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Study the pattern :
1 \times 8 + 1 = 9                       1234 \times 8 + 4 = 9876
12 \times 8 + 2 = 98                   12345 \times 8 + 5 = 98765
123 \times 8 + 3 = 987               
Write the next two steps. Can you say how the pattern works?
(Hint: 12345 = 11111 + 1111 + 111 + 11 + 1).

123456 8 + 6 = 987648 + 6 = 987654 1234567 8 + 7 = 9876536 + 7 = 9876543 Yes, the pattern works. As 123456 = 111111 + 11111 + 1111 + 111 + 11 + 1, 123456  8 = (111111 + 11111 + 1111 + 111 + 11 + 1) 8                    = 111111 8 + 11111 8 + 1111  8 + 111 8 + 11 8 + 1  8                    = 888888 + 88888 + 8888 + 888 + 88 + 8                    = 987648 And, 123456 8 + 6 = 987648 + 6 = 987648

Find using distributive property :
(a) 728 \times 101       (b) 5437 \times 1001      (c) 824 \times 25     (d) 4275 \times 125     (e) 504 \times 35

(a) 728 101= 728 (100 + 1)                      = 728 100 + 728 1                      = 72800 + 728                      = 73528 (b) 5437 1001 = 5437 (1000 + 1)                           = 5437  1000 + 5437  1                           = 5437000 + 5437                           = 5442437 (c) 824 25 (800 + 24) 25 = (800 + 25 - 1) 25                                              =800 ...

3. If the product of two whole numbers is 1, can we say that one or both of them will be 1? Justify through examples.

If the product of 2 numbers is 1, then both the numbers have to equal to 1. For example, 1 x 1 = 1 However, 1 x 6 = 6 Clearly, the product of two whole numbers will be 1 in the situation when both numbers to be multiplied are 1.

2. If the product of two whole numbers is zero, can we say that one or both of them will be zero? Justify through examples.

If the product of 2 whole numbers is zero, then one of them is definitely zero. For example, 0 x 2 = 0 and 17 x 0 = 0 If the product of 2 whole numbers is zero, then both of them may be zero. 0 x 0 = 0 However, 2 x 3 = 6 (Since numbers to be multiplied are not equal to zero, the result of the product will also be non-zero.)

1.  Which of the following will not represent zero:
(a) 1 + 0       (b) 0 × 0     (c) 0/ 2     (d) (10-10)/2

(a) 1 + 0 It does not represent zero.   (b) 0 × 0 It represents zero.   (c) It represents zero.   (d) It represents zero.

5. Some numbers can be shown by two rectangles, for example
, 
Give at least five other such examples.

We can represent a number by two rectangles. for example 12 = 3 x 4 or 2 x 6 five other such examples are : 24 = 12 x 2  or 24 = 6 x 4  18 = 9 x 2 or 18 = 3 x 6 15 = 15 x 1 or 15 = 3 x 5 30 = 10 x 3 or 30 = 5 x 6  40 = 10 x 4 or 40 = 5 x 8.  

4. Write down the first seven numbers that can be arranged as triangles, e.g. 3, 6, ...

3, 6, 10, 15, 21, 28, 36.

3. Which can be shown as rectangles?

 can be shown as rectangles.  (Note: We are not counting squares as recatangles here)  

2. Which can be shown as squares?

and  can be shown as squares. 4: 2 rows and 2 columns. 9: 3 rows and 3 columns  

1. Which numbers can be shown only as a line?

 can be shown only as a line. They cannot be shown as rectangle or square or triangle.

8. Which of the following statements are true (T) and which are false (F) ?
(a) Zero is the smallest natural number.
(b) 400 is the predecessor of 399.
(c) Zero is the smallest whole number.
(d) 600 is the successor of 599.
(e) All natural numbers are whole numbers.
(f) All whole numbers are natural numbers.
(g) The predecessor of a two digit number is never a single digit number.
(h) 1 is the smallest whole number.
(i) The natural number 1 has no predecessor.
(j) The whole number 1 has no predecessor.
(k) The whole number 13 lies between 11 and 12.
(l) The whole number 0 has no predecessor.
(m) The successor of a two digit number is always a two digit number

(a) Zero is the smallest natural number. - False. 0 is not a natural number. (b) 400 is the predecessor of 399. - False. 400 is the successor of 399. (c) Zero is the smallest whole number. - True. (d) 600 is the successor of 599. - True (e) All natural numbers are the whole numbers.- True. (f) All whole numbers are natural numbers.-False. 0 is a whole number but not a natural number. (g)...

7. Match the following:
(i) 425 × 136 = 425 × (6 + 30 +100)          (a) Commutativity under multiplication.
(ii) 2 × 49 × 50 = 2 × 50 × 49                     (b) Commutativity under addition.
(iii) 80 + 2005 + 20 = 80 + 20 + 2005        (c) Distributivity of multiplication over addition.

(i) (c) Distributivity of multiplication over addition. (ii) (a) Commutativity under multiplication. (iii)   (b) Commutativity under addition.  

6. A vendor supplies 32 litres of milk to a hotel in the morning and 68 litres of milk in the evening. If the milk costs rupees 45 per litre, how much money is due to the vendor per day?

Amount of milk supplied in the morning =  Amount of milk supplied in the evening =  Total amount of petrol =  Cost of 1 litre of milk =   Cost of  of milk = 

5. A taxidriver 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 rupees 44 per litre, how much did he spend in all on petrol?

Amount of petrol filled on Monday =  Amount of petrol filled on Tuesday =  Total amount of petrol =  Cost of 1 litre of petrol =   Cost of  of petrol = 

4. Find the product using suitable properties.
(a) 738 × 103                (b) 854 × 102
(c) 258 × 1008              (d) 1005 × 168

The product of the folllowing using suitable properties are: (a) Using distributive law.                  (b) Using distributive law. (c) Using Distributive law. (d) Using Distributive law.  

3. Find the value of the following:
(a) 297 × 17 + 297 × 3                    (b) 54279 × 92 + 8 × 54279
(c) 81265 × 169 – 81265 × 69        (d) 3845 × 5 × 782 + 769 × 25 × 218

(a) Using Distributive law.                       (b) Using Commutative under multiplication Using Distributive law.   (c)  Using Distributive law.       (d) Using distributive law.  

2. Find the product by suitable rearrangement:
(a) 2 × 1768 × 50               (b) 4 × 166 × 25
(c) 8 × 291 × 125               (d) 625 × 279 × 16
(e) 285 × 5 × 60                 (f) 125 × 40 × 8 × 25

The product of the following by suitable rearrangement are: (a)   (b) (c)                (d)   (e)                     (f)

1. Find the sum by suitable rearrangement:
(a) 837 + 208 + 363                        (b) 1962 + 453 + 1538 + 647

Sum by suitable rearrangement: (a) 837 + 208 + 363   (b) 1962 + 453 + 1538 + 647

7. In each of the following pairs of numbers, state which whole number is on the left of the other number on the number line. Also write them with the appropriate sign (>, <) between them.
(a) 530, 503        (b) 370, 307         (c) 98765, 56789       (d) 9830415, 10023001

The number on the left on the number line is smaller than the number that is on the right on the number line. (a) 530, 503         is on the left.   (b) 370, 307       is on the left.   (c) 98765, 56789         is on the left.   (d) 9830415, 10023001  is on the left.

6.  Write the predecessor of :
(a) 94    (b) 10000     (c) 208090      (d) 7654321

The predecessor of the following numbers are: (a) 94   (b) 10000        (c) 208090        (d) 7654321
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