--- title: "NCERT Solutions for Class 8 Maths Chapter 3 A Story of Numbers" url: https://www.swavid.com/maths/class/8/chapter/a-story-of-numbers/ncert-solutions dateModified: 2026-10-07T15:19:57+00:00 --- # NCERT Solutions for Class 8 Maths Chapter 3 A Story of Numbers This chapter's questions cover the historical evolution of numbers, counting methods, early number systems, bases, and place value systems. Free PDF (15 pages): https://www.swavid.com/api/seo/pdf/ncert/maths/class-8/swavid-ncert-solutions-class-8-maths-chapter-3-a-story-of-numbers-2d42aec9e5.pdf ## Figure it Out ### Question 1 *3 marks · Short answer* Suppose you are using the number system that uses sticks to represent numbers, as in Method 1. Without using either the number names or the numerals of the Hindu number system, give a method for adding, subtracting, multiplying and dividing two numbers or two collections of sticks. **Solution** 1. Count the sticks in each given collection to determine the initial numbers. 2. To add, put all the sticks from both collections together at one place and count the total number of sticks. 3. To subtract, take out sticks equal to the second collection from the first collection and count the remaining sticks. 4. To multiply, count groups of equal numbers of sticks repeatedly, and to divide, group the total sticks into sets of a required size. **Answer:** Operations are performed using physical collections of sticks through counting, combining, removing, grouping, and regrouping. > Common mistake: Confusing the process of grouping equal collections for multiplication with simple addition of the two collections. ### Question 2 *3 marks · Short answer* One way of extending the number system in Method 2 is by using strings with more than one letter—for example, we could use 'aa' for 27. How can you extend this system to represent all the numbers? There are many ways of doing it! **Solution** 1. Recognize that the standard English alphabet provides 26 unique symbols from 'a' to 'z' for numbers 1 to 26. 2. Extend the system for numbers beyond 26 by repeating letters such as 'aa', 'bb', 'cc', ..., 'zz' for the next set of numbers. 3. Continue the pattern further with combinations like 'aaa', 'bbb', etc., to ensure an unending standard sequence of letter strings. **Answer:** The system can be extended by repeating letters such as 'aa', 'bb', 'cc', ..., 'zz' and continuing with longer strings of identical letters. > Common mistake: Using random letter combinations instead of following a systematic repeating order like 'aa', 'bb', 'cc'. ### Question 3 *3 marks · Short answer* Try making your own number system. **Solution** 1. Select a set of unique shapes or symbols to act as basic numerals, such as geometric figures. 2. Define a base or grouping rule, such as grouping every 4 or 5 units to form a new symbol. 3. Demonstrate the representation of initial numbers using the chosen symbols and grouping rule. **Answer:** A custom number system can be created by assigning unique symbols for basic units and establishing a fixed grouping rule for larger numbers. > Common mistake: Failing to define a clear rule for how numbers beyond the initial set of symbols are represented. ## Figure it Out ### Question 1 *3 marks · Short answer* Represent the following numbers in the Roman system. (i) 1222 (ii) 2999 (iii) 302 (iv) 715 **Part (i)** 1. $1222 = 1000 + 200 + 20 + 2$ 2. Writing in Roman numerals using landmark numbers: $\text{M} + \text{CC} + \text{XX} + \text{II}$ 3. So, $1222$ in Roman numerals is $\text{MCCXXII}$. Answer (i): MCCXXII **Part (ii)** 1. $2999 = 2000 + 900 + 90 + 9$ 2. Writing $900$ as $\text{CM}$, $90$ as $\text{XC}$, and $9$ as $\text{IX}$ 3. So, $2999$ in Roman numerals is $\text{MMCMXCIX}$. Answer (ii): MMCMXCIX **Part (iii)** 1. $302 = 300 + 2$ 2. Writing $300$ as $\text{CCC}$ and $2$ as $\text{II}$ 3. So, $302$ in Roman numerals is $\text{CCCII}$. Answer (iii): CCCII **Part (iv)** 1. $715 = 500 + 200 + 10 + 5$ 2. Writing $500$ as $\text{D}$, $200$ as $\text{CC}$, $10$ as $\text{X}$, and $5$ as $\text{V}$ 3. So, $715$ in Roman numerals is $\text{DCCXV}$. Answer (iv): DCCXV **Answer:** (i) MCCXXII, (ii) MMCMXCIX, (iii) CCCII, (iv) DCCXV > Common mistake: Incorrectly grouping numbers without following landmark values or wrong subtraction placement like writing $900$ as $\text{CCCCCCCCC}$ instead of $\text{CM}$. ## Figure it Out ### Question 1 *3 marks · Short answer* A group of indigenous people in a Pacific island use different sequences of number names to count different objects. Why do you think they do this? **Solution** 1. Different objects in daily life held different levels of importance or sacredness for indigenous communities. 2. Counting certain special objects like canoes, sacred animals, or ritual items often required specific restricted or ceremonial counting words. 3. This resulted in the use of different sequences of number names for counting different types of objects. **Answer:** Different sequences of number names were used for different objects due to cultural practices, varying importance, or sacredness associated with those objects. > Common mistake: Thinking that they did not know how to count properly, ignoring cultural and contextual reasons. ### Question 2 *3 marks · Short answer* Consider the extension of the Gumulgal number system beyond 6 in the same way of counting by 2s. Come up with ways of performing the different arithmetic operations (+, –, ×, ÷) for numbers occurring in this system, without using Hindu numerals. Use this to evaluate the following: (i) (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar-ukasar-urapon) (ii) (ukasar-ukasar-ukasar-ukasar-urapon) – (ukasar-ukasar-ukasar) (iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar) (iv) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar) **Part (i)** 1. Given expression: (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar-ukasar-urapon) 2. Convert to values: $9 + 7$ 3. Perform addition: $9 + 7 = 16$, which is represented in counting by 2s as eight ukasars. Answer (i): ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar **Part (ii)** 1. Given expression: (ukasar-ukasar-ukasar-ukasar-urapon) – (ukasar-ukasar-ukasar) 2. Convert to values: $9 - 6$ 3. Perform subtraction: $9 - 6 = 3$, which is represented as ukasar-urapon. Answer (ii): ukasar-urapon **Part (iii)** 1. Given expression: (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar) 2. Convert to values: $9 \times 2$ 3. Perform multiplication: $9 \times 2 = 18$, represented as 18 ukasars. Answer (iii): ukasar-...-ukasar (18 times) **Part (iv)** 1. Given expression: (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar) 2. Convert to values: $16 \div 2$ 3. Perform division: $16 \div 2 = 8$, represented as four ukasars. Answer (iv): ukasar-ukasar-ukasar-ukasar **Answer:** Evaluated using the Gumulgal system where urapon = 1 and ukasar = 2. > Common mistake: Confusing the number of times ukasar is repeated in counting by 2s. ### Question 3 *3 marks · Short answer* Identify the features of the Hindu number system that make it efficient when compared to the Roman number system. **Solution** 1. The Hindu number system is a place value system where the position of a digit determines its value, unlike the Roman system which uses fixed values for symbols. 2. The Hindu system includes the concept and symbol for zero (0), which acts both as a placeholder and a number on par with others; the Roman system has no zero. 3. Because of place value and zero, arithmetic calculations (+, -, ×, ÷) are extremely easy and efficient in the Hindu system compared to the Roman system. **Answer:** Features making the Hindu system efficient are the place value system, the inclusion of zero (0), and ease of performing arithmetic operations. > Common mistake: Forgetting to mention the place value structure and zero as the core reasons for efficiency. ### Question 4 *3 marks · Short answer* Using the ideas discussed in this section, try refining the number system you might have made earlier. **Solution** 1. Earlier number systems could be refined by introducing a fixed base (such as base 4 or base 5) instead of relying solely on tally marks or repetitive names. 2. Using landmark numbers as powers of the chosen base simplifies writing larger quantities. 3. Incorporating a place value structure along with a placeholder symbol for zero makes the refined number system unending and unambiguous. **Answer:** We can refine our number system by applying the concept of a fixed base, powers as landmark numbers, and a place value structure with zero. > Common mistake: Not including the place value concept when attempting to refine a primitive number system. ## Figure it Out ### Question 1 *3 marks · Short answer* Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707. **Part (i)** 1. 10458 = 10,000 + 400 + 50 + 8 2. Represent 10,000 with one lotus flower, 400 with four scrolls, 50 with five heel bones, and 8 with eight strokes. Answer (i): $10458 = 10000 + 400 + 50 + 8$ **Part (ii)** 1. 1023 = 1000 + 20 + 3 2. Represent 1000 with one water lily, 20 with two heel bones, and 3 with three strokes. Answer (ii): $1023 = 1000 + 20 + 3$ **Part (iii)** 1. 2660 = 2000 + 600 + 60 2. Represent 2000 with two water lilies, 600 with six scrolls, and 60 with six heel bones. Answer (iii): $2660 = 2000 + 600 + 60$ **Part (iv)** 1. 784 = 700 + 80 + 4 2. Represent 700 with seven scrolls, 80 with eight heel bones, and 4 with four strokes. Answer (iv): $784 = 700 + 80 + 4$ **Part (v)** 1. 1111 = 1000 + 100 + 10 + 1 2. Represent 1000 with one water lily, 100 with one scroll, 10 with one heel bone, and 1 with one stroke. Answer (v): $1111 = 1000 + 100 + 10 + 1$ **Part (vi)** 1. 70707 = 70,000 + 700 + 7 2. Represent 70,000 with seven lotus flowers, 700 with seven scrolls, and 7 with seven strokes. Answer (vi): $70707 = 70000 + 700 + 7$ **Answer:** Representations of the given numbers in the Egyptian system. > Common mistake: Confusing the symbols for different powers of 10 such as scrolls for 100 and heel bones for 10. ### Question 2 *3 marks · Short answer* What numbers do these numerals stand for? (i) [Egyptian numeral graphic] (ii) [Egyptian numeral graphic] **Part (i)** 1. Count the symbols: two scrolls ($2 \times 100 = 200$), seven heel bones ($7 \times 10 = 70$), and six strokes ($6 \times 1 = 6$). 2. Add the values: $200 + 70 + 6 = 276$. Answer (i): $276$ **Part (ii)** 1. Count the symbols: four water lilies ($4 \times 1000 = 4000$), three scrolls ($3 \times 100 = 300$), two heel bones ($2 \times 10 = 20$), and two strokes ($2 \times 1 = 2$). 2. Add the values: $4000 + 300 + 20 + 2 = 4322$. Answer (ii): $4322$ **Answer:** Numerical values for the given Egyptian numerals. > Common mistake: Miscounting the number of identical symbols grouped together. ## Figure it Out ### Question 1 *3 marks · Short answer* Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651. **Part (i)** 1. $15 = 2 \times 5 + 5 = 3 \times 5$ 2. $15 = 125$ (not possible, so $5^2$ coefficient is 0), $15 = 3 \times 5^1 + 0 \times 5^0$ Answer (i): $15 = \triangle \triangle \triangle$ **Part (ii)** 1. $50 = 2 \times 25 = 2 \times 5^2$ 2. $50 = 2 \times 5^2 + 0 \times 5^1 + 0 \times 5^0$ Answer (ii): $50 = \circ \circ$ **Part (iii)** 1. $137 = 1 \times 125 + 0 \times 25 + 2 \times 5 + 2 \times 1$ 2. $137 = 1 \times 5^3 + 0 \times 5^2 + 2 \times 5^1 + 2 \times 5^0$ Answer (iii): $137 = \bigcirc \square \triangle \triangle$ **Answer:** The numbers 15, 50, 137, 293, and 651 are represented in the base-5 system as shown in the sub-parts. > Common mistake: Confusing the powers of 5 or the symbols corresponding to each power. ### Question 2 *3 marks · Short answer* Is there a number that cannot be represented in our base-5 system above? Why or why not? **Solution** 1. In the base-5 system discussed in the chapter, the landmark numbers are powers of 5: $5^0 = 1$, $5^1 = 5$, $5^2 = 25$, $5^3 = 125$, and so on, represented by specific symbols. 2. To express any number, we group it into these landmark numbers starting from the largest one smaller than the number. 3. However, there is no symbol for zero in this specific base-5 symbol system introduced on page 63, which means numbers like 0 or numbers containing 0 in their place value expansion cannot be represented using these symbols. **Answer:** Yes, zero (0) cannot be represented because there is no symbol for it in the base-5 system introduced in the chapter. > Common mistake: Thinking that all positive whole numbers can be represented without needing a symbol for zero. ### Question 3 *3 marks · Short answer* Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-$n$ system? **Solution** 1. For a base-7 system, the landmark numbers are powers of 7 starting from $7^0$. 2. Computing the powers: $7^0 = 1$, $7^1 = 7$, $7^2 = 49$, $7^3 = 343$, and so on. 3. In general, the landmark numbers of a base-$n$ system are the powers of $n$: $n^0, n^1, n^2, n^3, \dots$ **Answer:** Landmark numbers of base-7 are $1, 7, 49, 343, \dots$; for base-$n$, they are $n^0, n^1, n^2, n^3, \dots$ > Common mistake: Starting the powers from $n^1$ instead of $n^0 = 1$. ## Figure it Out ### Question 1 *3 marks · Short answer* Add the following Egyptian numerals: (i) [Egyptian numeral graphic] and [Egyptian numeral graphic] (ii) [Egyptian numeral graphic] and [Egyptian numeral graphic] **Part (i)** 1. Count the total number of strokes and coils in both Egyptian numerals. 2. Combine them to get 15 coils and 15 strokes. 3. Regroup every 10 coils into a water lily symbol, and every 10 strokes into a coil, leaving 1 water lily, 6 coils, and 5 strokes. Answer (i): $1 \text{ water lily}, 6 \text{ coils}, 5 \text{ strokes}$ **Part (ii)** 1. Count the total number of coils and strokes in the given base-10 Egyptian numerals. 2. Combine the symbols to obtain 9 coils and 6 strokes. 3. Since there are fewer than 10 of each symbol, no further regrouping is required. Answer (ii): $9 \text{ coils}, 6 \text{ strokes}$ **Answer:** Parts (i) and (ii) solved using the rules of Egyptian grouping and regrouping. > Common mistake: Failing to regroup 10 smaller symbols into the next higher landmark number symbol. ### Question 2 *3 marks · Short answer* Add the following numerals that are in the base-5 system that we created: [Base-5 numeral graphic] + [Base-5 numeral graphic] **Solution** 1. Count the total number of each shape (triangles, squares, circles) from both collections of base-5 numerals. 2. Combine the symbols to get 7 triangles, 2 squares, and 3 circles. 3. Since 5 triangles make 1 square, regroup 5 triangles into 1 square to get 2 triangles and 3 squares, and 3 circles. **Answer:** $3 \text{ circles}, 3 \text{ squares}, 2 \text{ triangles}$ > Common mistake: Not regrouping every 5 lower-order landmark symbols into the next higher symbol. ## Figure it Out ### Question 1 *3 marks · Short answer* Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not? **Solution** 1. In the Egyptian system, 10 collections of any landmark number give the next higher landmark number. 2. Therefore, a symbol can never occur 10 or more times because any group of 10 identical symbols is immediately replaced by a single symbol of the next higher landmark number. **Answer:** No, because 10 times any landmark number gives the next landmark number. > Common mistake: Confusing the replacement rule of 10 with other bases. ### Question 2 *3 marks · Short answer* Create your own number system of base 4, and represent numbers from 1 to 16. **Solution** 1. In a base-4 system, the landmark numbers are powers of 4: $4^0 = 1$, $4^1 = 4$, and $4^2 = 16$. 2. Let the symbols for $1$, $4$, and $16$ be $\lrcorner$, $\Delta$, and $\square$ respectively, following the textbook convention. 3. Numbers from $1$ to $16$ are represented by grouping them into powers of $4$ as $1 = \lrcorner$, $4 = \Delta$, $5 = \Delta \lrcorner$, and $16 = \square$. **Answer:** Base-4 representation for $1$ to $16$ using symbols for $1, 4, 16$. > Common mistake: Using base-10 grouping instead of powers of 4. ### Question 3 *3 marks · Short answer* Give a simple rule to multiply a given number by 5 in the base-5 system that we created. **Solution** 1. In a base-n place value system, multiplying any number by the base $n$ increases the exponent of each place value by 1. 2. Since the base is 5, multiplying a number by 5 shifts each digit one position to the left. 3. Therefore, the simple rule is to append a zero ($0$) to the right of the base-5 representation of the number. **Answer:** To multiply a given number by 5 in the base-5 system, append a zero to the right of the number. > Common mistake: Forgetting that appending zero in base-5 means multiplying the whole value by 5, similar to multiplying by 10 in the decimal system. ## Figure it Out ### Question 1 *3 marks · Short answer* Represent the following numbers in the Mesopotamian system — (i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605 **Part (i)** 1. $63 = 1 \times 60 + 3$ 2. In the Mesopotamian system, this is represented using one symbol for 60 and three symbols for 1. Answer (i): $1 \times 60 + 3$ **Part (ii)** 1. $132 = 2 \times 60 + 10 + 2$ 2. In the Mesopotamian system, this is represented using two symbols for 60, one symbol for 10, and two symbols for 1. Answer (ii): $2 \times 60 + 10 + 2$ **Part (iii)** 1. $200 = 3 \times 60 + 20$ 2. In the Mesopotamian system, this is represented using three symbols for 60 and two symbols for 10. Answer (iii): $3 \times 60 + 20$ **Part (iv)** 1. $60 = 1 \times 60$ 2. In the Mesopotamian system, this is represented using one symbol for 60. Answer (iv): $1 \times 60$ **Part (v)** 1. $3605 = 1 \times 3600 + 5$ 2. In the Mesopotamian system, this is represented using one symbol for 3600 and five symbols for 1. Answer (v): $1 \times 3600 + 5$ **Answer:** Represented as sums of powers of 60 and corresponding Mesopotamian symbols. > Common mistake: Confusing the landmark numbers of the base-60 system (1, 60, 3600). ## Figure it Out ### Question 1 *3 marks · Short answer* Represent the following numbers using the Mayan system: (i) 77 (ii) 100 (iii) 361 (iv) 721 **Part (i)** 1. Group the number 77 using the Mayan landmark positions: $77 = (3) \times 20 + (17) \times 1$. 2. Represent the symbols vertically with 3 dots and a bar for 20s, and three bars and two dots for 1s. Answer (i): $77 = (3) \times 20 + (17) \times 1$ **Part (ii)** 1. Group the number 100 using the Mayan landmark positions: $100 = (5) \times 20 + (0) \times 1$. 2. Represent the symbols vertically with a bar for 5 groups of 20, and a seashell for 0 at the 1s position. Answer (ii): $100 = (5) \times 20 + (0) \times 1$ **Part (iii)** 1. Group the number 361 using the Mayan landmark positions: $361 = (1) \times 360 + (0) \times 20 + (1) \times 1$. 2. Represent the symbols vertically with a dot for 360, a seashell for 0 at 20s, and a dot for 1. Answer (iii): $361 = (1) \times 360 + (0) \times 20 + (1) \times 1$ **Part (iv)** 1. Group the number 721 using the Mayan landmark positions: $721 = (2) \times 360 + (0) \times 20 + (1) \times 1$. 2. Represent the symbols vertically with two dots for 360s, a seashell for 0 at 20s, and a dot for 1. Answer (iv): $721 = (2) \times 360 + (0) \times 20 + (1) \times 1$ **Answer:** Represented numbers using the Mayan system with landmark positions 360, 20, and 1. > Common mistake: Confusing the third landmark number as 400 instead of 360 in the Mayan system. ## Figure it Out ### Question 1 *3 marks · Short answer* Why do you think the Chinese alternated between the Zong and Heng symbols? If only the Zong symbols were to be used, how would 41 be represented? Could this numeral be interpreted in any other way if there is no significant space between two successive positions? **Solution** 1. The Chinese alternated between the Zong (vertical) and Heng (horizontal) rod numerals to distinguish between adjacent place value positions. 2. If only Zong symbols were to be used, 41 would be represented as four vertical lines for 4 followed by one vertical line for 1, looking like 5 vertical lines. 3. Without significant spacing, this numeral could be misinterpreted as 23, 32, or 122. **Answer:** The Chinese alternated Zong and Heng symbols to distinguish adjacent places; using only Zong symbols, 41 would look like 5 vertical lines and could be misinterpreted as 23, 32, or 122. > Common mistake: Forgetting that alternating symbols prevent ambiguity between adjacent place value positions. ### Question 2 *3 marks · Short answer* Form a base-2 place value system using 'ukasar' and 'urapon' as the digits. Compare this system with that of the Gumulgal's. **Solution** 1. In a base-2 place value system, we use 'urapon' for 0 and 'ukasar' for 1, employing positions as powers of 2. 2. The Gumulgal system is a cumulative system using repetition of number names where 3 is ukasar-urapon and 4 is ukasar-ukasar. 3. The base-2 place value system uses position to represent arbitrarily large numbers compactly, whereas the Gumulgal system uses names only up to 6 before calling it ras. **Answer:** Base-2 uses position and symbols for 0 and 1 to represent numbers efficiently, whereas the Gumulgal system uses additive number names up to 6. > Common mistake: Confusing a place value system with an additive word-based number system. ### Question 3 *3 marks · Short answer* Where in your daily lives, and in which professions, do the Hindu numerals, and 0, play an important role? How might our lives have been different if our number system and 0 hadn't been invented or conceived of? **Solution** 1. Hindu numerals and zero play essential roles in daily life, commerce, science, engineering, and digital computing. 2. Zero acts both as a crucial placeholder in our decimal place value system and as a number in its own right. 3. Without our number system and zero, advanced mathematics, modern technology, and global communication would have been extremely difficult to develop. **Answer:** Hindu numerals and zero are fundamental to commerce, science, and technology; without them, modern technological and scientific progress would have been hindered. > Common mistake: Omitting the dual role of zero as both a placeholder and a number. ### Question 4 *3 marks · Short answer* The ancient Indians likely used base 10 for the Hindu number system because humans have 10 fingers, and so we can use our fingers to count. But what if we had only 8 fingers? How would we be writing numbers then? What would the Hindu numerals look like if we were using base 8 instead? Base 5? Try writing the base-10 Hindu numeral 25 as base-8 and base-5 Hindu numerals, respectively. Can you write it in base-2? **Solution** 1. In base-8, the number 25 is expressed as $25 = (3) \times 8^1 + (1) \times 8^0$, giving the representation 31. 2. In base-5, the number 25 is expressed as $25 = (1) \times 5^2 + (0) \times 5^1 + (0) \times 5^0$, giving the representation 100. 3. In base-2, 25 is expressed as $16 + 8 + 1$, which equals $(1) \times 2^4 + (1) \times 2^3 + (0) \times 2^2 + (0) \times 2^1 + (1) \times 2^0$, giving the representation 11001. **Answer:** The base-8 representation of 25 is 31, the base-5 representation is 100, and the base-2 representation is 11001. > Common mistake: Incorrectly dividing by the base during successive divisions or miscalculating powers of the base. ## Frequently asked questions ### How many exercises or questions are there in NCERT Solutions for Class 8 Maths Chapter 3 A Story of Numbers? The chapter features a total of 10 'Figure it Out' sections containing a combined 24 short answer questions based on the new NCERT book for the 2026-27 session. You can find step-by-step solutions for all these questions in SwaVid's free PDF available on this page only. ### Which topics do the questions cover in Class 8 Maths Chapter 3? The questions cover counting with physical sticks, creating personal number systems, and exploring ancient systems like Roman, Egyptian, Mesopotamian, Mayan, and Chinese Rod numerals. Other topics include base conversions, base-5 and base-4 arithmetic, and the importance of Hindu numerals and zero. ### What are the hardest question types in this chapter and how should I approach them? Base conversion problems and arithmetic operations in non-decimal systems like base-5 and base-4 are often considered the trickiest. To approach them, carefully understand the grouping principles and place value rules of the specific base before attempting addition or multiplication. ### How should I write my answers to score full marks in Class 8 Maths Chapter 3? To score full marks, clearly show the grouping logic, base conversions, and intermediate steps rather than just writing the final numeral. SwaVid's free PDF on this page only provides well-structured, step-by-step solutions to help you learn the exact method of presentation. ### Is the free PDF for Class 8 Maths Chapter 3 A Story of Numbers available online? Yes, SwaVid provides a comprehensive and free PDF containing complete solutions aligned with the new NCERT book for the 2026-27 session. You can access these detailed step-by-step explanations directly on this page only to support your exam preparation. ## Related pages - [Class 8 Maths chapters](https://www.swavid.com/maths/class/8) Solutions written by SwaVid, a personal AI tutor for Class 6 to 10 Maths and Science. Practise this chapter free: https://www.swavid.com/start/student