CBSE Class 6 Mathematics Revision Worksheet Set 15

Choose correct option in questions 1 to 7.

Question 1. Which is greatest?
(a) 234
(b) 543
(c) 657
(d) 456
Answer: (c) 657
Comparing the hundreds place of all four numbers: 2, 5, 6, and 4. Since 6 is the largest digit in the hundreds place, 657 is the greatest number.
In simple words: 657 is bigger than all the other given numbers.

Exam Tip: Compare digits from left to right (highest place value first) to quickly determine magnitude.

Question 2. Which is smallest?
(a) 4567
(b) 3456
(c) 2345
(d) 1234
Answer: (d) 1234
All numbers have 4 digits. Comparing the thousands place: 4, 3, 2, and 1. The smallest digit is 1, so 1234 is the smallest number.
In simple words: 1234 has the smallest thousands digit, making it the lowest number.

Exam Tip: For numbers with the same number of digits, look at the leftmost digit first.

Question 3. What is 100 - 1?
(a) 9
(b) 99
(c) 999
(d) 9999
Answer: (b) 99
Subtracting 1 from 100 gives 99.
In simple words: Taking 1 away from 100 leaves 99.

Exam Tip: Subtracting 1 from any power of 10 gives a string of nines.

Question 4. What is the place value of 5 in '4567'?
(a) 50
(b) 5
(c) 500
(d) 5000
Answer: (c) 500
In the number 4567, the digit 5 is in the hundreds place. Therefore, its place value is \(5 \times 100 = 500\).
In simple words: Since 5 is in the hundreds column, its value is 500.

Exam Tip: Place value is the digit multiplied by its position value (ones, tens, hundreds, thousands).

Question 5. What is the sum of 567 and 843?
(a) 567
(b) 843
(c) 1410
(d) 1500
Answer: (c) 1410
Adding the two numbers:
\(567 + 843 = 1410\)
In simple words: Add 567 and 843 together to get 1410.

Exam Tip: Align numbers vertically by their place values before adding.

Question 6. The greatest four digit number using 3, 0, 5, 4 without repetition is
(a) 3054
(b) 4035
(c) 5403
(d) 5043
Answer: (c) 5403
To form the greatest number, arrange the given digits in descending order from left to right: 5, 4, 3, 0. Thus, the number is 5403.
In simple words: Put the biggest digits first to make the largest possible number.

Question 7. 1 crore = _____ million
(a) 10
(b) 100
(c) 1000
(d) 1
Answer: (b) 100
1 crore = 1,00,00,000 (7 zeros)
1 million = 10,00,000 (6 zeros)
Therefore, \(1 \text{ crore} = 100 \text{ million}\).
In simple words: One crore is equal to one hundred million.

Exam Tip: Remember conversion ratios: 1 lakh = 100 thousand, 1 crore = 10 million (Wait, 1 crore = 10 million in international system? Let us verify: 1 crore = 10,000,000. 1 million = 1,000,000. So 1 crore = 10 millions? Wait, 1,00,00,000 / 10,00,000 = 10. Let us check standard Indian numbering: 1 million = 10 lakhs, 10 million = 1 crore. Therefore, 1 crore = 10 million. Let us correct option (b) to (a) 10).
Correction: 1 crore = 10 million. Option (a) is 10.

Fill in the blanks:

Question 8. Number name of 45678 is __________.
Answer: Forty-five thousand six hundred seventy-eight
In simple words: Write out the digits in words according to the Indian numbering system.

Exam Tip: Use commas correctly in the Indian system: 45,678.

Question 9. Place value of 4 in '56743' is __________.
Answer: 40
In simple words: The digit 4 is in the tens place, so its place value is 40.

Exam Tip: Identify the position of the digit from the right (ones, tens, hundreds...).

Question 10. Expanded form of 6549 is ___________.
Answer: \(6000 + 500 + 40 + 9\)
In simple words: Break the number down by the place value of each digit.

Exam Tip: Expanded form is the sum of the place values of all digits in a number.

Question 11. Number name of 756432 in international system of numeration is _________.
Answer: Seven hundred fifty-six thousand four hundred thirty-two
In simple words: In the international system, group digits in threes from the right: 756,432.

Exam Tip: In the international system, periods are separated by groups of three digits.

Answer the following:

Question 12. A box contains 5,00,000 medicine tablets each weighing 50 mg. What is the total weight of all the tablets in the box in grams and in kilograms?
Answer:
Number of tablets = 5,00,000
Weight of 1 tablet = 50 mg
Total weight in mg = \(5,00,000 \times 50 = 2,50,00,000 \text{ mg}\)
Converting to grams (since \(1 \text{ g} = 1000 \text{ mg}\)):
Total weight in grams = \(\frac{2,50,00,000}{1000} = 25,000 \text{ g}\)
Converting to kilograms (since \(1 \text{ kg} = 1000 \text{ g}\)):
Total weight in kilograms = \(\frac{25,000}{1000} = 25 \text{ kg}\)
In simple words: Multiply the number of tablets by the weight of one tablet, then convert milligrams to grams and kilograms.

Exam Tip: Remember conversion factors: \(1 \text{ g} = 1000 \text{ mg}\) and \(1 \text{ kg} = 1000 \text{ g}\).

Question 13. Population of Shivaji park was 2,35,471 in the year 2002. In the year 2012 it was found to be increased by 72,958. What was the population of the city in 2012?
Answer:
Population in 2002 = 2,35,471
Increase in population = 72,958
Population in 2012 = 2,35,471 + 72,958
\(2,35,471 + 72,958 = 3,08,429\)
In simple words: Add the increase to the original population to find the new population.

Exam Tip: Keywords like "increased by" indicate addition.

Question 14. The town newspaper is published every day. One copy has 12 pages. Everyday 12,000 copies are printed. How many total pages are printed every day?
Answer:
Pages in one copy = 12
Number of copies printed daily = 12,000
Total pages printed every day = \(12,000 \times 12 = 1,44,000\)
In simple words: Multiply the number of pages per copy by the total number of copies printed.

Exam Tip: For total items when given units and quantities, use multiplication.

Question 15. Find the difference between the greatest and the least number that can be written using the digits 6, 2, 8, 5, 4 each only once.
Answer:
Given digits: 6, 2, 8, 5, 4
Greatest 5-digit number (digits in descending order): 86542
Least 5-digit number (digits in ascending order): 24568
Difference = \(86542 - 24568\)
\(86542 - 24568 = 61974\)
In simple words: Form the largest and smallest numbers using the given digits, then subtract them.

Exam Tip: Arrange digits descending for the largest number and ascending for the smallest number.

Question 16. The distance between the school and the house of Shreya is 1 km 375m. Every day she walks both the ways. Find the total distance covered by her in five days.
Answer:
Distance from house to school = 1 km 375 m = 1375 m
Walking both ways in a day = \(1375 \text{ m} \times 2 = 2750 \text{ m}\)
Total distance covered in 5 days = \(2750 \text{ m} \times 5 = 13750 \text{ m} = 13 \text{ km } 750 \text{ m}\)
In simple words: Multiply one-way distance by 2 for a round trip, then multiply by 5 for five days.

Exam Tip: Convert all units to meters first to avoid calculation errors during addition and multiplication.

Question 17. Give a rough estimate by rounding off to nearest hundreds:
a. 12904 + 2888
b. 8325 - 4317
c. 5281 x 592
Answer:
a. Rounding 12904 to nearest hundred = 12900
Rounding 2888 to nearest hundred = 2900
Estimated sum = \(12900 + 2900 = 15800\)
b. Rounding 8325 to nearest hundred = 8300
Rounding 4317 to nearest hundred = 4300
Estimated difference = \(8300 - 4300 = 4000\)
c. Rounding 5281 to nearest hundred = 5300
Rounding 592 to nearest hundred = 600
Estimated product = \(5300 \times 600 = 3180000\)
In simple words: Round each number to the nearest hundred first, then perform the indicated operation.

Exam Tip: Look at the tens digit: if it is 5 or more, round up; if less than 5, round down.

Question 18. Write in Roman numerals:
a. 69
b. 98
Answer:
a. \(69 = 60 + 9 = \text{LX} + \text{IX} = \text{LXIX}\)
b. \(98 = 90 + 8 = \text{XC} + \text{VIII} = \text{XCVIII}\)
In simple words: Break the numbers into tens and ones and convert them into Roman symbols.

Exam Tip: Remember basic Roman symbols: I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

CLASS - VI Mathematics (Knowing our Numbers)

Worksheet-2

Choose correct option in questions 1 to 5.

Question 1. 23456 ____ 78945
(a) >
(b) <
(c) =
(d) none of these
Answer: (b) <
Comparing 23456 and 78945, the thousands/ten-thousands digit 2 is less than 7. Therefore, \(23456 < 78945\).
In simple words: 23456 is smaller than 78945.

Exam Tip: The open side of the inequality sign (> or <) always faces the greater number.

Question 2. 5678 ____ 6754
(a) <
(b) >
(c) =
(d) none of these
Answer: (a) <
Comparing the thousands digits: 5 is less than 6. Hence, \(5678 < 6754\).
In simple words: 5678 is less than 6754.

Exam Tip: Compare digits from left to right to determine which number is smaller.

Question 3. What is 10000 - 1?
(a) 9
(b) 99
(c) 999
(d) 9999
Answer: (d) 9999
\(10000 - 1 = 9999\)
In simple words: Subtracting 1 from 10000 gives 9999.

Exam Tip: Subtracting 1 from a power of 10 results in a number consisting entirely of nines.

Question 4. What is the product of place value and face value of 5 in '4567'?
(a) 50
(b) 5
(c) 500
(d) 2500
Answer: (d) 2500
In '4567', the digit is 5.
Face value of 5 = 5
Place value of 5 = 500
Product = \(500 \times 5 = 2500\)
In simple words: Multiply the face value (5) by the place value (500) to get 2500.

Exam Tip: Face value is the digit itself; place value depends on its position in the number.

Question 5. What is the product of 784 and 300?
(a) 235200
(b) 253200
(c) 230000
(d) 210000
Answer: (a) 235200
\(784 \times 300 = 784 \times 3 \times 100 = 2352 \times 100 = 235200\)
In simple words: Multiply 784 by 3 and attach two zeros at the end.

Exam Tip: When multiplying by multiples of 10, multiply the non-zero digits first and then append the zeros.

Fill in the blanks:

Question 6. Number name of 2834567 in Indian system of numeration is __________.
Answer: Twenty-eight lakh thirty-four thousand five hundred sixty-seven
In simple words: Read the number according to the Indian place value chart (lakhs, thousands, hundreds).

Exam Tip: In the Indian system, commas are placed after 3 digits from the right, then after every 2 digits.

Question 7. Place value of 9 in '796745' is __________.
Answer: 90,000
In simple words: 9 is in the ten-thousands place, so its place value is 90000.

Exam Tip: Count the digits to the right of 9 to determine how many zeros to append.

Question 8. Expanded form of 2345678 is ___________.
Answer: \(2000000 + 300000 + 40000 + 5000 + 600 + 70 + 8\)
In simple words: Write out the value of each digit based on its position.

Exam Tip: Each term in the expanded form corresponds to a non-zero digit multiplied by its place value.

Question 9. Number name of 6789053 in international system of numeration is _________.
Answer: Six million seven hundred eighty-nine thousand fifty-three
In simple words: Group digits in threes from the right in the international system.

Question 10. Starting from the smallest 8-digit number, write the next five numbers in ascending order.
Answer:
Smallest 8-digit number = 1,00,00,000 (Ten million / One crore)
Next five numbers in ascending order:
1. 1,00,00,001
2. 1,00,00,002
3. 1,00,00,003
4. 1,00,00,004
5. 1,00,00,005
In simple words: Add 1 successively to the starting number to get the next five numbers.

Exam Tip: Ascending order means arranging numbers from smallest to largest.

Question 11. Write 8945673 using commas in Indian as well as International System of Numeration.
Answer:
Indian System: 89,45,673
International System: 8,945,673
In simple words: Place commas after every two digits (starting from hundreds) for the Indian system, and after every three digits for the international system.

Exam Tip: Memorize comma placement: Indian (3, 2, 2) vs International (3, 3, 3).

Question 12. The number of sheets of paper available for making notebooks is 25,000. Each sheet makes 12 pages of a notebook. Each notebook contains 300 pages. How many notebooks can be made from the paper available?
Answer:
Total sheets available = 25,000
Pages per sheet = 12
Total pages available = \(25,000 \times 12 = 3,00,000 \text{ pages}\)
Pages per notebook = 300
Total notebooks that can be made = \(\frac{3,00,000}{300} = 1,000 \text{ notebooks}\)
In simple words: Find total pages by multiplying sheets by 12, then divide by 300 to find the number of notebooks.

Exam Tip: Always calculate total units before dividing by the size of each final product.

CLASS - VI Mathematics (Whole Numbers)

Worksheet-3

Choose correct option in questions 1 to 7.

Question 1. What is the predecessor of 3456?
(a) 3455
(b) 3454
(c) 3453
(d) 3452
Answer: (a) 3455
Predecessor means 1 less than the given number: \(3456 - 1 = 3455\).
In simple words: Subtract 1 to find the predecessor.

Exam Tip: Predecessor = Number - 1.

Question 2. What is the successor of 678?
(a) 677
(b) 679
(c) 680
(d) 681
Answer: (b) 679
Successor means 1 more than the given number: \(678 + 1 = 679\).
In simple words: Add 1 to find the successor.

Exam Tip: Successor = Number + 1.

Question 3. 430 ___ 403
(a) >
(b) <
(c) =
(d) none of these
Answer: (a) >
Comparing 430 and 403, 430 is greater than 403. Therefore, \(430 > 403\).
In simple words: 430 is bigger than 403.

Question 4. 12345 _____ 45678
(a) >
(b) <
(c) =
(d) none of these
Answer: (b) <
Comparing 12345 and 45678, the thousands/ten-thousands digit 1 is less than 4. Hence, \(12345 < 45678\).
In simple words: 12345 is smaller than 45678.

Exam Tip: Compare the highest place value first.

Question 5. What is 9 x 3?
(a) 21
(b) 27
(c) 6
(d) 12
Answer: (b) 27
\(9 \times 3 = 27\)
In simple words: 9 multiplied by 3 equals 27.

Exam Tip: Memorize multiplication tables up to 20 for speed.

Question 6. The whole number ____ has no predecessor.
(a) 0
(b) 1
(c) 2
(d) none of these
Answer: (a) 0
In whole numbers, 0 is the smallest number and has no predecessor in whole numbers.
In simple words: Whole numbers start at 0, so 0 has no whole number before it.

Exam Tip: Natural numbers have no predecessor for 1, but whole numbers have no predecessor for 0.

Question 7. Which of the following will not represent zero?
(a) \(1 + 0\)
(b) \(0 \times 0\)
(c) \(\frac{0}{2}\)
(d) \(\frac{10 - 10}{2}\)
Answer: (a) \(1 + 0\)
\(1 + 0 = 1\), which is not zero. The other options evaluate to 0.
In simple words: Adding 0 to 1 gives 1, not 0.

Exam Tip: Test each option mathematically to check its result.

Fill in the blanks:

Question 6. If the product of two whole numbers is zero, then ______ of them will be zero.
Answer: at least one
In simple words: If multiplication gives 0, one or both numbers must be 0.

Exam Tip: Zero property of multiplication: \(a \times 0 = 0\).

Question 7. Every natural number except _____ has a predecessor.
Answer: 1
In simple words: 1 is the first natural number, so it has no natural number before it.

Exam Tip: Natural numbers start from 1.

Question 8. If we add the number _____ to the collection of natural numbers, we get the collection of whole numbers.
Answer: 0
In simple words: Whole numbers are natural numbers plus zero.

Exam Tip: Natural numbers = {1, 2, 3...}, Whole numbers = {0, 1, 2, 3...}.

Question 9. 7 x 4 = ____
Answer: 28
In simple words: 7 times 4 is 28.

Question 10. You are traveling at a constant speed of 55 miles per hour. How long will it take to travel 220 miles?
Answer:
Speed = 55 miles per hour
Total distance = 220 miles
Time = \(\frac{\text{Distance}}{\text{Speed}}\)
Time = \(\frac{220}{55} = 4 \text{ hours}\)
In simple words: Divide total distance by speed to get the time taken.

Exam Tip: Formula: \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\).

Question 11. Every classroom in a school contains 32 students. How many students would 5 classrooms hold?
Answer:
Students in 1 classroom = 32
Number of classrooms = 5
Total students = \(32 \times 5 = 160 \text{ students}\)
In simple words: Multiply students per classroom by the number of classrooms.

Exam Tip: For equal groups, use multiplication.

Question 12. A book contains 365 pages. You are currently on page 123. How much more do you have to read?
Answer:
Total pages = 365
Current page = 123
Pages remaining to read = \(365 - 123 = 242\)
In simple words: Subtract the current page number from the total pages.

Exam Tip: Finding the remaining amount requires subtraction.

Question 13. Find the sum by suitable rearrangement: 837 + 208 + 363.
Answer:
\(837 + 208 + 363\)
Using commutative and associative properties, group numbers that make a round sum (837 + 363 = 1200):
\(= (837 + 363) + 208\)
\(= 1200 + 208\)
\(= 1408\)
In simple words: Rearrange the numbers to add 837 and 363 first, then add 208.

Exam Tip: Look for units digits that add up to 10 to simplify addition.

Question 14. Find the product by suitable rearrangement: 2 x 1768 x 50.
Answer:
\(2 \times 1768 \times 50\)
Group 2 and 50 first because \(2 \times 50 = 100\):
\(= (2 \times 50) \times 1768\)
\(= 100 \times 1768\)
\(= 176800\)
In simple words: Multiply 2 and 50 to get 100, then multiply by 1768.

Exam Tip: Group pairs like (2, 50) or (4, 25) or (8, 125) to make multiplication easier.

CLASS - VI Mathematics (Whole Numbers)

Worksheet-4

Choose correct option in questions 1 to 5.

Question 1. What is the predecessor of 1678?
(a) 1677
(b) 1676
(c) 1675
(d) 1674
Answer: (a) 1677
\(1678 - 1 = 1677\)
In simple words: Subtract 1 from 1678.

Exam Tip: Predecessor is the number immediately preceding.

Question 2. What is the successor of 456?
(a) 455
(b) 457
(c) 458
(d) 454
Answer: (b) 457
\(456 + 1 = 457\)
In simple words: Add 1 to 456.

Exam Tip: Successor is the number immediately following.

Question 3. 460 ___ 406
(a) >
(b) <
(c) =
(d) none of these
Answer: (a) >
460 is greater than 406, so \(460 > 406\).
In simple words: 460 is bigger than 406.

Exam Tip: Compare digits from left to right.

Question 4. 3456 _____ 4356
(a) >
(b) <
(c) =
(d) none of these
Answer: (b) <
Comparing thousands digits: 3 is less than 4, so \(3456 < 4356\).
In simple words: 3456 is smaller than 4356.

Exam Tip: Look at the highest place value to compare.

Question 5. What is 7 - 5?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2
\(7 - 5 = 2\)
In simple words: Taking 5 away from 7 leaves 2.

Exam Tip: Basic subtraction fact.

Fill in the blanks:

Question 6. All natural numbers are __________ numbers.
Answer: whole
In simple words: Every natural number is also a whole number.

Exam Tip: The collection of whole numbers includes all natural numbers plus zero.

Question 7. 600 is the successor of _____.
Answer: 599
In simple words: 600 comes right after 599.

Exam Tip: To find the number whose successor is given, subtract 1.

Question 8. A number remains unchanged when multiplied to ______.
Answer: 1
In simple words: Multiplying any number by 1 gives the same number.

Exam Tip: 1 is the multiplicative identity.

Question 9. 7 x 3 = ____
Answer: 21
In simple words: 7 times 3 is 21.

Exam Tip: Multiplication table of 7.

Answer the following:

Question 10. A taxi driver filled his car petrol tank with 40 litres of petrol on Monday. The next day, he filled the tank with 50 litres of petrol. If the petrol costs Rs 45 per litre, how much did he spend in all on petrol?
Answer:
Petrol filled on Monday = 40 litres
Petrol filled on next day = 50 litres
Total petrol filled = \(40 + 50 = 90 \text{ litres}\)
Cost per litre = Rs 45
Total money spent = \(90 \times 45 = \text{Rs } 4050\)
In simple words: Add the total litres of petrol and multiply by the cost per litre. Alternatively, use distributive property: \(45 \times (40 + 50) = 45 \times 90 = 4050\).

Exam Tip: You can add quantities first and then multiply by the unit price, or use the distributive property.

Question 11. If the product of two whole numbers is 1, can we say that one or both of them will be 1? Justify through examples.
Answer:
Yes, both of them must be equal to 1.
Example: \(1 \times 1 = 1\).
If any number other than 1 (say 2) is multiplied by 1, the product is 2, not 1. Thus, to get a product of 1, both numbers must be 1.
In simple words: The only way to multiply two whole numbers and get 1 is if both numbers are 1.

Exam Tip: Multiplicative identity property states that \(a \times 1 = a\); hence for the product to remain 1, the multiplier must be 1.

Question 12. Find:
a. 12 x 35
b. 34 x 99
Answer:
a. \(12 \times 35 = 12 \times (30 + 5) = (12 \times 30) + (12 \times 5) = 360 + 60 = 420\)
b. \(34 \times 99 = 34 \times (100 - 1) = (34 \times 100) - (34 \times 1) = 3400 - 34 = 3366\)
In simple words: Use the distributive property to break numbers into tens and ones or round numbers like 99 to (100 - 1).

Exam Tip: Splitting numbers using distributivity makes mental math much easier.

Question 13. Find the value using suitable properties: 52465 x 93 + 7 x 52465.
Answer:
Using distributive property \(a \times b + a \times c = a \times (b + c)\):
\(52465 \times 93 + 7 \times 52465\)
\(= 52465 \times (93 + 7)\)
\(= 52465 \times 100\)
\(= 5246500\)
In simple words: Factor out 52465 and add 93 and 7 to get 100, then multiply.

Exam Tip: Look for common factors to apply the distributive property in reverse.

Question 14. Find the product using suitable properties: 854 x 103.
Answer:
\(854 \times 103 = 854 \times (100 + 3)\)
Using distributive property:
\(= (854 \times 100) + (854 \times 3)\)
\(= 85400 + 2562\)
\(= 87962\)
In simple words: Write 103 as \(100 + 3\) and distribute 854 over the sum.

Exam Tip: Convert numbers near multiples of 10 into a sum or difference.

Question 15. Match the following:
i. 425 x 136 = 425 x (6 + 30 + 100)
ii. 2 x 49 x 50 = 2 x 50 x 49
iii. 80 + 2005 + 20 = 80 + 20 + 2005
a. Commutativity under multiplication
b. Commutativity under addition
c. Distributivity of multiplication over addition
Answer:
(i) matches (c) Distributivity of multiplication over addition
(ii) matches (a) Commutativity under multiplication
(iii) matches (b) Commutativity under addition
In simple words: Match each mathematical property with its correct example equation.

Exam Tip: Commutative property changes order; distributive property spreads multiplication over addition.

CLASS - VI Mathematics (Playing with Numbers)

Worksheet-5

Choose correct option in questions 1 to 10.

Question 1. _____ is the factor of 50.
(a) 10
(b) 3
(c) 7
(d) 6
Answer: (a) 10
Since \(50 \div 10 = 5\) without a remainder, 10 is a factor of 50.
In simple words: 10 divides 50 completely, so it is a factor.

Exam Tip: A factor divides a number completely leaving a remainder of 0.

Question 2. Sixth multiple of 11 is _____.
(a) 55
(b) 66
(c) 77
(d) 88
Answer: (b) 66
Sixth multiple of 11 = \(11 \times 6 = 66\).
In simple words: Multiply 11 by 6 to get the sixth multiple.

Exam Tip: The nth multiple of a number is found by multiplying the number by n.

Question 3. Which of them is a prime number?
(a) 27
(b) 21
(c) 23
(d) 25
Answer: (c) 23
23 has only two factors: 1 and 23. The others (27, 21, 25) have additional factors.
In simple words: 23 cannot be divided evenly by any number other than 1 and itself.

Exam Tip: Prime numbers have exactly two distinct factors: 1 and themselves.

Question 4. Which of them is a composite number?
(a) 23
(b) 29
(c) 31
(d) 33
Answer: (d) 33
33 has factors 1, 3, 11, and 33, making it composite. The others are prime.
In simple words: 33 has more than two factors, so it is composite.

Exam Tip: Composite numbers have more than two factors.

Question 5. _____ is a factor of every number.
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1
1 divides every number completely, so it is a universal factor.
In simple words: Every number can be divided by 1.

Exam Tip: 1 is a factor of every number, and every number is a multiple of 1.

Question 6. The number of multiples of a given number is
(a) 1
(b) 10
(c) 100
(d) infinite
Answer: (d) infinite
Multiples continue endlessly because you can multiply a number by any whole number.
In simple words: You can never list all multiples because they go on forever.

Exam Tip: Factors are finite, but multiples are infinite.

Question 7. The only even prime number is
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (a) 2
2 is prime because its only factors are 1 and 2, and it is even. All other even numbers are divisible by 2 and thus composite.
In simple words: 2 is the only even number that is also prime.

Question 8. The smallest composite number is
(a) 4
(b) 1
(c) 9
(d) 6
Answer: (a) 4
The factors of 4 are 1, 2, and 4. Since it has more than two factors and is smaller than 6 and 9, 4 is the smallest composite number.
In simple words: 4 is the first number with more than two factors.

Exam Tip: 1 is neither prime nor composite; 2 and 3 are prime.

Question 9. The smallest odd composite number is
(a) 5
(b) 15
(c) 9
(d) 3
Answer: (c) 9
The odd numbers are 1, 3, 5, 7, 9... 1 is neither, 3 and 5 and 7 are prime. 9 has factors 1, 3, and 9, making it the smallest odd composite number.
In simple words: 9 is the first odd number that is not prime.

Exam Tip: Check odd numbers starting from 3 for factors.

Question 10. Two numbers having only 1 as a common factor are called ________ numbers.
(a) co-prime numbers
(b) twin prime numbers
(c) composite numbers
(d) prime numbers
Answer: (a) co-prime numbers
Co-prime numbers have a Greatest Common Divisor (GCD) of 1.
In simple words: Numbers that share no common factors other than 1 are co-prime.

Exam Tip: Co-prime numbers do not need to be prime individually.

Fill in the blanks:

Question 11. Every multiple of a given number is greater than or equal to that ________.
Answer: number
In simple words: Multiples start from the number itself and go higher.

Exam Tip: Factors are less than or equal to the number; multiples are greater than or equal to the number.

Question 12. The number _____ is the smallest prime number and is even.
Answer: 2
In simple words: 2 is the starting prime number.

Exam Tip: 1 is not prime.

Question 13. HCF of 12 and 16 is _____.
Answer: 4
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16
Common factors: 1, 2, 4. Highest is 4.
In simple words: 4 is the largest number that divides both 12 and 16.

Exam Tip: HCF stands for Highest Common Factor.

Question 14. First three multiples of 10 are ________.
Answer: 10, 20, 30
In simple words: Multiply 10 by 1, 2, and 3.

Answer the following:

Question 15. Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 7 in each case.
Answer:
First, find the LCM of 12, 16, 24, and 36.
Prime factorizations:
\(12 = 2^2 \times 3\)
\(16 = 2^4\)
\(24 = 2^3 \times 3\)
\(36 = 2^2 \times 3^2\)
LCM = \(2^4 \times 3^2 = 16 \times 9 = 144\)
Since it leaves a remainder 7 in each case, add 7 to the LCM:
Required number = \(144 + 7 = 151\)
In simple words: Find the LCM of the divisors and add the remainder to it.

Exam Tip: When a common remainder is required, find the LCM of the divisors and add the remainder.

Question 16. In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that all can cover the same distance in complete steps?
Answer:
To find the minimum distance, we need to find the LCM of 80, 85, and 90.
Prime factorizations:
\(80 = 2^4 \times 5\)
\(85 = 5 \times 17\)
\(90 = 2 \times 3^2 \times 5\)
LCM = \(2^4 \times 3^2 \times 5 \times 17 = 16 \times 9 \times 5 \times 17 = 12240 \text{ cm} = 122 \text{ m } 40 \text{ cm}\)
In simple words: Calculate the LCM of step lengths to find the smallest distance they can all cover in exact steps.

Exam Tip: Problems asking for "minimum distance" or "together again" typically require finding the LCM.

Question 17. Find the LCM of the following numbers:
a. 9 and 4
b. 6 and 5
Answer:
a. Since 9 and 4 are co-prime numbers, their LCM is their product: \(9 \times 4 = 36\).
b. Since 6 and 5 are co-prime numbers, their LCM is their product: \(6 \times 5 = 30\).
In simple words: For co-prime numbers, multiply them together to find their LCM.

Exam Tip: The LCM of two co-prime numbers is always equal to their product.

CLASS - VI Mathematics (Playing with Numbers)

Worksheet-6

Choose correct option in questions 1 to 11.

Question 1. _____ is the factor of 81.
(a) 9
(b) 8
(c) 7
(d) 5
Answer: (a) 9
Since \(81 \div 9 = 9\), 9 is a factor of 81.
In simple words: 9 divides 81 evenly.

Exam Tip: Check division without remainders.

Question 2. Fifth multiple of 9 is _____.
(a) 54
(b) 45
(c) 36
(d) 27
Answer: (b) 45
\(9 \times 5 = 45\)
In simple words: Multiply 9 by 5.

Exam Tip: Multiple = Number x n.

Question 3. Which of them is not a prime number?
(a) 7
(b) 11
(c) 10
(d) 13
Answer: (c) 10
10 is a composite number because its factors are 1, 2, 5, and 10. The others are prime.
In simple words: 10 can be divided by 2 and 5, so it is not prime.

Exam Tip: Even numbers greater than 2 are never prime.

Question 4. Which of them is not a composite number?
(a) 14
(b) 16
(c) 8
(d) 13
Answer: (d) 13
13 is a prime number, not composite. 14, 16, and 8 are composite.
In simple words: 13 is prime, while the others are composite.

Exam Tip: Prime numbers have only two factors.

Question 5. Two numbers having only _____ as a common factor are called co-prime numbers.
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1
Co-prime numbers share only 1 as a common factor.
In simple words: Their HCF is 1.

Exam Tip: Definition of co-prime numbers.

Question 6. The HCF of two consecutive numbers is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1
Consecutive numbers (like 4 and 5) always share only 1 as a common factor.
In simple words: Any two numbers coming right after each other have an HCF of 1.

Exam Tip: Consecutive integers are always co-prime.

Question 7. The HCF of two consecutive odd numbers is
(a) 3
(b) 2
(c) 1
(d) 0
Answer: (c) 1
Consecutive odd numbers (like 3 and 5, or 15 and 17) have an HCF of 1.
In simple words: Odd numbers in a row share no common factors other than 1.

Exam Tip: Consecutive odd numbers are co-prime.

Question 8. The HCF of two consecutive even numbers is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (c) 2
Consecutive even numbers (like 4 and 6, or 10 and 12) are both divisible by 2, and their HCF is always 2.
In simple words: Since both numbers are even, they can both be divided by 2.

Question 9. The HCF of two co-prime numbers is
(a) 0
(b) 3
(c) 2
(d) 1
Answer: (d) 1
By definition, co-prime numbers have an HCF of 1.
In simple words: Co-prime numbers only share 1 as a common factor.

Exam Tip: HCF of co-prime numbers is always 1.

Question 10. The LCM of two consecutive numbers is
(a) their sum
(b) their product
(c) their difference
(d) none of these
Answer: (b) their product
Since consecutive numbers are co-prime, their LCM is their product.
In simple words: Multiply the two consecutive numbers to find their LCM.

Exam Tip: LCM of co-prime numbers equals their product.

Question 11. The LCM of two co-prime numbers is
(a) their sum
(b) their product
(c) their difference
(d) none of these
Answer: (b) their product
By definition, the LCM of co-prime numbers is their product.
In simple words: Co-prime numbers have an LCM equal to their multiplication.

Exam Tip: Product rule for LCM of co-prime numbers.

Fill in the blanks:

Question 12. If two given numbers are divisible by a number, then their _________ are also divisible by that number.
Answer: sum and difference
In simple words: If a number divides two numbers, it also divides their sum and difference.

Exam Tip: Divisibility property of sum and difference.

Question 13. A number is divisible by ______, if it is divisible by both 3 and 6.
Answer: 18 (Wait, if a number is divisible by both 3 and 6, it is divisible by their LCM, which is 6. Let us check standard phrasing: if a number is divisible by both 3 and 6, is it divisible by 18? Not necessarily, e.g., 6 is divisible by 3 and 6, but not 18. Wait, if a number is divisible by 6, it is automatically divisible by 3. Let us re-read: "A number is divisible by _____, if it is divisible by both 3 and 6" - actually, divisibility by 6 already implies divisibility by 3. If a number is divisible by 6, it is divisible by 6. Let us put 6 or check if it means something else. Alternatively, 18 if co-prime, but 3 and 6 are not co-prime. Let us state 6).
Answer: 6
In simple words: Divisibility by 6 covers divisibility by 3.

Exam Tip: If a number is divisible by a larger number, it is divisible by all its factors.

Question 14. HCF of 8 and 12 is _____.
Answer: 4
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
Highest common factor is 4.
In simple words: 4 is the largest number dividing both 8 and 12.

Exam Tip: List factors to find the HCF.

Question 15. First two multiples of 9 are ________.
Answer: 9, 18
In simple words: 9 x 1 = 9 and 9 x 2 = 18.

Answer the following:

Question 16. Sort out even and odd numbers: 43, 48, 61, 69, 80, 155, 332, 264, 89, 19, 76, 125, 64
Answer:
Even numbers: 48, 80, 332, 264, 76, 64
Odd numbers: 43, 61, 69, 155, 89, 19, 125
In simple words: Even numbers end in 0, 2, 4, 6, 8; odd numbers end in 1, 3, 5, 7, 9.

Question 17. Match the items in column I and column II.
COLUMN - I          COLUMN - II
(i) 45                         (A) multiple of 3
(ii) 15                        (B) factor of 40
(iii) 24                       (C) multiple of 7
(iv) 20                       (D) factor of 30
(v) 35                        (E) multiple of 9
Answer:
(i) 45 matches (E) multiple of 9
(ii) 15 matches (A) multiple of 3 (or factor of 30? Wait, 15 is a factor of 30. Let us check: 15 is a factor of 30. Let us re-verify: factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. So (ii) 15 matches (D) factor of 30).
(iii) 24 matches (A) multiple of 3
(iv) 20 matches (B) factor of 40
(v) 35 matches (C) multiple of 7
Let us re-map correctly:
(i) 45 - (E) multiple of 9
(ii) 15 - (D) factor of 30
(iii) 24 - (A) multiple of 3
(iv) 20 - (B) factor of 40
(v) 35 - (C) multiple of 7
In simple words: Match each number with its correct mathematical property.

Exam Tip: Verify factors by division and multiples by multiplication.

Question 18. Find the multiples of 7 which is greater than 56 but less than 77.
Answer:
Multiples of 7 around this range: \(7 \times 8 = 56\), \(7 \times 9 = 63\), \(7 \times 10 = 70\), \(7 \times 11 = 77\).
Numbers strictly greater than 56 and less than 77 are 63 and 70.
In simple words: The multiples of 7 between 56 and 77 are 63 and 70.

Exam Tip: List multiplication table of 7 to find values in a given range.

Question 19. Find the HCF of 70, 105, 175.
Answer:
Prime factorizations:
\(70 = 2 \times 5 \times 7\)
\(105 = 3 \times 5 \times 7\)
\(175 = 5^2 \times 7\)
Common prime factors are 5 and 7.
HCF = \(5 \times 7 = 35\)
In simple words: 35 is the largest number that divides 70, 105, and 175 completely.

Exam Tip: HCF is the product of the lowest powers of common prime factors.

Question 20. Determine the smallest three digit number which is exactly divisible by 6, 8 and 12.
Answer:
First, find the LCM of 6, 8, and 12.
\(6 = 2 \times 3\)
\(8 = 2^3\)
\(12 = 2^2 \times 3\)
LCM = \(2^3 \times 3 = 24\)
Now, find the smallest 3-digit multiple of 24.
Multiples of 24: 24, 48, 72, 96, 120.
The smallest 3-digit multiple is 120.
In simple words: Find the LCM of the numbers, then find the smallest 3-digit multiple of that LCM.

Exam Tip: Any number divisible by 6, 8, and 12 must be a multiple of their LCM.

Question 21. Is 70169308 divisible by 11 (use divisibility test).
Answer:
Number: 7 0 1 6 9 3 0 8
Positions from right: 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th.
Sum of digits at odd places (from right): \(8 + 3 + 1 + 7 = 19\)
Sum of digits at even places (from right): \(0 + 9 + 6 + 0 = 15\)
Difference = \(19 - 15 = 4\)
Since 4 is not divisible by 11, the number 70169308 is not divisible by 11.
In simple words: Subtract the sum of even-position digits from the sum of odd-position digits; if the difference is not a multiple of 11, the number is not divisible by 11.

Question 22. Find the prime factorization of 980.
Answer:
\(980 \div 2 = 490\)
\(490 \div 2 = 245\)
\(245 \div 5 = 49\)
\(49 \div 7 = 7\)
\(7 \div 7 = 1\)
Prime factorization of 980 = \(2 \times 2 \times 5 \times 7 \times 7 = 2^2 \times 5 \times 7^2\)
In simple words: Divide the number repeatedly by prime numbers until you reach 1.

Exam Tip: Express prime factorization using exponents when factors repeat.

Question 23. Express 53 as the sum of three odd primes.
Answer:
\(53 = 13 + 13 + 27\) (27 is not prime)
Let us try: \(53 = 13 + 17 + 23\)
Check: \(13 + 17 = 30\); \(30 + 23 = 53\). All three (13, 17, 23) are odd prime numbers.
In simple words: Find three odd prime numbers that add up to 53.

Exam Tip: Test combinations of small odd prime numbers (3, 5, 7, 11, 13, 17, 19, 23...).

Question 24. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
Answer:
The sequence is: 90, 91, 92, 93, 94, 95, 96
Let us verify:
90 (composite), 91 (\(7 \times 13\)), 92 (\(2 \times 46\)), 93 (\(3 \times 31\)), 94 (\(2 \times 47\)), 95 (\(5 \times 19\)), 96 (\(2 \times 48\)).
All seven numbers are composite and less than 100.
In simple words: Find a block of seven numbers in a row where every single one is composite.

Exam Tip: Remember this standard textbook sequence (90 to 96) for consecutive composite numbers.

Question 25. The length, breadth and height of a room are 825cm, 675cm and 450cm respectively. Find the longest tape which can measure the three dimensions of the room exactly.
Answer:
To find the longest tape, we need to calculate the HCF of 825, 675, and 450.
Prime factorizations:
\(825 = 3 \times 5^2 \times 11\)
\(675 = 3^3 \times 5^2\)
\(450 = 2 \times 3^2 \times 5^2\)
Common prime factors with lowest powers: \(3^1 \times 5^2 = 3 \times 25 = 75\)
HCF = 75 cm
In simple words: Calculate the HCF of the three dimensions to find the length of the longest measuring tape.

Exam Tip: Questions asking for the "longest tape", "greatest capacity", or "maximum length" require finding the HCF.

CLASS - VI Mathematics (Integers)

Worksheet-7

Choose correct option in questions 1 to 10.

Question 1. Subtract (-6) from (-11).
(a) -5
(b) 5
(c) -17
(d) 17
Answer: (a) -5
\((-11) - (-6) = -11 + 6 = -5\)
In simple words: Subtracting negative 6 is the same as adding 6 to -11.

Exam Tip: Subtracting a negative number is equivalent to adding its positive counterpart.

Question 2. Write the integer which is 3 more than 5.
(a) -8
(b) 8
(c) 2
(d) -2
Answer: (b) 8
\(5 + 3 = 8\)
In simple words: Add 3 to 5 to get 8.

Exam Tip: "More than" means addition.

Question 3. Write the integer which is 4 more than -1.
(a) 5
(b) -5
(c) 3
(d) -3
Answer: (c) 3
\(-1 + 4 = 3\)
In simple words: Adding 4 to -1 gives 3.

Exam Tip: Moving to the right on the number line represents addition.

Question 4. (-9) + (+13) = _______
(a) -22
(b) 22
(c) -4
(d) 4
Answer: (d) 4
\(-9 + 13 = 4\)
In simple words: Combine -9 and 13 to get 4.

Exam Tip: Subtract the smaller absolute value from the larger and keep the sign of the larger number.

Question 5. (+12) + (-7) = _____
(a) 5
(b) -5
(c) -19
(d) 19
Answer: (a) 5
\(12 - 7 = 5\)
In simple words: 12 minus 7 equals 5.

Exam Tip: Adding a negative integer is the same as subtraction.

Question 6. Write with appropriate sign: 15°C below 0°C temperature
(a) -15
(b) 15
(c) 5
(d) -5
Answer: (a) -15
Temperatures below zero are represented with a negative sign.
In simple words: "Below zero" means negative, so it is -15.

Question 7. -1 _____ 0
(a) >
(b) <
(c) =
(d) none of these
Answer: (b) <
On the number line, -1 is to the left of 0, so \(-1 < 0\).
In simple words: Any negative number is smaller than zero.

Exam Tip: Negative numbers are always less than zero.

Question 8. -5 ______ -7
(a) =
(b) <
(c) >
(d) none of these
Answer: (c) >
On the number line, -5 is to the right of -7, so \(-5 > -7\).
In simple words: A smaller negative number has a greater value.

Exam Tip: For negative numbers, the number with the smaller magnitude is greater.

Question 9. Sum of (-9) and 15.
(a) -24
(b) 24
(c) -6
(d) 6
Answer: (d) 6
\(-9 + 15 = 6\)
In simple words: Add -9 and 15 to get 6.

Exam Tip: Follow rules of integer addition.

Question 10. Subtract (-12) from 7.
(a) 19
(b) -19
(c) -5
(d) 5
Answer: (a) 19
\(7 - (-12) = 7 + 12 = 19\)
In simple words: 7 minus negative 12 becomes \(7 + 12 = 19\).

Exam Tip: "Subtract A from B" means \(B - A\).

Fill in the blanks:

Question 11. 13 + _____ = 0
Answer: -13
In simple words: Add the additive inverse (-13) to get zero.

Exam Tip: A number plus its opposite equals 0.

Question 12. – 1, – 2, – 3, – 4, ... called negative numbers are ___________ integers.
Answer: negative
In simple words: These are negative integers.

Exam Tip: Integers consist of positive integers, zero, and negative integers.

Question 13. One less than given number gives ___________.
Answer: predecessor
In simple words: Subtracting 1 gives the predecessor.

Exam Tip: Predecessor is number minus 1.

Question 14. When two negative integers are added, we get a ___________ integer.
Answer: negative
In simple words: Adding two negative numbers always results in a negative number.

Exam Tip: Same signs add up and keep the common sign.

Answer the following:

Question 15. Find the value of (30) + (– 23) + (– 63) + (+ 55).
Answer:
\(= 30 - 23 - 63 + 55\)
Combine positive terms: \(30 + 55 = 85\)
Combine negative terms: \(-23 - 63 = -86\)
\(= 85 - 86 = -1\)
In simple words: Add all positive numbers together and all negative numbers together, then find the difference.

Exam Tip: Group positives and negatives separately to simplify calculations.

Question 16. Add without using number line:
a. (– 10) + (+ 19)
b. (– 250) + (+ 150)
Answer:
a. \(-10 + 19 = 9\)
b. \(-250 + 150 = -100\)
In simple words: Find the difference between the absolute values and assign the sign of the larger number.

Exam Tip: Different signs mean subtraction of absolute values.

Question 17. Which integers lie between – 10 and – 5?
Answer:
The integers between -10 and -5 are -9, -8, -7, and -6.
In simple words: List the whole number steps between -10 and -5 on the number line.

Exam Tip: Do not include the endpoints (-10 and -5) when asked for integers "between" them.

Question 18. Fill in the blanks with >, < or = sign.
a. (– 3) + (– 6) ______ (– 3) – (– 6)
b. (– 21) – (– 10) _____ (– 31) + (– 11)
Answer:
a. Left side: \(-3 + (-6) = -9\)
Right side: \(-3 - (-6) = -3 + 6 = 3\)
Comparing -9 and 3: \(-9 < 3\)
b. Left side: \(-21 - (-10) = -21 + 10 = -11\)
Right side: \(-31 + (-11) = -42\)
Comparing -11 and -42: \(-11 > -42\)
In simple words: Simplify both sides of each statement first, then compare the resulting integers.

Exam Tip: Always evaluate expressions on both sides before placing inequality symbols.

CLASS - VI Mathematics (Integers)

Worksheet-8

Choose correct option in questions 1 to 5.

Question 1. What is opposite of ’50 km of south’?
(a) 50 km of north
(b) 50 km of east
(c) 50 km of west
(d) none of these
Answer: (a) 50 km of north
The opposite direction of south is north.
In simple words: North is the exact opposite of south.

Exam Tip: Opposites pair up cardinal directions (North-South, East-West).

Question 2. Which is the right form of integer ‘20°C above 0°C’?
(a) -20
(b) 20
(c) -30
(d) 30
Answer: (b) 20
Temperatures above zero are represented as positive integers.
In simple words: "Above zero" means positive +20.

Question 3. Which number is to the right of the other on the number line? 0, -1
(a) 0 is on the left of 1
(b) 0 is on the right of 1
(c) 0 is on the right of -1
(d) 0 is on the left of -1
Answer: (c) 0 is on the right of -1
On a number line, numbers increase from left to right. Since 0 is greater than -1, 0 is to the right of -1.
In simple words: 0 is to the right of -1 on the number line.

Exam Tip: Greater numbers always lie to the right of smaller numbers on the number line.

Question 4. (– 11) + (– 12) = ______
(a) -1
(b) 23
(c) 1
(d) -23
Answer: (d) -23
\(-11 + (-12) = -(11 + 12) = -23\)
In simple words: Add 11 and 12 and keep the negative sign.

Exam Tip: Sum of two negative numbers is negative.

Question 5. (– 4) + (+ 3) = _____
(a) -1
(b) 1
(c) -7
(d) 7
Answer: (a) -1
\(-4 + 3 = -1\)
In simple words: Combine -4 and 3 to get -1.

Exam Tip: Take the difference of absolute values and give the sign of the larger number.

Fill in the blanks:

Question 6. Every positive integer is larger than every ________ integer.
Answer: negative
In simple words: Positive numbers are always greater than negative numbers.

Exam Tip: Any positive number > 0 > any negative number.

Question 7. Farther a number from zero on the right, _________ is its value.
Answer: greater
In simple words: Moving right on the number line increases the value.

Exam Tip: Value increases from left to right.

Question 8. _______ is less than every positive integer.
Answer: Zero (or every negative integer)
In simple words: Negative integers and zero are less than positive integers.

Question 9. – 26 is _________ than – 25.
Answer: less
In simple words: -26 is further to the left than -25, so it is smaller.

Exam Tip: For negative numbers, larger absolute value means smaller value.

Question 10. (-24) + _____ = 0
Answer: 24
In simple words: Add 24 to -24 to get zero.

Question 11. -8 ______ -16
Answer: >
In simple words: -8 is greater than -16.

Question 12. The subtraction of an integer is the same as the addition of its ___________.
Answer: additive inverse (or opposite sign)
In simple words: Subtracting a number is the same as adding its opposite.

Question 13. The collection of _________ can be written as..., –5, –4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5...
Answer: integers
In simple words: This set represents all integers.

Answer the following:

Question 14. Which integers lie between – 6 and – 1? Which is the largest integer and the smallest integer among them?
Answer:
Integers between -6 and -1 are -5, -4, -3, and -2.
Largest integer among them = -2
Smallest integer among them = -5
In simple words: The numbers between -6 and -1 are -5, -4, -3, -2, with -2 being the largest and -5 the smallest.

Exam Tip: On the number line, numbers to the right are greater.

Question 15. Represent the following numbers as integers with appropriate signs.
a. A submarine is moving at a depth, eight hundred metre below the sea level.
b. A deposit of rupees two hundred.
Answer:
a. -800 m
b. +Rs 200 (or 200)
In simple words: Depths and withdrawals use negative signs; deposits and heights above sea level use positive signs.

Question 16. Write the following numbers with appropriate signs:
a. 150 m below sea level.
b. 30°C above 0°C temperature.
Answer:
a. -150 m
b. +30°C
In simple words: Below sea level is negative, above zero temperature is positive.

Question 17. Compare the following pairs of numbers using > or <.
a. 0 _____ -15
b. -13 _____ 13
Answer:
a. \(0 > -15\)
b. \(-13 < 13\)
In simple words: Zero is greater than any negative number, and negative numbers are smaller than positive numbers.

Question 18. Which integers lie between – 8 and – 2?
Answer:
The integers between -8 and -2 are -7, -6, -5, -4, and -3.
In simple words: List all whole integer values between -8 and -2.

CLASS - VI Mathematics (Basic Geometrical Ideas)

Worksheet-9

Choose correct option in questions 1 to 8.

Question 1. Which of them is not a line segment?
(a) an edge of a box
(b) a tube light
(c) the edge of a post card
(d) rail lines
Answer: (d) rail lines
Rail lines extend indefinitely and represent lines, whereas the others have fixed endpoints representing line segments.
In simple words: Railway tracks go on without ending, unlike finite line segments.

Exam Tip: Line segments have two definite endpoints.

Question 2. Which of them is a ray?
(a) an edge of a box
(b) sun rays
(c) the edge of a post card
(d) rail lines
Answer: (b) sun rays
Sun rays start from a source (the sun) and travel infinitely in one direction, which defines a ray.
In simple words: Rays start at a point and go on infinitely in one direction.

Exam Tip: A ray has one starting point and extends infinitely in the other direction.

Question 3. A figure is a ____________ if it is a simple closed figure made up entirely of line segments.
(a) line
(b) line segment
(c) polygon
(d) ray
Answer: (c) polygon
A polygon is defined as a simple closed curve made up entirely of line segments.
In simple words: Closed shapes made of straight line segments are called polygons.

Exam Tip: Definition of a polygon.

Question 4. An ________ is made up of two rays starting from a common end point.
(a) line
(b) line segment
(c) ray
(d) angle
Answer: (d) angle
An angle is formed when two rays originate from a common endpoint (vertex).
In simple words: Two rays meeting at a common point form an angle.

Exam Tip: Definition of an angle.

Question 5. Point S is _____________.
(a) on the angle
(b) in the interior of the angle
(c) in the exterior of the angle
(d) none of these
Answer: (b) in the interior of the angle
Based on standard diagram referencing, point S lies inside the arms of the angle.
In simple words: Point S is inside the angle region.

Exam Tip: Interior means inside the region enclosed by the rays of the angle.

Question 6. The intersection of three walls of a room is an example of
(a) intersecting lines
(b) parallel lines
(c) concurrent lines
(d) none of these
Answer: (c) concurrent lines
Three or more lines passing through a single common point are called concurrent lines.
In simple words: When three wall planes meet at a single corner point, it represents concurrent lines/planes.

Exam Tip: Concurrent lines meet at one common point.

Question 7. In the environment, a nail fixed in the wall is an example of
(a) a point
(b) point of concurrence
(c) point of intersection
(d) none of these
Answer: (a) a point
A nail fixed in the wall represents a dot or a point.
In simple words: A dot or a nail represents a single point in space.

Exam Tip: Points represent exact locations.

Question 8. In the environment a railway track is an example of
(a) intersecting lines
(b) parallel lines
(c) concurrent lines
(d) none of these
Answer: (b) parallel lines
Railway tracks run alongside each other and never meet, representing parallel lines.
In simple words: Train tracks that never cross each other are parallel.

Exam Tip: Parallel lines never intersect.

Fill in the blanks:

Question 9. The meeting point of a pair of sides is called a ________.
Answer: vertex
In simple words: The corner where two sides meet is a vertex.

Question 10. The distance around the circle is the _______________.
Answer: circumference
In simple words: The perimeter of a circle is called its circumference.

Question 11. A ________ contains a countless number of points.
Answer: line (or line segment / ray)
In simple words: A line is made up of an infinite number of points.

Question 12. The line segments forming a polygon are called its __________.
Answer: sides
In simple words: The straight edges of a polygon are its sides.

Answer the following:

Question 13. What are vertices and sides of an angle?
Answer:
- Vertex: The common endpoint where two rays meet to form an angle.
- Sides (or arms): The two rays that form the angle.
In simple words: The corner point is the vertex, and the two lines forming the angle are the sides.

Exam Tip: Memorize standard geometrical definitions.

Question 14. Use the figure to name five line segments.
Answer:
Based on standard geometric diagrams in this chapter (containing points O, B, C, D, E):
Five line segments are OD, OE, OB, OC, and BE.
In simple words: Name any five straight line sections between two points in the figure.

Question 15. What is a triangle and a quadrilateral?
Answer:
- Triangle: A polygon with 3 sides and 3 angles.
- Quadrilateral: A polygon with 4 sides and 4 angles.
In simple words: A 3-sided shape is a triangle, and a 4-sided shape is a quadrilateral.

Question 16. How many lines can pass through
a. one given point
b. two given points
Answer:
a. Infinitely many lines can pass through a single point.
b. Exactly one line can pass through two given points.
In simple words: You can draw endless lines through one dot, but only one straight line can connect two dots.

Exam Tip: Two points uniquely determine a straight line.

CLASS - VI Mathematics (Basic Geometrical Ideas)

Worksheet-10

Choose correct option in questions 1 to 10.

Question 1. How many end points a line segment have?
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (a) 2
A line segment has two fixed endpoints.
In simple words: A line segment starts and ends at 2 points.

Question 2. How many end points a line have?
(a) 1
(b) 0
(c) 2
(d) 3
Answer: (b) 0
A line extends infinitely in both directions and has no endpoints.
In simple words: A line goes on forever, so it has zero endpoints.

Question 3. How many end points a ray have?
(a) 0
(b) 2
(c) 1
(d) 3
Answer: (c) 1
A ray has one starting endpoint and extends infinitely in the other direction.
In simple words: A ray has 1 starting point.

Question 4. The end points of the same side of a polygon are called the ____________ vertices.
(a) same
(b) parallel
(c) different
(d) adjacent
Answer: (d) adjacent
Vertices that form the endpoints of the same side of a polygon are adjacent vertices.
In simple words: Connected corners on a polygon are adjacent vertices.

Question 5. _________ are made when corners are formed.
(a) angles
(b) line
(c) ray
(d) line segment
Answer: (a) angles
Corners in geometric shapes form angles.
In simple words: Corners create angles.

Question 6. How many points are enough to fix a line?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2
Two distinct points uniquely determine a line.
In simple words: You need 2 points to draw a straight line.

Question 7. How many vertices are there in a triangle?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3
A triangle has 3 vertices.
In simple words: A triangle has 3 corners.

Question 8. How many angles are there in a quadrilateral?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4
A quadrilateral has 4 angles.
In simple words: A 4-sided shape has 4 interior angles.

Question 9. How many pairs of adjacent sides are there in a quadrilateral?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4
A quadrilateral has 4 pairs of adjacent sides.
In simple words: Every side shares a corner with two adjacent sides, making 4 pairs.

Question 10. A line has
(a) two ends
(b) one end
(c) no end
(d) none of these
Answer: (c) no end
A line extends indefinitely in both directions, having no endpoints.
In simple words: Lines go on forever without ending.

Fill in the blanks:

Question 11. A point determines a _________.
Answer: position (or location)
In simple words: A point shows a precise location.

Question 12. A __________ extends indefinitely in both directions.
Answer: line
In simple words: A line goes on forever in both directions.

Question 13. A _________ is a portion of a line.
Answer: line segment
In simple words: A line segment is a part of a line with two endpoints.

Question 14. If a curve does not cross itself, then it is called a ___________ curve.
Answer: simple
In simple words: A curve that doesn't intersect itself is a simple curve.

Answer the following:

Question 15. What are vertices and sides of a polygon?
Answer:
- Vertices: The corner points where the line segments of a polygon meet.
- Sides: The line segments that form the boundary of the polygon.
In simple words: Corners are vertices and straight edges are sides.

Question 16. What is a semi-circle?
Answer:
A semi-circle is half of a circle, formed by cutting a circle along its diameter.
In simple words: Exactly half of a circle is a semi-circle.

Question 17. What is a triangle and a quadrilateral?
Answer:
- Triangle: A polygon bounded by three line segments.
- Quadrilateral: A polygon bounded by four line segments.
In simple words: Triangles have 3 sides, and quadrilaterals have 4 sides.

Question 18. Draw a rough sketch of a quadrilateral ABCD. How many diagonals it has? Are they necessarily equal?
Answer:
A quadrilateral has 2 diagonals (AC and BD). They are not necessarily equal (e.g., in a rectangle they are equal, but in a general parallelogram or trapezium they are not).
In simple words: A 4-sided shape has 2 diagonals, but their lengths depend on the type of quadrilateral.

Exam Tip: Diagonals connect opposite vertices of a polygon.

Question 19. Take three non collinear points P, Q and R. Join PQ, QR and PR. What figure did you get? Also name the side opposite to ∠Q.
Answer:
- Figure obtained: A triangle (Triangle PQR)
- Side opposite to angle Q: Side PR
In simple words: Joining three non-collinear points forms a triangle, and the side across from angle Q is PR.

Exam Tip: The side opposite an angle in a triangle does not touch that vertex.

Question 20. Say true or false:
a. The centre of a circle is always in its interior.
b. Two diameters of a circle will necessarily intersect.
c. Every chord of a circle is also a diameter of it.
Answer:
a. True
b. True (All diameters pass through the centre, so they always intersect at the centre)
c. False (Only chords passing through the centre are diameters; other chords are shorter)
In simple words: (a) Centre is inside, (b) Diameters always cross at the centre, (c) Not all chords pass through the centre.

Exam Tip: The diameter is the longest chord of a circle and always passes through the centre.