Algebra PlayClass 8 Mathematics NCERT Solutions

The original game's steps are:

The final answer is half the number added in step 3. In the original game, 4 was added, so the answer is $\frac{4}{2} = 2$.

To get a final answer of 3, we need $\frac{N}{2} = 3$, which means $N=6$. So, we should change step 3 to "Add six".

To get a final answer of 5, we need $\frac{N}{2} = 5$, which means $N=10$. So, we should change step 3 to "Add ten".

Here is an example trick where the answer is always 7.

Final Answer: The result will always be 7, no matter what number you start with.

Given: The final answer from the date trick is 1390. The formula derived in the text is: Final Answer $= 100M + 165 + D$, where $M$ is the month number and $D$ is the day.

To Find: The date (Month and Day) Mukta thought of.

Solution: We can find the value of $100M + D$ by subtracting 165 from the final answer. $100M + D = \text{Final Answer} - 165$ $100M + D = 1390 - 165$ $100M + D = 1225$

In the expression $100M + D$, the value of $D$ (the day) will be the last two digits, and the value of $M$ (the month) will be the preceding digit(s). From $1225$, we can see:

The 12th month is December.

Final Answer: The date Mukta started with was December 25th.

Yes, we can change the steps and still find the date. The key is to construct the steps so that the final algebraic expression is of the form $100M + D + K$, where $K$ is a constant.

Let's create a new trick.

Let's call the final answer $A$. So, $A = 100M + 70 + D$. To find the date, we would need to subtract 70 from the final answer. $100M + D = A - 70$

Example: Suppose the date is May 20th ($M=5, D=20$).

The final answer is 590.

To find the date from the answer 590:

Final Answer: Yes, the trick can be changed. For the example steps provided, you would subtract 70 from the final answer to find the date.

I predict your answer is the number you first thought of. Am I right?

Final Answer: The final result is $x$, which is the original number.

Given: A 3-row number pyramid with the bottom row [2, 5, 1]. The rule is that each number is the sum of the two numbers directly below it.

Solution: Bottom row: [2, 5, 1]

Final Pyramid: Top row: [13] Second row: [7, 6] Bottom row: [2, 5, 1]

Final Answer: The numbers to fill the pyramid, from bottom to top, are: second row is [7, 6] and the top number is 13.

Given: A 3-row number pyramid with the bottom row [3, 7, 4]. The rule is that each number is the sum of the two numbers directly below it.

Solution: Bottom row: [3, 7, 4]

Final Pyramid: Top row: [21] Second row: [10, 11] Bottom row: [3, 7, 4]

Final Answer: The numbers to fill the pyramid, from bottom to top, are: second row is [10, 11] and the top number is 21.

Given: A 4-row number pyramid. The top number is 100. The bottom row is [25, $x$, $y$, 15], where $x$ and $y$ are unknown.

Let: The bottom row be [25, $x$, $y$, 15].

Solution: Let's build the pyramid upwards algebraically.

We are given that the top number is 100. $40 + 3x + 3y = 100$ $3x + 3y = 100 - 40$ $3(x + y) = 60$ $x + y = 20$

This problem does not have a unique solution, as any two numbers $x$ and $y$ that sum to 20 will work. Let's choose a simple pair, for example, $x=10$ and $y=10$.

This is one possible solution.

Final Answer: The problem has multiple solutions. We need to find two numbers for the middle of the bottom row that sum to 20. One possible solution for the complete bottom row is [25, 10, 10, 15].

Given: A 3-row number pyramid. The top number is 60. The bottom row is [12, $c$, 8], where $c$ is unknown.

Let: The bottom row be [12, $c$, 8]. The second row be [$a$, $b$]. The top number is 60.

Solution: From the rules of the pyramid, we can form equations:

Now, substitute the expressions for $a$ and $b$ from equations (2) and (3) into equation (1): $(12 + c) + (c + 8) = 60$ $20 + 2c = 60$ $2c = 60 - 20$ $2c = 40$ $c = 20$

Now that we have $c$, we can find $a$ and $b$.

Let's check: $a + b = 32 + 28 = 60$. This matches the top number.

Final Pyramid: Top row: [60] Second row: [32, 28] Bottom row: [12, 20, 8]

Final Answer: We start by representing the unknown middle number in the bottom row with a variable, $c$. The value of $c$ is 20. The second row is [32, 28].

Given: A 4-row pyramid with some values filled. Let's denote the cells by R(row)C(column) from the top.

Let's use variables for the empty cells: Row 4 (bottom): [$a, 5, b, c$] Row 3: [$d, e, f$] Row 2: [$g, 22$] Row 1 (top): [48]

Forming equations from the pyramid rules: From Row 2 to Row 1:

From Row 3 to Row 2: 2. $d + e = g = 26$ 3. $e + f = 22$

From Row 4 to Row 3: 4. $a + 5 = d$ 5. $5 + b = e$ 6. $b + c = f$

We know $f=12$ is given (R3C3=12). Substitute $f=12$ into equation 3: $e + 12 = 22 \implies e = 10$

Now substitute $e=10$ into equation 2: $d + 10 = 26 \implies d = 16$

Now use the bottom row equations: From equation 5: $5 + b = e = 10 \implies b = 5$ From equation 4: $a + 5 = d = 16 \implies a = 11$ From equation 6: $b + c = f = 12 \implies 5 + c = 12 \implies c = 7$

The filled pyramid: Row 4 (bottom): [11, 5, 5, 7] Row 3: [16, 10, 12] Row 2: [26, 22] Row 1 (top): [48]

Final Answer: The filled pyramid numbers are:

Given: A 4-row pyramid. The top number is 100. The third row is [15, $x$, 25], where $x$ is unknown.

Let's use variables: Row 4 (bottom): [$a, b, c, d$] Row 3: [15, $x$, 25] Row 2: [$y, z$] Row 1 (top): [100]

Forming equations from the pyramid rules: From Row 3 to Row 2:

From Row 2 to Row 1: 3. $y + z = 100$

Substitute equations 1 and 2 into equation 3: $(15 + x) + (x + 25) = 100$ $2x + 40 = 100$ $2x = 60$ $x = 30$

So, the middle number in the third row is 30. Now we can find the second row:

The problem only provides enough information to fill the top three rows. The bottom row cannot be uniquely determined. For example: $a+b=15$, $b+c=30$, $c+d=25$. There are infinite solutions for $a, b, c, d$. Let's assume the question asks to fill the parts that can be determined.

Final Filled Pyramid (top three rows): Row 3: [15, 30, 25] Row 2: [45, 55] Row 1 (top): [100]

Final Answer: The numbers that can be determined are: the middle number in the third row is 30, and the second row is [45, 55]. The bottom row cannot be uniquely determined.

Given: A 4-row pyramid with some values.

Let's use variables: Row 4 (bottom): [1.2, $a$, $b$, 3.4] Row 3: [$c$, 2.5, $d$] Row 2: [$e, f$] Row 1 (top): [$g$]

Forming equations from the pyramid rules: From Row 4 to Row 3:

There is not enough information to find a unique solution for the variables $a, b, c, d, e, f, g$. The problem is underspecified. For any choice of $a$ (such that $a < 2.5$), we can find a corresponding $b$, and then the rest of the pyramid can be filled.

Example Solution: Let's choose $a=1$. From (2): $1 + b = 2.5 \implies b = 1.5$ From (1): $c = 1.2 + 1 = 2.2$ From (3): $d = 1.5 + 3.4 = 4.9$

Filled pyramid with a=1: Row 4: [1.2, 1, 1.5, 3.4] Row 3: [2.2, 2.5, 4.9] Row 2: $e = 2.2+2.5 = 4.7, f = 2.5+4.9 = 7.4$. Row 2 is [4.7, 7.4] Row 1: $g = 4.7+7.4 = 12.1$. Row 1 is [12.1]

Final Answer: The problem does not have a unique solution. One possible completion is: Bottom row [1.2, 1, 1.5, 3.4], Third row [2.2, 2.5, 4.9], Second row [4.7, 7.4], Top [12.1].