = 7 × 1/100 = 0.07.
- Tenths ↔ fraction: x tenths = x/10 = 0.x (for a single-digit x).
- Hundredths ↔ fraction: y hundredths = y/100 = 0.0y (use leading zero if needed).
- Decimal → fraction: Move decimal digits to numerator and use 10, 100, 1000... as denominator then simplify. Example: 0.45 = 45/100 = 9/20.
- To add/subtract decimals: align decimal points; if needed, pad with zeros to equalize places (e.g., 0.4 → 0.40).
Decimal Notation and Place Value
What is decimal notation? Decimal notation is a system to write numbers that are not whole by using a decimal point (.) to separate the whole part from the fractional part. The digits to the right of the decimal point represent parts of one (tenths, hundredths, thousandths, etc.) in powers of 10.
Place value for decimals
- To the left of the decimal point: ... hundreds, tens, ones (units).
- The decimal point (.) separates the whole number part from the fractional part.
- To the right of the decimal point: tenths (1/10), hundredths (1/100), thousandths (1/1000), and so on.
Place-value table (example)
... | 100 | 10 | 1 | . | 1/10 | 1/100 | 1/1000 | ...
Example: 532.47 = 5×100 + 3×10 + 2×1 + 4×(1/10) + 7×(1/100)
Expanded form and reading
Expanded form writes each digit with its place value (see example above). To read decimals, say the whole number part, say "point" for the decimal point, then read each digit of the fractional part separately (e.g., 3.04 = "three point zero four"). For common denominators (tenths, hundredths) you can also say "and fifty-seven hundredths" for 0.57.
Relating decimals to fractions
- Digits after decimal are fractions whose denominators are powers of 10: 0.3 = 3/10, 0.25 = 25/100 = 1/4, 0.125 = 125/1000 = 1/8.
- To convert a decimal to a fraction, place the decimal digits over the corresponding power of 10 and simplify.
Comparing and ordering decimals
Compare digits from left to right starting at the largest place value. If needed, add zeros to the right to make the same number of places (e.g., 2.5 = 2.50).
Uses and intuition
Decimals give a compact way to express parts of a unit in money, measurement and data. They fit the base-10 system and make calculation with fractions that have denominators 10, 100, 1000 easy.
- Money: Rs 12.75 means 12 rupees and 75 paise = 12 + 75/100 = 12 + 0.75.
- Measurement: 2.3 m = 2 metres and 3 tenths of a metre = 2 + 3/10 metres.
- Convert fraction to decimal: 3/10 = 0.3 , 7/100 = 0.07 , 125/1000 = 0.125.
- Expanded form: 46.208 = 4×10 + 6×1 + 2×(1/10) + 0×(1/100) + 8×(1/1000).
- Compare decimals: 4.56 and 4.506 → compare tenths (5=5), compare hundredths (6 > 0) so 4.56 > 4.506.
- Convert decimal to fraction: 0.875 = 875/1000 = 7/8 after simplification.
- Value of a digit in nth place after decimal = digit × 10^(-n) (e.g., in 0.432, 4 is 4×10^-1 = 4/10).
- Decimal to fraction: move fractional digits over denominator 10^n, then simplify (e.g., 0.36 = 36/100 = 9/25).
- Fraction with denominator 10^n to decimal: write numerator with decimal point placed n digits from right (e.g., 7/1000 = 0.007).
- Expanded form: sum of each digit × its place value (e.g., 123.45 = 1×100 + 2×10 + 3×1 + 4×1/10 + 5×1/100).
- Percent conversion: percent = decimal × 100 (e.g., 0.25 = 25%).
- Rounding to n decimal places: look at (n+1)th digit; if ≥5 round up the nth digit, otherwise keep it.
Decimals on the Number Line
What is a decimal on the number line?
A decimal number represents a part of a whole and can be shown on a number line just like whole numbers. To place a decimal on the number line, find the two consecutive integers between which the decimal lies, then divide that interval into equal parts (tenths, hundredths, etc.) according to the number of decimal places.
Steps to plot a decimal
- Locate the integers that bound the decimal. Example: 0.7 is between 0 and 1; 1.25 is between 1 and 2.
- Decide the place-value partition needed. For one decimal place (tenths) divide each unit interval into 10 equal parts. For two decimal places (hundredths) divide into 100 equal parts, and so on.
- Count the required parts from the left endpoint and mark the decimal. Example: 0.7 is the 7th mark when 0–1 is divided into ten equal parts; 1.25 is 25 hundredths after 1 (or the 25th mark if 1–2 is divided into 100 parts).
Place value reminder
Each decimal place represents a fraction of 1: the first place after the decimal point is tenths (1/10), the second is hundredths (1/100), the third is thousandths (1/1000), etc. So 0.34 = 3×(1/10) + 4×(1/100).
Comparing decimals on the number line
The one that lies to the right is larger. To compare decimals easily, align digits by place value (add trailing zeros if needed) and compare digits from left to right. Example: 0.47 > 0.403 because 0.47 = 0.470 and 0.470 > 0.403.
Converting fractions to decimals for placement
To place a fraction on the number line, convert it to a decimal by division (numerator ÷ denominator) or by writing an equivalent fraction with denominator 10, 100, 1000, ... Example: 3/10 = 0.3 so it is the 3rd tenth from 0.
Useful idea: zooming in
If two decimals are very close (e.g., 0.37 and 0.38), draw a smaller number line only from 0.37 to 0.38 and subdivide that interval to show hundredths or thousandths. This helps visualize tiny differences.
- Money: ₹12.50 is 12 rupees and 50 paise = 12 + 50/100 = 12.50; on a number line it lies between 12 and 13, halfway at 12.5.
- Length: 0.75 m = 75 cm. On a 0–1 number line divided into 100 parts, 0.75 is at the 75th mark (three quarters).
- Placing 0.3: divide interval 0–1 into 10 equal parts (tenths); 0.3 is at the 3rd mark from 0 = 3/10.
- Placing 1.25: between 1 and 2, divide into 100 equal parts for hundredths (or first into tenths then hundredths); 1.25 is 25 hundredths after 1.
- Negative decimal: -0.6 is on the left side of 0; on -1 to 0 interval divided into 10 parts, -0.6 is 6th mark from 0 toward -1.
- Place-value: digit at tenths = digit × 1/10; hundredths = digit × 1/100; thousandths = digit × 1/1000.
- Decimal to fraction: move decimal point right n places (n = number of decimal digits) and divide by 10^n. Example: 0.46 = 46/100 = 23/50.
- Fraction to decimal: divide numerator by denominator (or convert to an equivalent denominator 10, 100, ...). Example: 3/4 = 3 ÷ 4 = 0.75.
- Compare decimals: equalize decimal places by adding trailing zeros, then compare digits left to right. Example: 0.5 = 0.50; compare 0.50 and 0.47 → 0.50 > 0.47.
- Distance between two decimals a and b on the number line: |a - b|.
Comparing and Ordering Decimals
What is a decimal? A decimal number is a way to represent parts of a whole using a decimal point. Each digit to the right of the decimal point has a place value: tenths (10⁻¹), hundredths (10⁻²), thousandths (10⁻³), and so on.
How to compare decimals (step-by-step)
- Step 1: Compare the whole-number (integer) parts. The number with the larger whole part is greater.
- Step 2: If the whole parts are equal, compare digits in the tenths place. The larger tenths digit makes the number larger.
- Step 3: If tenths are equal, move to the hundredths place, then thousandths, etc., comparing digit by digit from left to right.
- Rule when digits run out: Treat missing digits as zeros. For example, 3.5 = 3.50, 0.6 = 0.60.
- Alternative method: Convert decimals to like (same number of decimal places) by adding zeros or multiply by a power of 10 to make them whole numbers and then compare those whole numbers.
Tips: Always align decimal points when writing decimals vertically. When ordering (ascending or descending), use the same comparison steps for each pair.
Why this works: Each place after the decimal is 1/10 of the previous place. Comparing left-to-right ensures you compare larger-value places first (tenths before hundredths), which determines which number is larger.
- Compare 4.37 and 4.305: Whole parts are equal (4). Compare tenths: 3 and 3 (equal). Compare hundredths: 7 and 0 → 7 > 0, so 4.37 > 4.305. (Or write 4.37 as 4.370 to compare digits directly.)
- Compare 0.6 and 0.59: Whole parts 0 = 0. Tenths: 6 and 5 → 6 > 5, so 0.6 > 0.59. (Also 0.6 = 0.60, hence 0.60 > 0.59.)
- Order the set {3.4, 3.04, 3.400, 3.399}: Convert or align zeros: 3.400 = 3.4 and 3.04 = 3.040. Compare: 3.399 < 3.4 = 3.400, and 3.04 is the smallest. So ascending: 3.04, 3.399, 3.4 (3.400).
- Using multiplication: To compare 0.72 and 0.705 multiply by 1000 → 720 and 705. Since 720 > 705, 0.72 > 0.705.
- Real-life: Shopping prices 149.50 and 149.45. Whole parts equal (149). Compare tenths (5 and 4) → 149.50 > 149.45, so the first price is costlier.
- \[Place value: value of a digit = digit × (10^{−n}) where n = position after decimal (tenths n=1\]\[hundredths n=2, ...).\]
- Make decimals like (same number of places): add trailing zeros, e.g., 2.5 = 2.50 = 2.500.
- Compare by converting to integers: multiply both decimals by 10^k (k = number of decimal places needed) to remove the decimal point, then compare the resulting whole numbers.
- Decimal to fraction: move decimal point right k places → numerator; denominator = 10^k. Example: 0.375 = 375/1000 = 3/8 (after simplification).
Conversion Between Fractions and Decimals
Conversion between fractions and decimals means expressing the same part of a whole in two different ways. A fraction a/b shows 'a' equal parts out of 'b' equal parts. A decimal shows parts of 1 using place values such as tenths, hundredths, thousandths, etc.
To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number). The result is either a terminating decimal (stops) or a repeating decimal (a digit or group of digits repeats).
To convert a decimal to a fraction, use the place value of the last decimal digit: write the decimal as an integer over 10, 100, 1000 (depending on tenths, hundredths, thousandths), then simplify the fraction. For repeating decimals, use a shortcut: put the repeating digits over as many 9s as there are repeating digits, then simplify.
Important ideas to remember:
- If the denominator of a fraction is a factor of 10, 100, 1000 (e.g., 2, 4, 5, 8, 10, 25), the decimal will terminate.
- Repeating decimals can be changed to fractions by placing the repeating block over 9, 99, 999 depending on how many digits repeat (then simplify).
- Always simplify the fraction after converting a decimal to a fraction.
- Convert 3/4 to a decimal: 3 ÷ 4 = 0.75.
- Convert 2/5 to a decimal: 2 ÷ 5 = 0.4 (or 0.40).
- Convert 1/8 to a decimal: 1 ÷ 8 = 0.125.
- Convert 1/3 to a decimal: 1 ÷ 3 = 0.333... (repeating). Write as 0.\u03051 (0.\u0305333...) often shown 0.\u0305333 or 0.3 repeating.
- Convert 0.75 to a fraction: 0.75 = 75/100 = 3/4 after simplifying (divide numerator and denominator by 25).
- Convert 0.125 to a fraction: 0.125 = 125/1000 = 1/8 (simplify by 125).
- Fraction → Decimal: a/b = a ÷ b (use long division).
- Decimal → Fraction (terminating): If decimal has n digits after point, decimal = integer / 10^n. Example: 0.36 = 36/100 = 9/25.
- Repeating decimal → Fraction (simple rule): If k digits repeat, place the repeating digits over k nines. Example: 0.777... = 7/9, 0.121212... = 12/99 (then simplify).
- General simplification: After forming a fraction, divide numerator and denominator by their greatest common divisor (GCD).
Addition and Subtraction of Decimals
What are decimals? Decimals represent parts of a whole using a decimal point. Each place to the right of the decimal point is a fraction of 10 (tenths, hundredths, thousandths, ...).
Basic idea of addition and subtraction of decimals
- Always line up the decimal points in the numbers you are adding or subtracting. This ensures like place values (tenths with tenths, hundredths with hundredths) are operated on.
- If needed, add zeros at the end of a number to make all numbers have the same number of decimal places (for example, 3.5 = 3.50 when adding with 2.37).
- Work from right to left (from the smallest place value to the left), just like in whole-number addition/subtraction. For subtraction, use borrowing (regrouping) when a smaller digit is subtracted from a larger place value digit.
- Place the decimal point in the answer directly under the decimal points of the numbers being added or subtracted.
Key points to remember
- Addition is commutative: a + b = b + a. You can change the order of decimals when adding.
- Subtraction is not commutative: a - b ≠ b - a, so keep the order correct.
- When adding or subtracting more than two decimals, line up all decimal points before performing operations.
Worked strategy (column method)
- Write each number one under the other aligning the decimal points.
- Add zeros to make the number of digits after the decimal equal for all numbers.
- Add or subtract digits column by column from right to left, carrying or borrowing as needed.
- Place the decimal point in the result under the other decimal points.
- Example 1 (Addition): 2.5 + 3.75 Step 1: Align decimals: 2.50\n 3.75 Step 2: Add hundredths: 0 + 5 = 5 (hundredths) Step 3: Add tenths: 5 + 7 = 12 → write 2, carry 1 to ones Step 4: Add ones: 2 + 3 + 1(carry) = 6 Answer: 6.25
- Example 2 (Subtraction with borrowing): 5.00 - 2.37 Step 1: Align decimals: 5.00\n 2.37 Step 2: Subtract hundredths: 0 - 7 → need to borrow from tenths. Tenths is 0, so borrow 1 from ones: 5 → 4 ones, tenths becomes 10 → after giving 1 to hundredths tenths becomes 9 and hundredths becomes 10. Now hundredths: 10 - 7 = 3 Tenths: 9 - 3 = 6 Ones: 4 - 2 = 2 Answer: 2.63
- Example 3 (Addition with different decimal lengths): 12.345 + 3.7 Step 1: Make decimal places equal: 12.345\n 3.700 Step 2: Add thousandths: 5 + 0 = 5 Hundredths: 4 + 0 = 4 Tenths: 3 + 7 = 10 → write 0, carry 1 to ones Ones: 2 + 3 + 1 = 6 Tens: 1 + 0 = 1 Answer: 16.045
- Example 4 (Subtraction giving a decimal less than 1): 0.8 - 0.345 Step 1: Align and add zeros: 0.800\n 0.345 Step 2: Subtract thousandths: 0 - 5 → borrow from hundredths (0) → borrow from tenths: 8 tenths → becomes 7 tenths, hundredths becomes 10 → then give 1 to thousandths: hundredths 9, thousandths 10. Now thousandths: 10 - 5 = 5 Hundredths: 9 - 4 = 5 Tenths: 7 - 3 = 4 Answer: 0.455
- Align decimal points before adding or subtracting.
- If decimal places differ, append zeros: e.g., 3.5 = 3.50 = 3.500 to match places.
- Column addition rule: add digits from right to left; carry to the next left column when sum ≥ 10.
- Column subtraction rule: subtract digits from right to left; borrow (regroup) from the next left column when needed.
- Commutative property for addition: a + b = b + a (helps to rearrange terms to simplify).
- Subtraction is not commutative: a - b ≠ b - a (order matters).
Applications and Word Problems
What this topic means
Applications and word problems with decimals teach how to use decimal numbers to solve everyday problems — for money, length, weight, capacity, area, time, temperature, etc. Students learn to read a word problem, convert quantities to decimals (if needed), set up the correct arithmetic (addition, subtraction, multiplication, division) and interpret the answer.
Steps to solve decimal word problems
- Read carefully: Identify what is given and what is asked.
- Convert units: Make sure all quantities use the same units (metres, centimetres, rupees, litres).
- Write numbers as decimals: Convert fractions or mixed numbers to decimals when needed.
- Choose operation: Decide whether to add, subtract, multiply or divide.
- Align and compute: For addition/subtraction align decimal points; for multiplication/division follow decimal rules (see formulas).
- Estimate and check: Round values to estimate the result and check the reasonableness of the answer.
Common contexts: shopping (prices, discounts), measurement (length, weight, liquid), area/volume, speed & time, sharing/dividing quantities, currency conversions.
Tips: Always line up decimal points for addition/subtraction; when multiplying, ignore decimals, multiply as whole numbers, then place the decimal point; when dividing by a decimal, shift decimal points to make the divisor a whole number.
- 1) Buying apples: A customer buys 2.75 kg of apples at Rs 40 per kg. What is the cost? Solution: Multiply 2.75 × 40. 2.75 × 40 = 2.75 × (4 × 10) = (2.75 × 4) × 10 = 11.00 × 10 = 110.00. Cost = Rs 110.
- 2) Cloth measurement (unit conversion and addition): Rina has two pieces of cloth: 1.85 m and 125 cm. Find the total length in metres. Solution: Convert 125 cm to metres: 125 cm = 1.25 m. Now add: 1.85 + 1.25 = 3.10 m. Total = 3.10 m (or 3.1 m).
- 3) Subtraction with decimals: A water tank had 50.00 L. After using 12.345 L, how much remains? Solution: 50.000 − 12.345 = 37.655 L. Remaining water = 37.655 L.
- 4) Multiplication (area): A rectangular garden is 3.6 m long and 2.5 m wide. Find the area. Solution: 3.6 × 2.5. Multiply as integers: 36 × 25 = 900. Total decimal places = 1 + 1 = 2, so area = 9.00 m² or 9 m².
- 5) Division (sharing): 7.5 kg of rice is shared equally among 5 families. How much does each get? Solution: 7.5 ÷ 5 = 1.5 kg per family.
- Place value: tenths (0.1), hundredths (0.01), thousandths (0.001). E.g., 3.456 = 3 + 4×0.1 + 5×0.01 + 6×0.001.
- Addition/Subtraction: align decimal points vertically before adding or subtracting.
- Multiplication by 10, 100, 1000: move decimal point right by 1, 2, 3 places respectively. E.g., 0.37 × 100 = 37.
- Division by 10, 100, 1000: move decimal point left by 1, 2, 3 places respectively. E.g., 12.5 ÷ 10 = 1.25.
- General multiplication of decimals: multiply as integers (ignore decimals), then place the decimal point so that the total number of decimal places equals the sum of decimal places in the factors. Example: 2.5 (1 dp) × 0.24 (2 dp) → 25 × 24 = 600 → result has 3 dp → 0.600 = 0.6.
- Division when divisor is a decimal: multiply dividend and divisor by the same power of 10 to make the divisor a whole number, then divide normally. Example: 3.75 ÷ 0.25 → multiply both by 100 → 375 ÷ 25 = 15.
Equivalent Forms and Place-value Tricks
What are Equivalent Forms? Equivalent forms of a decimal are different ways of writing the same numerical value. Common equivalent forms include:
- Standard decimal form (e.g., 4.5)
- Decimal with trailing zeros (e.g., 4.50, 4.500)
- Expanded form showing place-value parts (e.g., 4 + 0.5 = 4 + 5/10)
- Fraction form (e.g., 9/2 for 4.5 or 3/5 for 0.6)
Important idea: Adding or removing trailing zeros after the last non-zero digit does not change the value (0.6 = 0.60 = 0.600).
Place-value Tricks are simple rules using the decimal place-value system to perform operations or comparisons quickly:
- Multiplying by 10, 100, 1000 shifts the decimal point to the right by 1, 2, 3 places respectively (e.g., 3.72 × 100 = 372).
- Dividing by 10, 100, 1000 shifts the decimal point to the left by 1, 2, 3 places respectively (e.g., 45.6 ÷ 10 = 4.56).
- To compare decimals, write them to the same number of decimal places by adding trailing zeros (e.g., 0.5 = 0.50, so 0.50 > 0.47).
- To convert a decimal to a fraction, write the decimal over the place-value denominator and simplify (e.g., 0.75 = 75/100 = 3/4).
- To convert a fraction with denominator 10, 100, 1000 to a decimal, place the numerator digits according to place value (e.g., 37/100 = 0.37).
Why this matters: These ideas make mental arithmetic, measurements, money calculations, and conversions easy and less error-prone for everyday tasks and exams.
- Trailing zeros: 4.5 = 4.50 = 4.500. All represent the same value because adding zeros after the last decimal digit doesn't change the number.
- Convert decimal to fraction: 0.75 = 75/100 = 3/4 (divide numerator and denominator by 25).
- Convert fraction to decimal: 37/100 = 0.37 (numerator 37 occupies hundredths place).
- Multiply by 100: 3.72 × 100 → shift decimal 2 places right → 372.
- Divide by 10: 45.6 ÷ 10 → shift decimal 1 place left → 4.56.
- Compare decimals: Which is larger, 0.5 or 0.47? Make same places: 0.50 vs 0.47 → 0.50 > 0.47.
- Place values: ones . tenths hundredths thousandths (e.g., 12.345 = 1×10 + 2×1 + 3×(1/10) + 4×(1/100) + 5×(1/1000)).
- Decimal → Fraction: Move decimal digits as numerator over 10^n. Example: x = 0.abcd → x = abcd / 10^4, then simplify.
- Fraction (denominator 10^n) → Decimal: Put numerator digits in the appropriate decimal places. Example: 7/10 = 0.7, 37/100 = 0.37.
- Multiply by 10^n: shift decimal point n places to the right. (a.bcd) × 10^n → decimal moves right n positions.
- Divide by 10^n: shift decimal point n places to the left. (a.bcd) ÷ 10^n → decimal moves left n positions.
- Comparing decimals: Equalize decimal places by adding trailing zeros, then compare digits from left to right.
Key Concepts
- Decimal number
- A number that has an integer part and a fractional part separated by a decimal point.
- Decimal point
- The dot (.) that separates the whole number part from the fractional part in a decimal.
- Place value (in decimals)
- The value of a digit depending on its position; positions right of the decimal are tenths, hundredths, thousandths, etc.
- Tenths
- The first place to the right of the decimal point; each tenth is 1/10 of a whole.
- Hundredths
- The second place to the right of the decimal point; each hundredth is 1/100 of a whole.
- Thousandths
- The third place to the right of the decimal point; each thousandth is 1/1000 of a whole.
- Decimal fraction
- A fraction whose denominator is a power of 10 (10, 100, 1000, ...); it can be written as a decimal.
- Terminating decimal
- A decimal that has a finite number of digits after the decimal point.
- Non-terminating decimal
- A decimal that has infinitely many digits after the decimal point and does not end.
- Repeating (Recurring) decimal
- A non-terminating decimal in which a pattern of one or more digits repeats forever.
- Equivalent decimals
- Different decimal representations that denote the same value.
- Expanded form of a decimal
- Writing a decimal as the sum of each digit multiplied by its place value.
- Converting fraction to decimal
- Divide the numerator by the denominator to express a fraction as a decimal.
- Converting decimal to fraction
- Write the decimal over the appropriate power of 10 and simplify the fraction.
- Comparing decimals
- Compare by aligning decimal points and checking digits from left to right (or convert to same place value).
- Rounding decimals
- Approximating a decimal to a specified place value by increasing or keeping the previous digit based on the next digit.
- Addition of decimals
- Line up decimal points, add digits column-wise, and place the decimal point in the sum under the others.
- Subtraction of decimals
- Line up decimal points, subtract digits column-wise, and place the decimal point in the difference under the others.
- Multiplication of decimals
- Multiply as whole numbers, then place the decimal point in the product so that total decimal places equals the sum of decimal places in the factors.
- Division of decimals
- Make the divisor a whole number by shifting the decimal point in both divisor and dividend the same places, then divide as usual.
End-of-Chapter Trial Paper & Test Questions
Topic-wise questions to test your understanding of every concept in this chapter.
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What is the place value of 7 in 4.375? / 4.375 में 7 का स्थानीय मान क्या है? (a) Ones / इकाई (b) Tenths / दशांश (c) Hundredths / शतांश (d) Thousandths / सहस्रांश
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(c) Hundredths / शतांश — In 4.375, digit positions after decimal: 3 is in tenths, 7 is in hundredths, 5 is in thousandths. So 7 is in the hundredths place (7 × 1/100 = 0.07). / 4.375 में दशमलव के बाद: 3 दशांश में, 7 शतांश में, 5 सहस्रांश में। अतः 7 शतांश स्थान पर है (7 × 1/100 = 0.07)।
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Which decimal is greater — 0.6 or 0.59? / 0.6 और 0.59 में कौन-सा दशमलव बड़ा है? (a) 0.59 / 0.59 (b) 0.6 / 0.6 (c) Both are equal / दोनों बराबर हैं (d) Cannot be determined / निर्धारित नहीं किया जा सकता
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(b) 0.6 / 0.6 — Rewrite 0.6 as 0.60. Compare digit by digit: tenths 6 > 5, so 0.60 > 0.59. / 0.6 को 0.60 लिखें। अंक दर अंक तुलना: दशांश 6 > 5, अतः 0.60 > 0.59।
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0.125 as a fraction in simplest form is: / 0.125 सरलतम भिन्न रूप में है: (a) 125/100 / 125/100 (b) 1/8 / 1/8 (c) 5/40 / 5/40 (d) 1/4 / 1/4
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(b) 1/8 / 1/8 — 0.125 = 125/1000. HCF(125, 1000) = 125. Divide: 125÷125 = 1 and 1000÷125 = 8. So 0.125 = 1/8. / 0.125 = 125/1000। HCF(125, 1000) = 125। भाग: 125÷125 = 1 और 1000÷125 = 8। अतः 0.125 = 1/8।
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3.45 + 1.8 = ______. / 3.45 + 1.8 = ______।
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5.25 / 5.25 — Write 1.8 as 1.80. Align decimal points and add column-wise: 45 + 80 = 125 hundredths (write 25, carry 1), 3 + 1 + 1 = 5 ones. Result: 5.25. / 1.8 को 1.80 लिखें। दशमलव बिंदु संरेखित करके जोड़ें: 45 + 80 = 125 (25 लिखें, 1 carry), 3 + 1 + 1 = 5। परिणाम: 5.25।
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6.50 ÷ 100 = ______. / 6.50 ÷ 100 = ______।
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0.065 / 0.065 — Dividing a decimal by 100 shifts the decimal point 2 places to the left: 6.50 → 0.0650 = 0.065. / दशमलव को 100 से भाग देने पर दशमलव बिंदु 2 स्थान बाईं ओर खिसकता है: 6.50 → 0.0650 = 0.065।
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True or False: 2.50 and 2.5 represent the same number. / सत्य या असत्य: 2.50 और 2.5 एक ही संख्या दर्शाते हैं।
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True / सत्य — Adding or removing trailing zeros after the last non-zero decimal digit does not change the value. 2.50 = 2.5 (both equal 2 and 5 tenths). / अंतिम गैर-शून्य दशमलव अंक के बाद शून्य जोड़ने या हटाने से मान नहीं बदलता। 2.50 = 2.5 (दोनों = 2 और 5 दशांश)।
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Riya has ₹25.75 in her piggy bank. She spends ₹13.50 on a notebook. How much money is left? / रिया की गुल्लक में ₹25.75 है। वह नोटबुक पर ₹13.50 खर्च करती है। कितने रुपये बचते हैं?
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₹12.25 / ₹12.25 — Align decimals and subtract: 25.75 − 13.50 = 12.25. Hundredths: 5 − 0 = 5; Tenths: 7 − 5 = 2; Ones: 5 − 3 = 2; Tens: 2 − 1 = 1. / दशमलव संरेखित करके घटाएँ: 25.75 − 13.50 = 12.25। शतांश: 5 − 0 = 5; दशांश: 7 − 5 = 2; इकाई: 5 − 3 = 2; दहाई: 2 − 1 = 1।
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A metal rod is 2.4 m long and another is 85 cm long. What is their total length in metres? / एक धातु की छड़ 2.4 मीटर लंबी है और दूसरी 85 सेमी। मीटर में उनकी कुल लंबाई क्या है?
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3.25 m / 3.25 मीटर — Convert 85 cm to metres: 85 ÷ 100 = 0.85 m. Add: 2.40 + 0.85 = 3.25 m. / 85 सेमी को मीटर में बदलें: 85 ÷ 100 = 0.85 मीटर। जोड़ें: 2.40 + 0.85 = 3.25 मीटर।
Related Laws & Principles
Explore allFoundational laws & principles behind this chapter. Each one opens a full page — what it says, why it matters, five practice questions and the mistakes to avoid.