Exercise 22C
Question 1
1. Replace each letter with the correct numeral :
(a) $\begin{aligned}4&A&7&2\\B&8&5&C\\ 9&2&D&6\\ \hline +E&7&7&7&5 \end{aligned}$
(b) $\begin{aligned} A&2&4&6 \\ -5&2&B&7 \\ \hline 1&C&D&E \end{aligned}$
(c) $\begin{aligned}1&2&A \\ \times B&1&5 \\ \hline C&2&5 \\ D&E&F&0 \\ 2&G&0&0&0 \\ \hline H&6&8&k&J \end{aligned}$
(d) <to be added>
2. What should come in place of the question mark (?) in the following number series ?
(a) $563 \quad 582 \quad 620 \quad 677 \quad 753 \quad 848$
(b) $17 \quad 17 \quad 51 \quad 255 \quad 1785 \quad 16065$ ?
3. What is the value of x in figure III.
(a) 4
(b) 16
(c) 25
(d) 36
<fig>
4. <fig> the values of $a$ and $b$ are : (A) 1,2
(B) 2,3
(C) 1,3
(D) None of these
5. The missing number is :
<fig>
(A) 17
(B) 30
(C) 71
(D) 16
6. In a digital display which digit can display all 10 digits from $\square$ to 9 .
(A) nine
(B) six
(C) zero
(D) eight
7. Complete the magic square so that the sum of the numbers in each row, each column and along the diagonals may be 15 .
<fig>
8. Write the nine, non-repetitive digits in the circles of this triangle in such a way as to have a total of 20 on each side.
9. The magic hexagon has a total of 19 hexagons within it. All sets of 3,4 and 5 hexagons in a row add to 38. Fill in the missing figures. No number should be repeated.
<fig>
10. Rumours : Rumours spread very fast. Sometimes an incident witnessed by just a few persons becomes the talk of the town within less than two hours. Suppose a man comes at about 8 a.m. to a town with a population of about 50,000 and conveys an interesting news to just three persons. This takes up, say, 15 minutes. Within 2 hours it becomes a talk of the town. Ṣhow, how it is possible.
11. Number Patterns
$\begin{aligned} 0 \times 9+1 &=1 \\ 1 \times 9+2 &=11 \\ 12 \times 9+3 &=111 \\ 123 \times 9+4 &=1111 \\ 1234 \times 9+5 &=11111 \\ 12345 \times 9+6 &=111111 \\ 123456 \times 9+7 &=1111111 \\ 1234567 \times 9+8 &=11111111 \\ 12345678 \times 9+9 &=111111111 \\ 123456789 \times 9+10 &=1111111111 \end{aligned}$
Investigate a similar number pattern where the first two lines are :
$
\begin{gathered}
1 \times 8+1=9 \\
12 \times 8+2=98
\end{gathered}
$
12. Choose any four digits, for example, $6,8,9$, and 2 .
<fig>
- Repeat the first digit at the end.
- Find the difference between the pairs of digits next to each other. Always repeat the first digit at the end.
- Continue the pattern until all the numbers are the same. Try the above Pattern with these numbers.
1. $4,5,5,2$
2. $9,4,3,8$
3. $4,8,7,6$
4. $8,6,3,2$
5. Try the same pattern using any four digits you wish.
13. You may want to try this with your calculator.
- Enter any three digits in your calculator.
- Repeat the three digits to get a six-digits number.
- Divide by $13 .$
Divide by $11 .$
- Divide by $7 .$
- You will find that your result is your original number.
$\begin{aligned}
&753 \\
&753753 \\
&57981 \\
&5271 \\
&753
\end{aligned}$
Try Pattern II with these numbers.
1. 347
2. 952
3. 885
4. 409
5. Try the same pattern using any three digits you wish.
14. Blaise Pascal(1623-1662) was a famous French mathematician and philosopher. One of his most important discoveries is known as Pascal's Triangle. See if you can discover the secret of this pattern of numbers.
Solve.
1. Write Row 8 of Pascal's Triangle.
2. Write Row 9 of Pascal's Triangle.
3. Write Row 10 of Pascal's Triangle.
4. What is the sum of the numbers in Row 1 of Pascal's Triangle?
5. What is the sum of the numbers in Row 3 of Pascal's Triangle?
6. Find the sum of the numbers in Row 8 of Pascal's Triangle without adding.
15. Sudoko: Fill in the grid so that every horizontal row, every vertical column and every $3 \times 3$ box contains the digits $1-9$, without repeating the numbers in the same row, column or box. You can't change the digits already given in the grid.
16. One way of expressing 10 as a combination of five 9 's is $9+\frac{99}{99}=10$. Find the further ways of doing so.
17. A clock takes three seconds to strike 3 O' clock. How long does it take to strike 6 O'clock?
18. What is the smallest integer that can be written with two digits?
19. Write 1 by using all the ten digits.
20. Write 10 with five 9 's. Do it in at least two ways?
21. Write 100 by using all the ten digits.
22. Show four different ways of writing 100 with five identical digits.
23. What is the biggest number that can be written with four 1's?
24. Calculate the length of the strip of all the millimetre squares in one square metre; if placed one alongside the other.
25. Calculate the height of a pole made up of all the millimetre cubes in one cubic metre, if placed one atop another.
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