CBSE Class 8 Mathematics A Story of Numbers Practice Paper Worksheet Set 02 with Answers

Here is Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Practice Paper 13 (40 marks, 20 questions in Sections A to E), solved step by step so you can check your own working. This paper tests the Lebombo bone and the origins of counting, the subtractive principle in numerals such as CD and XC, converting 1111 from binary and 13 into base 4, adding Roman and Egyptian numerals, the place value of each digit in 5,005, and converting 20 into the base-8 system.

Section A: Multiple Choice Questions (1 Mark Each)

Section B: Short Answer Questions (2 Marks Each)

Question 11. Convert the Roman numeral CD to its equivalent in the Hindu number system.
Answer:
C = 100 and D = 500.
C is smaller and comes before D, so it is subtracted.
CD = 500 − 100 = 400.
In simple words: One hundred less than five hundred is four hundred.

Teacher's Note:
1. 400 is never written as CCCC in standard Roman numerals.
2. Compare: DC = 500 + 100 = 600.

Question 12. Convert the binary number 1111 to its decimal (Hindu) equivalent. Show your working.
Answer:
Place values in base 2 from the right: 1, 2, 4, 8.
1111 = 1 × 8 + 1 × 4 + 1 × 2 + 1 × 1
= 8 + 4 + 2 + 1 = 15.
In simple words: All four places hold 1, so add 8 + 4 + 2 + 1 = 15.

Teacher's Note:
1. The largest 4-digit binary number is 1111 = 15.
2. Adding 1 more gives 10000 in binary, which is 16.

Question 13. Convert the Hindu number 63 into its Roman numeral equivalent.
Answer:
Split by place value: 63 = 60 + 3.
60 = 50 + 10 = LX.
3 = 1 + 1 + 1 = III.
So 63 = LXIII.
In simple words: Sixty is LX and three is III, so 63 is LXIII.

Teacher's Note:
1. Write tens first, then ones.
2. I can be repeated up to three times.

Question 14. In a base-10 number system, what would be the value of the landmark number 10⁵?
Answer:
10⁵ = 10 × 10 × 10 × 10 × 10.
= 1,00,000 (one lakh), which is 100,000 or one hundred thousand in the international system.
In simple words: Write 1 followed by five zeros.

Teacher's Note:
1. The power tells you how many zeros follow the 1.
2. 1,00,000 is the place value of the sixth digit from the right.

Section C: Short Answer Questions (3 Marks Each)

Question 15. Calculate the difference between the Roman numerals L and XXX. Write the answer in Roman numerals and Hindu numerals.
Answer:
L = 50 and XXX = 10 + 10 + 10 = 30.
Difference = 50 − 30 = 20.
20 = 10 + 10 = XX.
Answer: XX (Roman) = 20 (Hindu).
Check: XXX + XX = XXXXX = L, since five tens are fifty. ✔
In simple words: Fifty minus thirty is twenty, written XX.

Teacher's Note:
1. Convert, calculate, then convert back.
2. Check the answer by adding it to the smaller number.

Question 16. An Egyptian scribe adds two groups: Group 1 has 5 symbols for 10, and Group 2 has 6 symbols for 10 and 3 symbols for 1. What is the total sum in the Hindu number system?
Answer:
Group 1 = 5 × 10 = 50.
Group 2 = 6 × 10 + 3 × 1 = 60 + 3 = 63.
Total = 50 + 63 = 113.
(In Egyptian symbols: 11 ten-symbols and 3 one-symbols; ten of the ten-symbols become one 100-symbol, giving 1 hundred, 1 ten and 3 ones.)
In simple words: Fifty plus sixty-three is one hundred and thirteen.

Teacher's Note:
1. Count each kind of symbol, then multiply by its value.
2. Regroup any ten equal symbols into the next symbol.

Question 17. Convert the Hindu number 13 into the base-4 number system.
Answer:
Landmark numbers in base 4: 1, 4, 16, …
16 is bigger than 13, so we start with 4.
13 ÷ 4 = 3 remainder 1, so 13 = 3 × 4 + 1 × 1.
So 13 in base 4 is 31, written (31)₄.
Check: 3 × 4 + 1 = 13. ✔
In simple words: 13 is three fours and one more, so it is 31 in base 4.

Teacher's Note:
1. Base-4 digits are only 0, 1, 2 and 3.
2. Write the base as a small subscript so 31 is not read as thirty-one.

Section D: Long Answer Question (5 Marks)

Question 18. A farmer counts DCL chickens and CX ducks. How many total animals does the farmer have? Write your final answer in both Roman and Hindu numerals, showing the addition process in Hindu numerals.
Answer:
Step 1: Convert the chickens. DCL = D + C + L = 500 + 100 + 50 = 650.
Step 2: Convert the ducks. CX = C + X = 100 + 10 = 110.
Step 3: Add in Hindu numerals. Ones: 0 + 0 = 0; Tens: 5 + 1 = 6; Hundreds: 6 + 1 = 7. So 650 + 110 = 760.
Step 4: Convert back. 760 = 500 + 100 + 100 + 50 + 10 = D + CC + L + X = DCCLX.
Total animals = 760 (DCCLX).
In simple words: 650 chickens and 110 ducks make 760 animals, written DCCLX.

Teacher's Note:
1. All the given numerals are additive, so convert by simple adding.
2. Line up places (ones, tens, hundreds) when adding in Hindu numerals.
3. Write the Roman answer from largest symbol to smallest.

Section E: Case Study Based Questions (4 Marks Each)

Question 19. Case Study-1: The Place Value Problem
Dr. Sunita Verma was explaining the elegance of the Hindu number system to her students, noting that the same numeral can hold vastly different values depending on its position. She wrote a seemingly simple number on the board: 5,005. She challenged the class to analyze this number, highlighting how the placeholders (the zeros) and the position of the two fives showcase the system's efficiency. Understanding this fundamental concept is crucial, she stressed, because it is the key reason why this system is considered one of history's greatest mathematical inventions and why it enables unambiguous writing and easy computation.
Consider the Hindu number 5,005.
(i) What is the place value of the leftmost digit 5? (1)
(ii) What is the actual value of the rightmost digit 5? (1)
(iii) What is the actual value of each digit 0? (1)
(iv) How many times greater is the value of the leftmost 5 than the value of the rightmost 5? (1)
Answer:
(i) The leftmost 5 is in the thousands place, so its place value is 5 × 1000 = 5000.
(ii) The rightmost 5 is in the ones place, so its value is 5 × 1 = 5.
(iii) The zeros are in the hundreds and tens places, and each has value 0 × 100 = 0 and 0 × 10 = 0, so 0. They only hold the places.
(iv) 5000 ÷ 5 = 1000 times greater.
In simple words: The same digit 5 is worth 5000 on the left and 5 on the right, a thousand times more.

Teacher's Note:
1. Place value = digit × value of its place.
2. Each step to the left multiplies the value by 10; three steps multiply it by 1000.

Question 20. Case Study-2: Base-8 Number System
Dr. Arjun then introduced the concept of different counting bases, shifting the focus from the familiar decimal system. He noted that while their everyday mathematics was in base-10, ancient systems often used other bases. To illustrate this, he proposed a thought experiment involving a base-8 system. He explained that any base system must define its landmark numbers and the digits it uses. He challenged the students to convert a regular number, 20 apples, into this new base by grouping them into powers of eight, thereby demonstrating the flexible nature of positional number systems.
A new base system is created with a base of 8.
(i) What is the value of the landmark number 8²? (1)
(ii) Which digits would be used in a base-8 system to represent all numbers? (1)
(iii) Convert the Hindu number 20 into this base-8 system. (2)
Answer:
(i) 8² = 8 × 8 = 64.
(ii) A base-8 system needs 8 digits, from 0 up to 8 − 1: 0, 1, 2, 3, 4, 5, 6, 7.
(iii) Landmarks: 1, 8, 64. Since 64 > 20, use 8s and 1s. 20 ÷ 8 = 2 remainder 4, so 20 = 2 × 8 + 4 × 1.
So 20 in base 8 is 24, written (24)₈: 2 groups of eight apples and 4 single apples. Check: 2 × 8 + 4 = 20. ✔
In simple words: Twenty apples make two bags of eight with four left over, so 20 is written 24 in base 8.

Teacher's Note:
1. There is no digit 8 in base 8; eight is written 10.
2. Find the biggest landmark not more than the number, then work down.