Question 1

Q1: The following are some equations along with the steps used to solve them to find the value of the letter-number. Go through each solution and decide w...

Accepted Answer

Problem 1: $4(4q+2)=50$    Mistake: In the second step, $4(4q)=50-2$, the term 2 was subtracted from the right side. This is incorrect. The 4 outside the parenthesis multiplies both terms inside, so it should be $16q + 8 = 50$. Alternatively, one could divide the entire right side by 4.    Correct S...

Question 2

Q1: 1. Fill in the blanks with integers. (a) $5 	imes \_ - 8=37$ (b) $37-(33-\_)=35$ (c) $-3 	imes(-11+\_)=45$

Accepted Answer

Let the blank be represented by x. (a) $5 	imes x - 8 = 37$ Add 8 to both sides: $5x = 37 + 8$ $5x = 45$ Divide by 5: $x = 9$ The integer is 9. (b) $37 - (33 - x) = 35$ Simplify the left side: $37 - 33 + x = 35$ $4 + x = 35$ Subtract 4 from both sides: $x = 35 - 4$ $x = 31$ The integer is 31. (c) $...

Question 3

Q10: 10. Given $28 p-36=98$, find the value of $14 p-19$ and $28 p-38$.

Accepted Answer

We are given the equation $28p - 36 = 98$. To find the value of $14p - 19$: Notice that the terms in $14p - 19$ are half of the terms $28p$ and close to half of $36$. Let's manipulate the original equation. $28p - 36 = 98$ Divide every term in the equation by 2: $\frac{28p}{2} - \frac{36}{2} = \frac...

Question 4

Q11: 11. The steps to solve three equations are shown below. Identify and correct any mistakes. (a) $6 x+9=66$ $x+9=11$ $x=11-9$ $x=2$ (b) $14 y+24=36$ $7...

Accepted Answer

(a) $6x + 9 = 66$    Mistake: In the second step, $x+9=11$, the entire left side was not divided by 6 correctly. The term 9 should also have been divided by 6. The step should be $\frac{6x+9}{6} = \frac{66}{6}$, which would lead to $x + \frac{9}{6} = 11$.    Correct Solution:     $6x + 9 = 66$     S...

Question 5

Q12: 12. Find the measures of the angles of these triangles.

Accepted Answer

The question requires figures of triangles with angles expressed in terms of variables. As these figures are not provided in the source content, this question cannot be answered.

Question 6

Q13: 13. Write 4 equations whose solution is $u=6$.

Accepted Answer

We can generate equations by starting with the solution $u=6$ and applying the same operation to both sides. 1.  Start with $u=6$. Add 4 to both sides:     $u + 4 = 6 + 4$     $u + 4 = 10$ 2.  Start with $u=6$. Subtract 10 from both sides:     $u - 10 = 6 - 10$     $u - 10 = -4$ 3.  Start with $u=6$...

Question 7

Q14: 14. The Bakhśhāli Manuscript (300 CE) mentions the following problem. The amount given to the first person is not known. The second person is given tw...

Accepted Answer

Let the amount given to the first person be x.    Amount for the second person = $2 	imes (	ext{first person's amount}) = 2x$.    Amount for the third person = $3 	imes (	ext{second person's amount}) = 3(2x) = 6x$. *   Amount for the fourth person = $4 	imes (	ext{third person's amount}) = 4(6x)...

Question 8

Q15: 15. The height of a giraffe is two and a half metres more than half its height. How tall is the giraffe?

Accepted Answer

Let the height of the giraffe be h metres. Two and a half metres can be written as 2.5 metres. According to the problem, we can form the equation: Height = (Half its height) + 2.5 metres $h = \frac{h}{2} + 2.5$ To solve for h, we subtract $\frac{h}{2}$ from both sides: $h - \frac{h}{2} = 2.5$ $\frac...

Question 9

Q16: 16. Two separate figures are given below. Each figure shows the first few positions in a sequence of arrangements made with sticks. Identify the patte...

Accepted Answer

Based on the figures in the textbook, there are two sequences. Figure 1: Sequence of Squares    Position 1: 1 square, 4 sticks.    Position 2: 2 squares, 7 sticks.    Position 3: 3 squares, 10 sticks.    Pattern: For position n, there are n squares. The number of sticks is given by the formula 3n +...

Question 10

Q17: 17. A number increased by 36 is equal to ten times itself. What is the number?

Accepted Answer

Let the number be x. According to the problem, we can form the equation: Number + 36 = 10 × Number $x + 36 = 10x$ Now, we solve for x: 1.  Subtract x from both sides:     $36 = 10x - x$     $36 = 9x$ 2.  Divide both sides by 9:     $x = \frac{36}{9}$     $x = 4$ Therefore, the number is 4.

Question 11

Q18: 18. Solve these equations: (a) $5(r+2)=10$ (b) $-3(u+2)=2(u-1)$ (c) $2(7-2 n)=-6$ (d) $2(x-4)=-16$ (e) $6(x-1)=2(x-1)-4$ (f) $3-7 s=7-3 s$ (g) $2 x+1=...

Accepted Answer

(a) $5(r+2)=10$ Divide by 5: $r+2 = 2$. Subtract 2: $r = 0$. (b) $-3(u+2)=2(u-1)$ Distribute: $-3u - 6 = 2u - 2$. Add 3u to both sides: $-6 = 5u - 2$. Add 2 to both sides: $-4 = 5u$. Divide by 5: $u = -\frac{4}{5}$. (c) $2(7-2n)=-6$ Divide by 2: $7 - 2n = -3$. Subtract 7: $-2n = -10$. Divide by -2:...

Question 12

Q19: 19. Solve the equations to find a path from Start to the End. Show your work in the given boxes provided and colour your path as you proceed.

Accepted Answer

To find the path, we must solve the equation in each box and move to the next box whose solution matches the variable's value from the current box. 1.  START -> The first choice is between two paths.        Path 1: $x-3=-5 \implies x = -2$.        Path 2: $x+5=2 \implies x = -3$.     We must now che...

Question 13

Q2: 2. Ranju is a daily wage labourer. She earns ₹ 750 a day. Her employer pays her in 50 and 100 rupee notes. If Ranju gets an equal number of 50 and 100...

Accepted Answer

Let n be the number of 50 rupee notes. Since she gets an equal number of 100 rupee notes, the number of 100 rupee notes is also n. The total value from 50 rupee notes is 50n. The total value from 100 rupee notes is 100n. Her total earnings are ₹ 750. We can set up an equation: (Value from 50 rupee n...

Question 14

Q20: 20. There are some children and donkeys on a beach. Together they have 28 heads and 80 feet. How many donkeys are there? How many children are there?

Accepted Answer

Let c be the number of children and d be the number of donkeys. Each child and each donkey has 1 head. The total number of heads is 28. Equation 1: $c + d = 28$ Each child has 2 feet and each donkey has 4 feet. The total number of feet is 80. Equation 2: $2c + 4d = 80$ We can solve this system of eq...

Question 15

Q3: 3. In the given picture, each black blob hides an equal number of blue dots. If there are 25 dots in total, how many dots are covered by one blob? Wri...

Accepted Answer

From the picture, we can see:    There are 5 black blobs.    There are 5 visible blue dots. *   The total number of dots (visible and hidden) is 25. Let x be the number of dots hidden under one black blob. The total number of hidden dots is the number of blobs multiplied by the number of dots under...

Question 16

Q4: 4. Here are machines that take an input, perform an operation on it and send out the result as an output. (a) Find the inputs in the following cases:...

Accepted Answer

(a) Machine: Multiply by 3, then Add 5 Let the input be x and the output be y. The equation for this machine is $y = 3x + 5$. To find the input x, we can rearrange the equation: $x = \frac{y-5}{3}$.    Case 1: Output = 20     $x = \frac{20-5}{3} = \frac{15}{3} = 5$. The input is 5.    Case 2: Output...

Question 17

Q5: 5. What are the inputs to these machines? [Machine 1: Input -> Add 4 -> Multiply by 5 -> Output] [Machine 2: Input from M1 -> Subtract 10 -> Divide by...

Accepted Answer

Let the initial input be x. We will trace the operations step by step. Machine 1: 1.  Input: x 2.  Add 4: x + 4 3.  Multiply by 5: 5(x + 4) The output of Machine 1 is 5(x + 4). This becomes the input for Machine 2. Machine 2: 1.  Input from M1: 5(x + 4) 2.  Subtract 10: 5(x + 4) - 10 3.  Divide by 2...

Question 18

Q6: 6. A taxi driver charges a fixed fee of ₹800 per day plus ₹20 for each kilometer traveled. If the total cost for a taxi ride is ₹2200, determine the n...

Accepted Answer

Let k be the number of kilometres traveled. The cost is calculated as: Total Cost = Fixed Fee + (Cost per km × Number of km) We are given:    Fixed Fee = ₹800    Cost per km = ₹20 *   Total Cost = ₹2200 We can form the equation: $2200 = 800 + 20k$ Now, we solve for k: 1.  Subtract 800 from both side...

Question 19

Q7: 7. The sum of two numbers is 76. One number is three times the other number. What are the numbers?

Accepted Answer

Let one number be x. Since the other number is three times the first, the second number is 3x. The sum of the two numbers is 76. We can write the equation: $x + 3x = 76$ Now, we solve for x: $4x = 76$ $x = \frac{76}{4}$ $x = 19$ The first number is 19. The second number is $3x = 3(19) = 57$. The two...

Question 20

Q8: 8. The figure shows the diagram for a window with a grill. What is the gap between two rods in the grill?

Accepted Answer

Based on the figure provided in the textbook:    Total width of the window = 100 cm.    Width of the frame on each side = 5 cm.    Number of vertical rods = 4.    Width of each rod = 10 cm. *   Number of gaps between the rods = 5. Let g be the width of one gap. First, we find the internal width of t...

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