Natural Numbers & Whole Numbers — Exercise 3(E)

For each pattern, given below, write the next three steps:

(i) 1 × 9 + 1 = 10

12 × 9 + 2 = 110

123 × 9 + 3 = 1110

(ii) 9 × 9 + 7 = 88

98 × 9 + 6 = 888

987 × 9 + 5 = 8888

(iii) 1 × 8 + 1 = 9

12 × 8 + 2 = 98

123 × 8 + 3 = 987

(iv) 111 ÷ 3 = 37

222 ÷ 6 = 37

333 ÷ 9 = 37

Answer

(i) The next three steps are:

1234 × 9 + 4 = 11110

12345 × 9 + 5 = 111110

123456 × 9 + 6 = 1111110

(ii) The next three steps are:

9876 × 9 + 4 = 88888

98765 × 9 + 3 = 888888

987654 × 9 + 2 = 8888888

(iii) The next three steps are:

1234 × 8 + 4 = 9876

12345 × 8 + 5 = 98765

123456 × 8 + 6 = 987654

(iv) The next three steps are:

444 ÷ 12 = 37

555 ÷ 15 = 37

666 ÷ 18 = 37

Complete each of the following magic squares:

(i)

(ii)

(iii)

Answer

In a magic square, the sum of the numbers in each row, each column and each diagonal is the same. This common sum is called the magic sum, and for a 3 × 3 magic square it equals 3 times the centre number.

(i) The centre number is 5, so the magic sum = 3 × 5 = 15.

Row 1: 6 + 7 + ............... = 15 ⇒ missing number = 2.

Row 2: ............... + 5 + 9 = 15 ⇒ missing number = 1.

Row 3: 8 + ............... + 4 = 15 ⇒ missing number = 3.

The missing numbers (row-wise) are 2, 1 and 3. The completed magic square is:

(ii) The centre number is 7, so the magic sum = 3 × 7 = 21.

Diagonal: 8 + 7 + ............... = 21

⇒ Bottom-left number = 6.

Row 1: 4 + ............... + 8 = 21

⇒ Missing number = 9.

Diagonal 2: 8 + 7 + ............... = 21

⇒ Missing number = 6.

Column 1: 4 + ............... + 6 = 21

⇒ Middle-left number = 11.

Row 2: 11 + 7 + ............... = 21

⇒ Middle-right number = 3.

Row 3: 6 + ............... + 10 = 21

⇒ Missing number = 5.

The missing numbers (row-wise) are 9, 11, 3, 6 and 5. The completed magic square is:

(iii) The centre number is 10, so the magic sum = 3 × 10 = 30.

Row 1: 16 + 2 + ............... = 30

⇒ Missing number = 12.

Diagonal: 12 + 10 + ............... = 30

⇒ Bottom-left number = 8.

Column 1: 16 + ............... + 8 = 30

⇒ Middle-left number = 6.

Row 2: 6 + 10 + ............... = 30

⇒ Middle-right number = 14.

Row 3: 8 + ............... + 4 = 30

⇒ Missing number = 18.

The missing numbers (row-wise) are 12, 6, 14, 8 and 18. The completed magic square is:

See the following pattern carefully:

(i) If n denotes the figure number and S denotes the number of matchsticks, find S in terms of n.

(ii) Find how many matchsticks are required to make the:

(1) 15th figure

(2) 40th figure

(iii) Write a description of the pattern in words.

Answer

From the pattern, the number of matchsticks for the first few figures is:

(i) Each new figure is formed by adding one square, and each added square shares one edge with the previous square.

So, each new figure requires 3 more matchsticks.

Therefore, for the nth figure,

S = 7 + 3(n - 1)

S = 7 + 3n - 3

S = 3n + 4

Hence, S = 3n + 4.

(ii) Calculating,

(1) For the 15th figure, n = 15:

S = 3 × 15 + 4 = 45 + 4 = 49 matchsticks.

(2) For the 40th figure, n = 40:

S = 3 × 40 + 4 = 120 + 4 = 124 matchsticks.

(iii) The number of matchsticks (S) is equal to 4 more than three times the figure number (n).

(i) In the following pattern, draw the next two figures.

(ii) Construct a table to describe the figures in the above pattern.

(iii) If n denotes the number of figures and L denotes the number of matchsticks, find L in terms of n.

(iv) Find how many matchsticks are required to make the:

(1) 12th figure (2) 20th figure

Answer

(i) The next two figures in the pattern are:

(ii) The table describing the figures is:

(iii) Each time the number of figures (n) increases by 1, the number of matchsticks (L) increases by 2.

For n = 1, 2n = 2 and L = 2.

Hence, L = 2n.

(iv) (1) For the 12th figure, n = 12:

L = 2 × 12 = 24 matchsticks.

(2) For the 20th figure, n = 20:

L = 2 × 20 = 40 matchsticks.