1. A number $x$ is multiplied by 4 and then 7 is added. Write the resulting expression.
Answer:
\(4x+7\)
2. A number $y$ is doubled and then 9 is subtracted. What is the resulting expression?
Answer:
\(2y-9\)
3. A number $p$ is multiplied by 5 and then increased by 3. Form the algebraic expression.
Answer:
\(5p+3\)
4. A number $n$ is multiplied by 3 and then 8 is subtracted. What is the final expression?
Answer:
\(3n-8\)
5. If all the variable terms in a number trick cancel each other, what type of value is left?
Answer:
A constant value is left.
6. Why may a variable disappear from the final result of a number trick?
Answer:
The variable may disappear because the algebraic operations cause its terms to cancel each other.
7. A number $x$ is first increased by 6 and then the entire result is multiplied by 2. Write the expression.
Answer:
\(2(x+6)\)
8. What do we call a quantity that does not change its value?
Answer:
A constant.
9. In a number pyramid, the upper block is obtained by adding the two blocks directly below it. If the lower blocks are 7 and 11, find the upper block.
Answer:
\(7+11=18\)
So, the upper block is 18.
10. The bottom row of a number pyramid is $4,9,12$. Find the row immediately above it.
Answer:
\(4+9=13\)
\(9+12=21\)
Therefore, the next row is 13, 21.
11. If two adjacent blocks contain $m$ and $n$, what will be the block above them?
Answer:
\(m+n\)
12. The bottom row of a pyramid is $3,5,8$. Find the middle row and the top number.
Answer:
Middle row:
\(3+5=8\)
\(5+8=13\)
Top:
\(8+13=21\)
Therefore, the pyramid gives 8, 13 in the middle row and 21 at the top.
13. Which operation is used to obtain an upper block from two adjacent lower blocks in the number pyramid?
Answer:
Addition.
14. Construct the next row of a number pyramid whose bottom row is $6,10,15$.
Answer:
\(6+10=16\)
\(10+15=25\)
The next row is 16, 25.
15. In a $2\times2$ calendar block, the upper-left number is $k$. Write the other three numbers in terms of $k$.
Answer:
The four numbers are:
\(k,\quad k+1,\quad k+7,\quad k+8\)
16. Why do numbers in a $2\times2$ calendar block have fixed differences?
Answer:
Because consecutive dates differ by 1, while dates in the same column of consecutive weeks differ by 7.
17. If the upper-left number of a $2\times2$ calendar block is 8, write all four numbers.
Answer:
\(8,\quad9,\quad15,\quad16\)
18. The four numbers in a $2\times2$ calendar block are $a$, $a+1$, $a+7$ and $a+8$. Find their sum.
Answer:
\(a+(a+1)+(a+7)+(a+8)\)
\(=4a+16\)
Therefore, the sum is:
\(\boxed{4a+16}\)
19. The sum of the four numbers in a $2\times2$ calendar block is 52. Find the smallest number.
Answer:
\(4a+16=52\)
\(4a=36\)
\(a=9\)
Therefore, the smallest number is 9.
20. If the upper-left number of a $2\times2$ calendar block is 12, find the sum of all four numbers.
Answer:
\(4a+16=4(12)+16\)
\(=48+16=64\)
Therefore, the sum is 64.
21. Why can four dates in a $2\times2$ calendar block be represented using only one variable?
Answer:
Once the first date is known, the other three dates are fixed by adding 1, 7 and 8. Therefore, one variable is sufficient to represent all four numbers.
22. What mathematical property makes calendar patterns suitable for algebraic representation?
Answer:
The fixed differences between dates allow us to represent the dates using a variable and constants.
23. Using 1, 4 and 6 exactly once, form the greatest possible three-digit number.
Answer:
The greatest number is:
\(641\)
24. Using 2, 5 and 7 exactly once, what is the greatest three-digit number that can be formed?
Answer:
\(752\)
25. Why is the position of a digit important in a three-digit number?
Answer:
Because the value of a digit depends on its place value. For example, 5 in the hundreds place represents 500, whereas 5 in the units place represents 5.
26. If $a$, $b$ and $c$ are the hundreds, tens and units digits respectively, write the corresponding three-digit number.
Answer:
\(100a+10b+c\)
27. Which place contributes the greatest place value in a three-digit number?
Answer:
The hundreds place.
28. A two-digit number is represented by $10a+b$. Write its reversed form.
Answer:
\(10b+a\)
29. Find the difference between 82 and its reverse.
Answer:
Reverse of 82 = 28
\(82-28=54\)
Therefore, the difference is 54.
30. What happens when the tens and units digits of a two-digit number are interchanged?
Answer:
The number is reversed. For example:
\(46\rightarrow64\)
31. Express 64 algebraically using its digits.
Answer:
\(64=10(6)+4\)
32. If a two-digit number is $10a+b$, derive the expression for its difference from its reverse.
Answer:
Original number:
\(10a+b\)
Reversed number:
\(10b+a\)
Difference:
\(10a+b-(10b+a)\)
\(=9a-9b\)
\(=9(a-b)\)
Thus, the difference is:
\(\boxed{9(a-b)}\)
33. Find the reverse of 53 and calculate the difference between the two numbers.
Answer:
Reverse of 53 = 35
\(53-35=18\)
Therefore, the difference is 18.
34. Why is the difference between a two-digit number and its reverse always divisible by 9?
Answer:
If the number is $10a+b$, its reverse is $10b+a$.
Their difference is:
\(9(a-b)\)
Since this expression contains 9 as a factor, the difference is always divisible by 9.
35. If the digits of a two-digit number are $p$ and $q$, what is the difference between the number and its reverse?
Answer:
Original number:
\(10p+q\)
Reverse:
\(10q+p\)
Difference:
\(9(p-q)\)
Hence, the difference is always a multiple of 9.
36. Consider the number 74. Explain algebraically why its difference from its reverse is divisible by 9.
Answer:
For 74:
\(a=7,\quad b=4\)
Difference:
\(9(a-b)=9(7-4)=27\)
Since 27 is divisible by 9, the result confirms the divisibility rule.
37. Prove algebraically that the difference between a two-digit number and its reverse is a multiple of 9.
Answer:
Let the number be:
\(10a+b\)
Its reverse is:
\(10b+a\)
Difference:
\(10a+b-(10b+a)\)
\(=9a-9b\)
\(=9(a-b)\)
Therefore, the difference is always a multiple of 9.
38. The difference between a two-digit number and its reverse is 36. What is the difference between its digits?
Answer:
\(9(a-b)=36\)
Therefore,
\(a-b=4\)
So, the difference between the digits is 4.
39. Can the difference between a two-digit number and its reverse be 45? Explain.
Answer:
Yes.
Since:
\(45=9\times5\)
the difference between the digits would have to be 5. For example:
\(72-27=45\)
Therefore, it is possible.
40. How does algebra help us prove divisibility patterns?
Answer:
Algebra allows us to represent an entire class of numbers using variables. We can then simplify the expression and identify common factors, such as the factor 9 in the difference of a two-digit number and its reverse.
41. How can algebra explain a mathematical magic trick?
Answer:
We represent the unknown starting number by a variable and perform all the stated operations algebraically. If the variable terms cancel, the final expression becomes a constant, explaining why the trick gives the same answer.
42. Why do we use a variable such as $x$ when discussing an unknown number?
Answer:
A variable represents an unknown or changing number, allowing us to describe and analyse many possible values at once.
43. How can we determine whether a number trick always gives the same result?
Answer:
Represent the starting number by a variable and simplify the complete sequence of operations. If the final expression contains no variable, the result is constant.
44. How does algebra help us understand patterns in a calendar?
Answer:
We can represent one date by a variable. The other dates can then be expressed using the fixed differences of 1, 7 and 8.
45. What is the algebraic form of a two-digit number whose tens digit is $x$ and units digit is $y$?
Answer:
\(10x+y\)
46. What is the main advantage of proving a pattern algebraically instead of checking only numerical examples?
Answer:
An algebraic proof shows that the pattern is true for all permissible values, rather than just for a few examples.
47. A student says, "If a number trick works for three different numbers, it will work for every number." Is this reasoning sufficient?
Answer:
No. Testing examples does not prove that a result will always hold. We should use algebraic reasoning to establish the general result.
48. Give an example of a pattern from this topic that can be represented using an algebraic expression.
Answer:
The sum of a $2\times2$ calendar block can be represented as:
\(4a+16\)
where $a$ is the upper-left number.
49. Differentiate between a variable and a constant with an example.
Answer:
A variable can change its value, such as $x$. A constant has a fixed value, such as 7.
For example, in:
\(2x+7\)
$x$ is the variable and 7 is the constant.
50. What common mathematical idea connects number pyramids, calendar patterns and digit-reversal tricks?
Answer:
All three involve recognising patterns and representing them using algebraic expressions. Algebra helps us describe the pattern generally and prove that it works.