CBSE Class 6 Mathematics Whole Numbers Assignment Set 08

VI - Mathematics Assignment No - 03. Whole Numbers.

Question 1. If \(a = 15\), \(b = 16\), \(c = 17\) verify the following properties for addition and multiplication:
(i) closure Property
(ii) Commutative Property
(iii) Associative property
(iv) Distributive property
Answer: Given values: \(a = 15\), \(b = 16\), \(c = 17\)
(i) Closure Property:
For addition: \(a + b = 15 + 16 = 31\), which is a whole number.
For multiplication: \(a \times b = 15 \times 16 = 240\), which is a whole number.
(ii) Commutative Property:
For addition: \(a + b = b + a \implies 15 + 16 = 16 + 15 \implies 31 = 31\) (Verified).
For multiplication: \(a \times b = b \times a \implies 15 \times 16 = 16 \times 15 \implies 240 = 240\) (Verified).
(iii) Associative Property:
For addition: \((a + b) + c = a + (b + c) \implies (15 + 16) + 17 = 15 + (16 + 17) \implies 31 + 17 = 15 + 33 \implies 48 = 48\) (Verified).
For multiplication: \((a \times b) \times c = a \times (b \times c) \implies (15 \times 16) \times 17 = 15 \times (16 \times 17) \implies 240 \times 17 = 15 \times 272 \implies 4080 = 4080\) (Verified).
(iv) Distributive Property (Multiplication over Addition):
\(a \times (b + c) = (a \times b) + (a \times c) \implies 15 \times (16 + 17) = (15 \times 16) + (15 \times 17) \implies 15 \times 33 = 240 + 255 \implies 495 = 495\) (Verified).
In simple words: Whole numbers follow closure, commutative, associative, and distributive laws for addition and multiplication.

Exam Tip: Substitute the given numbers carefully into each algebraic identity and solve both sides independently to verify equality.

Question 2. Are the above properties true for Subtraction? If the answer is no then verify by taking any example of your choice.
Answer: No, the properties are not true for subtraction.
Verification with an example:
Let \(a = 5\) and \(b = 3\).
Closure property: \(5 - 3 = 2\) (Whole number), but if \(a = 3\) and \(b = 5\), then \(3 - 5 = -2\) (Not a whole number).
Commutative property: \(a - b \neq b - a \implies 5 - 3 \neq 3 - 5 \implies 2 \neq -2\) (Not true).
Associative property: \((5 - 3) - 1 = 2 - 1 = 1\), and \(5 - (3 - 1) = 5 - 2 = 3\). Since \(1 \neq 3\), it does not hold.
In simple words: Subtraction does not follow closure, commutative, or associative properties in whole numbers.

Exam Tip: Remember that subtraction is neither commutative nor associative for whole numbers.

Question 3. Verify and state the property:
(i) \(2 - 3 \neq 3 - 2\)
(ii) \(2 - (3 - 4) \neq (2 - 3) - 4\)
(iii) \((+2) \times (-3 + 4) = (+2) \times (-3) + (+2) \times 4\)
(iv) \(102 \times 70 = 70 \times 102\)
Answer:
(i) Commutative property of subtraction does not hold.
(ii) Associative property of subtraction does not hold.
(iii) Distributive property of multiplication over addition holds.
(iv) Commutative property of multiplication holds.
In simple words: Identify whether the equation shows equality or inequality and match it with standard arithmetic properties.

Exam Tip: Look at the operation symbols and grouping symbols to quickly identify the property name.

Question 4. Fill the gap to make the following statements true.
(i) \(395 + 602 = 602 + \dots\dots\)
(ii) \(475 + 1325 = 1325 + \dots\dots\)
(iii) \(47 + (52 + 93) = (47 + \dots) + 93\)
(iv) \(37 \times (45 + 27) = (37 \times \dots) + (37 \times \dots)\)
(v) \(12345 + 0 = \dots\dots\)
(vi) \(5672 \times 1 = \dots\dots\)
Answer:
(i) 395 (Commutative property of addition)
(ii) 475 (Commutative property of addition)
(iii) 52 (Associative property of addition)
(iv) 45, 27 (Distributive property of multiplication over addition)
(v) 12345 (Additive identity property)
(vi) 5672 (Multiplicative identity property)
In simple words: Fill in the missing numbers by applying basic properties of whole numbers like commutativity and identity elements.

Exam Tip: Check both sides of the equals sign to ensure the exact same numbers and operations balance out.

Question 5. Replace * by the correct digit in each of the following:

(i)

\[ \begin{array}{r} 917\\ -5**\\ \hline 5*8 \end{array} \]

(ii)

\[ \begin{array}{r} 6173\\ +*2*4\\ \hline 7*0* \end{array} \]

Answer:

(i)

\[ \begin{array}{r} 917\\ -359\\ \hline 558 \end{array} \]

Missing digits: 3, 5, 9, 5, 5, 8 (as per the completed subtraction).

(ii)

\[ \begin{array}{r} 6173\\ +1234\\ \hline 7407 \end{array} \]

In simple words: Use reverse operations (addition and subtraction) to find the missing digits in vertical arithmetic columns.

Exam Tip: Always verify borrowing and carrying carefully when working with missing digit puzzles.

Question 6. Add 5, 8, 10 by grouping them in as many ways as you can. Give the property you use at each step.
Answer:
Way 1: Grouping first two numbers:
\((5 + 8) + 10 = 13 + 10 = 23\)
Property used: Associative Property of Addition.
Way 2: Grouping last two numbers:
\(5 + (8 + 10) = 5 + 18 = 23\)
Property used: Associative Property of Addition.
Way 3: Grouping first and last numbers:
\((5 + 10) + 8 = 15 + 8 = 23\)
Property used: Commutative and Associative Properties of Addition.
In simple words: Change the order and grouping of numbers while keeping the sum same.

Question 7. Add the following numbers and check by reversing the order of addends.
(i) \(2345 + 1245\)
(ii) \(3757 + 1204\)
Answer:
(i) \(2345 + 1245 = 3590\)
Reversing order: \(1245 + 2345 = 3590\)
Both sums are equal, verifying the commutative property.
(ii) \(3757 + 1204 = 4961\)
Reversing order: \(1204 + 3757 = 4961\)
Both sums are equal, verifying the commutative property.
In simple words: Addition gives the same result even if the order of numbers is reversed.

Exam Tip: Use the reverse addition method as a quick self-check during exams.

Question 8. Perform the following subtractions. Check your result by corresponding additions.
(i) \(5294 - 1024\)
(ii) \(7809 - 1567\)
Answer:
(i) \(5294 - 1024 = 4270\)
Check by addition: \(4270 + 1024 = 5294\) (Correct).
(ii) \(7809 - 1567 = 6242\)
Check by addition: \(6242 + 1567 = 7809\) (Correct).
In simple words: Add the difference to the subtrahend to get back the original number.

Exam Tip: Always check subtraction results by adding the result back to the second number.

Question 9. Arun opened an account in a bank by depositing Rs 25000. Next day he deposited Rs 1274 (cheque) and the next day he withdraws Rs 5000. Find his balance in the bank.
Answer: Initial deposit = Rs 25000
Amount deposited next day = Rs 1274
Total amount in account = \(25000 + 1274 =\) Rs 26274
Amount withdrawn = Rs 5000
Final balance = \(26274 - 5000 =\) Rs 21274
In simple words: Add all deposits to the initial amount and subtract withdrawals to find the final bank balance.

Exam Tip: Read word problems carefully to distinguish between additions (deposits) and subtractions (withdrawals).

Question 10. Find the whole number represented by \(x\) in the following statements:
(i) \(x + 4 = 9\)
(ii) \(x - 2 = 11\)
(iii) \(x + 2 = 20\)
(iv) \(2x - 5 = 9\)
Answer:
(i) \(x + 4 = 9 \implies x = 9 - 4 = 5\)
(ii) \(x - 2 = 11 \implies x = 11 + 2 = 13\)
(iii) \(x + 2 = 20 \implies x = 20 - 2 = 18\)
(iv) \(2x - 5 = 9 \implies 2x = 9 + 5 = 14 \implies x = \frac{14}{2} = 7\)
In simple words: Transpose numbers to the other side of the equals sign with opposite operations to isolate \(x\).

Exam Tip: When moving a term across the equals sign, change plus to minus and minus to plus.

Question 11. Find the product in a easier way:
(i) \(4 \times 225 \times 1992\)
(ii) \(2757 \times 50 \times 9 \times 2\)
Answer:
(i) Rearrange using commutative and associative properties:
\((4 \times 225) \times 1992 = 900 \times 1992 = 1792800\)
(ii) Rearrange to form easy multiples of 100:
\((50 \times 2) \times 2757 \times 9 = 100 \times 2757 \times 9 = 275700 \times 9 = 2481300\)
In simple words: Group numbers that multiply to form powers of 10 (like 100 or 1000) to simplify calculations.

Exam Tip: Look for pairs like \(2 \times 50\) or \(4 \times 25\) to simplify multiplication instantly.

Question 12. Multiply by using Distributive property:
(i) \(968 \times 73 + 968 \times 27\)
(ii) \(454 \times 25 + 454 \times 75\)
Answer:
(i) \(968 \times 73 + 968 \times 27 = 968 \times (73 + 27) = 968 \times 100 = 96800\)
(ii) \(454 \times 25 + 454 \times 75 = 454 \times (25 + 75) = 454 \times 100 = 45400\)
In simple words: Take out the common multiplier and add the remaining numbers using the distributive law in reverse.

Exam Tip: The distributive property \(a \times b + a \times c = a \times (b + c)\) is extremely useful for mental math when a common factor repeats.