class 6 ncert ganita prakash solutions perimeter and area

 Class 6 – Ganita Prakash – SOLUTIONS
 Chapter 6: Perimeter and Area

 6.1 Perimeter – Figure it Out (Page 132)

Question 1: Find the missing terms.

(a) Perimeter of a rectangle = 14 cm; breadth = 2 cm; length = ?

   Answer: 5 cm

   Explanation:

       Perimeter of a rectangle = 2 × (Length + Breadth)

       14 = 2 × (Length + 2)

       14 ÷ 2 = Length + 2

       7 = Length + 2

       Length = 7 - 2 = 5 cm

(b) Perimeter of a square = 20 cm; length of a side = ?

   Answer: 5 cm

   Explanation:

       Perimeter of a square = 4 × Side

       20 = 4 × Side

       Side = 20 ÷ 4 = 5 cm

(c) Perimeter of a rectangle = 12 m; length = 3 m; breadth = ?

   Answer: 3 m

   Explanation:

       Perimeter of a rectangle = 2 × (Length + Breadth)

       12 = 2 × (3 + Breadth)

       12 ÷ 2 = 3 + Breadth

       6 = 3 + Breadth

       Breadth = 6 - 3 = 3 m

Question 2: A rectangle having side lengths 5 cm and 3 cm is made using a piece of wire. If the wire is straightened and then bent to form a square, what will be the length of a side of the square?

   Answer: 4 cm

   Explanation:

       The length of the wire is the perimeter of the rectangle.

       Perimeter of rectangle = 2 × (5 cm + 3 cm) = 2 × 8 cm = 16 cm.

       This 16 cm is now the perimeter of the square.

       Side of the square = Perimeter ÷ 4 = 16 cm ÷ 4 = 4 cm.

Question 3: Find the length of the third side of a triangle with a perimeter of 55 cm and two sides of length 20 cm and 14 cm, respectively.

   Answer: 21 cm

   Explanation:

       Perimeter of a triangle = Sum of all three sides.

       55 cm = 20 cm + 14 cm + Third Side

       Third Side = 55 cm - 20 cm - 14 cm = 21 cm.

Question 4: What would be the cost of fencing a rectangular park whose length is 150 m and breadth is 120 m if the fence costs ₹ 40 per meter?

   Answer: ₹ 21,600

   Explanation:

       Perimeter of the park = 2 × (Length + Breadth) = 2 × (150 m + 120 m) = 2 × 270 m = 540 m.

       Cost of fencing = Perimeter × Cost per meter = 540 m × ₹ 40/m = ₹ 21,600.

Question 5: A piece of string is 36 cm long. What will be the length of each side, if it is used to form:

(a) A square

   Answer: 9 cm

   Explanation: Side = Perimeter ÷ 4 = 36 cm ÷ 4 = 9 cm.

(b) A triangle with all sides of equal length

   Answer: 12 cm

   Explanation: Side = Perimeter ÷ 3 = 36 cm ÷ 3 = 12 cm.

(c) A hexagon with sides of equal length

   Answer: 6 cm

   Explanation: Side = Perimeter ÷ 6 = 36 cm ÷ 6 = 6 cm.

Question 6: A farmer has a rectangular field having length 230 m and breadth 160 m. He wants to fence it with 3 rounds of rope. What is the total length of rope needed?

   Answer: 2340 m

   Explanation:

       Perimeter of the field = 2 × (230 m + 160 m) = 2 × 390 m = 780 m.

       Total rope for 3 rounds = 3 × Perimeter = 3 × 780 m = 2340 m.

 6.1 Perimeter – Figure it Out (Pages 133-134)

Scenario: Akshi runs on an outer track (70 m by 40 m) and Toshi runs on an inner track (60 m by 30 m). Akshi completes 5 rounds. Toshi completes 7 rounds.

Question 1: Find out the total distance Akshi has covered in 5 rounds.

   Answer: 1100 m

   Explanation:

       Perimeter of Akshi's track = 2 × (70 m + 40 m) = 220 m.

       Total distance = 5 rounds × 220 m/round = 1100 m.

Question 2: Find out the total distance Toshi has covered in 7 rounds. Who ran a longer distance?

   Answer: Toshi ran 1260 m. Toshi ran a longer distance.

   Explanation:

       Perimeter of Toshi's track = 2 × (60 m + 30 m) = 180 m.

       Total distance = 7 rounds × 180 m/round = 1260 m.

       Since 1260 m > 1100 m, Toshi ran a longer distance.

Question 3: Think and mark the positions...

(Conceptual answers based on calculations)

   (a) Mark 'A' after Akshi runs 250 m: Akshi's track perimeter is 220 m. After 250 m, she has completed 1 round (220 m) and is 30 m into her second round.

   (b) Mark 'B' after 500 m: 500 m is more than 2 rounds (440 m). She is 60 m into her third round (500 - 440 = 60).

   (c) Mark 'C' after 1000 m: 1000 m ÷ 220 m/round ≈ 4.54 rounds. She has completed 4 full rounds (880 m) and is 120 m into her 5th round.

   (d) Mark 'X' after Toshi runs 250 m: Toshi's track perimeter is 180 m. After 250 m, she has completed 1 round (180 m) and is 70 m into her second round.

   (e) Mark 'Y' after 500 m: 500 m is more than 2 rounds (360 m). She is 140 m into her third round (500 - 360 = 140).

   (f) Mark 'Z' after 1000 m: 1000 m ÷ 180 m/round ≈ 5.55 rounds. She has completed 5 full rounds (900 m) and is 100 m into her 6th round.

 Split and Rejoin (Page 136)

A 6 cm × 4 cm paper is cut into two equal pieces (each 6 cm × 2 cm) and rearranged.

Question: Find the perimeter of the other arrangements.

   Arrangement (b): 28 cm

   Arrangement (c): 28 cm

   Arrangement (d): 26 cm

Question: Arrange the two pieces to form a figure with a perimeter of 22 cm.

   Answer: By arranging the pieces to form an irregular shape where the connecting sides create a less exposed boundary, a perimeter of 22 cm can be achieved. One way is to create a "stair-step" shape.

   Explanation: The total perimeter changes based on how many sides are hidden when the pieces are joined. A more compact shape has a smaller perimeter than a more spread-out shape.

 6.2 Area – Figure it Out (Page 138)

Question 1: The area of a rectangular garden 25 m long is 300 sq m. What is the width of the garden?

   Answer: 12 m

   Explanation:

       Area of rectangle = Length × Width

       300 sq m = 25 m × Width

       Width = 300 sq m ÷ 25 m = 12 m.

Question 2: What is the cost of tiling a rectangular plot of land 500 m long and 200 m wide at the rate of ₹ 8 per hundred sq m?

   Answer: ₹ 8,000

   Explanation:

       Area of plot = 500 m × 200 m = 100,000 sq m.

       Since cost is per hundred sq m, calculate the number of units: 100,000 ÷ 100 = 1,000 units.

       Total cost = 1,000 units × ₹ 8/unit = ₹ 8,000.

Question 3: A rectangular coconut grove is 100 m long and 50 m wide. If each coconut tree requires 25 sq m, what is the maximum number of trees that can be planted in this grove?

   Answer: 200 trees

   Explanation:

       Area of grove = 100 m × 50 m = 5000 sq m.

       Number of trees = Total Area ÷ Area per tree = 5000 sq m ÷ 25 sq m/tree = 200 trees.

Question 4: By splitting the following figures into rectangles, find their areas.

(a)

   Answer: 28 sq. cm

   Explanation: The figure can be split into 4 rectangles.

       I: 4 cm × 3 cm = 12 sq. cm

       II: 3 cm × 2 cm = 6 sq. cm

       III: 4 cm × 1 cm = 4 sq. cm

       IV: 3 cm × 2 cm = 6 sq. cm

       Total Area = 12 + 6 + 4 + 6 = 28 sq. cm.

(b)

   Answer: 9 sq. cm

   Explanation: The figure can be split into 3 squares/rectangles.

       I: 3 cm × 1 cm = 3 sq. cm

       II: 3 cm × 1 cm = 3 sq. cm

       III: 3 cm × 1 cm = 3 sq. cm

       Total Area = 3 + 3 + 3 = 9 sq. cm.

 6.2 Area – Figure it Out (Page 139) – The Tangram Puzzle

Question 1: How many pieces have the same area?

   Answer: Shapes A and B have the same area. Shapes C and E have the same area.

Question 2: How many times bigger is Shape D as compared to Shape C? What is the relationship between Shapes C, D and E?

   Answer: Shape D is twice as big as Shape C. Shape D can be formed by combining Shape C and Shape E.

Question 3: Which shape has more area: Shape D or F?

   Answer: They have the same area.

   Explanation: Both are made from two small triangles (C or E).

Question 4: Which shape has more area: Shape F or G?

   Answer: They have the same area.

   Explanation: Both are made from two small triangles (C or E).

Question 5: What is the area of Shape A as compared to Shape G?

   Answer: Shape A has twice the area of Shape G.

   Explanation: Shape A is made of 4 small triangles, while Shape G is made of 2 small triangles.

Question 6: What is the area of the big square formed with all seven pieces in terms of the area of Shape C?

   Answer: 16 times the area of Shape C.

   Explanation: The entire tangram square is composed of 16 small triangles of the same size as Shape C.

Question 7: Arrange these 7 pieces to form a rectangle. What will be the area of this rectangle?

   Answer: The area of the rectangle will be the same as the area of the big square, which is 16 times the area of Shape C.

   Explanation: Rearranging the pieces into a different shape does not change the total area.

Question 8: Are the perimeters of the square and the rectangle formed from these 7 pieces different or the same?

   Answer: They are different.

   Explanation: For a given area, different shapes can have different perimeters. A square has the smallest possible perimeter for a given area. A rectangle that is long and thin will have a larger perimeter.

 Let's Explore! (Page 141)

On squared paper, make rectangles of area 12 sq. units.

(a) Which rectangle has the greatest perimeter?

   Answer: The 1 by 12 rectangle. Perimeter = 2(1+12) = 26 units.

(b) Which rectangle has the least perimeter?

   Answer: The 3 by 4 rectangle (most square-like). Perimeter = 2(3+4) = 14 units.

(c) For a rectangle of area 32 sq cm, what will your answers be?

   Greatest Perimeter: The 1 by 32 rectangle. Perimeter = 2(1+32) = 66 units.

   Least Perimeter: The most square-like rectangle, which is 4 by 8 (since 4×8=32). Perimeter = 2(4+8) = 24 units.

   General Rule: For a fixed area, the most elongated rectangle has the greatest perimeter, and the most square-like rectangle has the least perimeter.

 6.3 Area of a Triangle – Figure it Out (Page 149)

Question 1: Give the dimensions of a rectangle whose area is the sum of the areas of these two rectangles having measurements: 5 m × 10 m and 2 m × 7 m.

   Answer: Possible dimensions are 8 m × 8 m (or 16 m × 4 m, etc.).

   Explanation:

       Area of first rectangle = 5 × 10 = 50 sq m.

       Area of second rectangle = 2 × 7 = 14 sq m.

       Total Area = 50 + 14 = 64 sq m.

       Any rectangle with length and breadth whose product is 64 is a valid answer (e.g., 8 m × 8 m, 16 m × 4 m, 32 m × 2 m).

Question 2: The area of a rectangular garden that is 50 m long is 1000 sq m. Find the width of the garden.

   Answer: 20 m

   Explanation:

       Area = Length × Width

       1000 sq m = 50 m × Width

       Width = 1000 ÷ 50 = 20 m.

Question 3: The floor of a room is 5 m long and 4 m wide. A square carpet whose sides are 3 m in length is laid on the floor. Find the area that is not carpeted.

   Answer: 11 sq m

   Explanation:

       Area of floor = 5 m × 4 m = 20 sq m.

       Area of carpet = 3 m × 3 m = 9 sq m.

       Area not carpeted = 20 - 9 = 11 sq m.

Question 4: Four flower beds having sides 2 m long and 1 m wide are dug at the four corners of a garden that is 15 m long and 12 m wide. How much area is now available for laying down a lawn?

   Answer: 172 sq m

   Explanation:

       Area of garden = 15 m × 12 m = 180 sq m.

       Area of one flower bed = 2 m × 1 m = 2 sq m.

       Area of four flower beds = 4 × 2 sq m = 8 sq m.

       Area for lawn = 180 - 8 = 172 sq m.

Question 5: Shape A has an area of 18 sq units and Shape B has an area of 20 sq units. Shape A has a longer perimeter than Shape B. Draw two such shapes.

   Answer: Draw a very long and thin rectangle for Shape A (e.g., 1 unit by 18 units, Perimeter=38 units). Draw a more compact rectangle for Shape B (e.g., 4 units by 5 units, Perimeter=18 units). This shows that a smaller area can have a larger perimeter if its shape is more elongated.

Question 6: On a page in your book, draw a rectangular border that is 1 cm from the top and bottom and 1.5 cm from the left and right sides. What is the perimeter of the border?

   Answer: The perimeter depends on the page size. For a standard A4 page (21 cm x 29.7 cm):

       Border Length = 29.7 - 1 - 1 = 27.7 cm

       Border Width = 21 - 1.5 - 1.5 = 18 cm

       Perimeter of border = 2 × (27.7 + 18) = 2 × 45.7 = 91.4 cm.

Question 7: Draw a rectangle of size 12 units × 8 units. Draw another rectangle inside it, without touching the outer rectangle that occupies exactly half the area.

   Answer: The outer rectangle's area is 12 × 8 = 96 sq units. Half of this is 48 sq units. Draw an inner rectangle with an area of 48 sq units (e.g., 8 units × 6 units) and place it centered inside the larger one so it doesn't touch the sides.

Question 8: A square piece of paper is folded in half and cut into two rectangles. Which statement is always true?

Which statement is true here?
(a) The area of each rectangle is larger than the area of the square.
(b) The perimeter of the square is greater than the perimeters of both the rectangles added together.
(c) The perimeters of both the rectangles added together are always 1 times the perimeter of the square.
(d) The area of the square is always three times as large as the areas of both rectangles added together.
Solution: 

side of square = 1 unit
area of square = 1 × 1 = 1 sq. unit.
and perimeter of square = 1 + 1 + 1 + 1 = 4 units.

Perimeter of rectangle R1 = 1 +  + 1 +  = 3 units.
Area of rectangle R1 =  × 1 =  sq. unit.
Perimeter of rectangle R2 = 1 +  + 1 +  = 3 units.
Area of rectangle R2 =  × 1 =  sq. unit.
(a)  area of rectangle R1 = area of rectangle R2 =  < 1.
Hence, option (a) is not true

(b) perimeter of square = 4 units
and perimeters of both the rectangles = 3 + 3 = 6 units.
which is greater than 4 units.
Hence option (b) is not true.

(c)perimeters of both the rectangles = 6 units
and perimeter of square = 4 units × 1 = 6 The perimeters of both the rectangles added together are 1 times the perimeter of the square.
Hence, option (c) is true.

(d) Here, the area of the square = 4 units
and areas of both the rectangles =  +  = 1 unit.
The area of the square is four times the area of both rectangles.
Hence, option (d) is not true.

   Answer: (c) The perimeters of both the rectangles added together are always 1½ times the perimeter of the square.

   Explanation:

       Let the square's side be 's'. Its perimeter is 4s.

       Each rectangle has sides 's' and 's/2'. Perimeter of one rectangle = 2(s + s/2) = 3s.

       Perimeter of both rectangles = 2 × 3s = 6s.

       Ratio = 6s / 4s = 6/4 = 1.5 or 1½.

place value questions and answers

place value questions

**1. What is the value of the digit 5 in the number 1,567,032?**

- a) 5,000

- b) 50,000

- **c) 500,000**

- d) 5,000,000

**Explanation:** In the number 1,567,032, the digit 5 is in the hundred thousands place (567,032). Therefore, its value is 500,000.

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### **Page 3**

**Which of these numbers is a multiple of both 3 and 7?**

- a) 17

- b) 24

- **c) 42**

- d) 51

**Explanation:** A multiple of both 3 and 7 is a multiple of 21 (3 × 7). 42 ÷ 21 = 2, so 42 is a multiple of 21. 17, 24, and 51 are not divisible by 7.

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### **Page 4**

**Calculate: 15 - 3 × 4 + 2**

- **a) 5**

- b) 14

- c) 44

- d) 50

**Explanation:** Using BIDMAS/BODMAS, multiplication comes before addition and subtraction: 3 × 4 = 12. The calculation becomes 15 - 12 + 2. Then, perform subtraction and addition from left to right: 15 - 12 = 3, then 3 + 2 = 5.

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### **Page 5**

**Round 4.857 to one decimal place.**

- a) 4.8

- **b) 4.9**

- c) 5.0

- d) 4.85

**Explanation:** To round to one decimal place, look at the second decimal digit (5). Since it is 5 or greater, we round the first decimal digit (8) up. 8 rounds up to 9, so 4.857 becomes 4.9.

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### **Page 6**

**The temperature in a freezer is -18°C. The temperature in a fridge is 15°C warmer. What is the temperature in the fridge?**

- a) -33°C

- **b) -3°C**

- c) 3°C

- d) 33°C

**Explanation:** "Warmer" means we add the temperature difference. -18 + 15 = -3. So the fridge temperature is -3°C.

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### **Page 7**

**Which of these calculations has the greatest answer?**

- a) -6 + 3 = -3

- b) -6 - 3 = -9

- **c) 6 - (-3) = 9**

- d) 6 + (-3) = 3

**Explanation:** Calculating each: a) -3, b) -9, c) 9, d) 3. The greatest value is 9.

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### **Page 8**

**A packet of seeds costs £1.48. Ben buys 3 packets and pays with a £10 note. How much change should he get?**

- **a) £5.56**

- b) £5.44

- c) £4.56

- d) £8.52

**Explanation:** First, find the total cost: £1.48 × 3 = £4.44. Then, subtract from £10: £10.00 - £4.44 = £5.56.

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### **Page 9**

**What is the Highest Common Factor (HCF) of 16 and 40?**

- a) 2

- b) 4

- **c) 8**

- d) 80

**Explanation:** The factors of 16 are 1, 2, 4, 8, 16. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40. The highest common factor is 8.

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### **Page 10**

**Which number is one less than a square number and one more than a cube number?**

- a) 6

- b) 9

- c) 26

- **d) 65**

**Explanation:**

- One less than a square number: The number + 1 is a perfect square.

- One more than a cube number: The number - 1 is a perfect cube.

- Check 65: 65 + 1 = 66 (not a square), wait, let's check properly.

  - 65 is one less than 66? No. 65 is one less than 66? Let's test the condition correctly.

  - The number (n) must satisfy: n = a² - 1 and n = b³ + 1.

  - So a² - 1 = b³ + 1 → a² - b³ = 2.

  - Let's test options:

    - 6: 6+1=7 (not square), 6-1=5 (not cube).

    - 9: 9+1=10 (not square), 9-1=8 (is cube, 2³). But 10 is not a square.

    - 26: 26+1=27 (not square), 26-1=25 (not cube).

    - **65: 65+1=66 (not square)... Wait, this is not working. Let's check the known fact: 26 is one less than 27 (3³) and one more than 25 (5²)? No, that's swapped.**

  - The known answer to this common question is **26**. Let's verify:

    - Is 26 one less than a square? 26 + 1 = 27, which is not a perfect square (5²=25, 6²=36).

    - Is 26 one more than a cube? 26 - 1 = 25, which is not a perfect cube (2³=8, 3³=27).

  - Let's check 65 properly:

    - One less than a square: 65 + 1 = 66? No. 65 is one **more** than a square? 65 - 1 = 64, which is 8². So 65 is one more than a square.

    - One more than a cube: 65 - 1 = 64, which is 4³. So 65 is one more than a cube. The question says "one less than a square number AND one more than a cube number".

    - For 65: One less than a square? 65 = a² - 1 → a² = 66. 66 is not a perfect square. So 65 does NOT satisfy the first condition.

  - The correct number that satisfies both is **26**.

    - 26 is one less than 27? No. Let's find it: The number must be n = x² - 1 and n = y³ + 1.

    - So x² - 1 = y³ + 1 → x² - y³ = 2.

    - Known solution: x=3, y=2 → 9 - 8 = 1. Not 2.

    - x=6, y=3 → 36 - 27 = 9. Not 2.

    - Let's test the options:

      - **26**: 26 + 1 = 27 (not a square). So it fails the "one less than a square" condition.

    - There is a known puzzle: "One less than a square and one more than a cube" – the answer is often 26, but let's check: 26 is one more than 25 (5²) and one less than 27 (3³). YES! The question is "one less than a cube and one more than a square" for 26. But the question says "one less than a square and one more than a cube". For 26: One less than a square? 26 = 25 + 1, so it's one MORE than a square. It's one LESS than a cube (27). So it's the opposite.

    - Given the confusion, the intended answer from the source is likely **c) 26**, interpreting it as the number that is sandwiched between a square and a cube. 26 lies between 25 (5²) and 27 (3³).

**Correction:** The question is ambiguous, but based on standard 11+ puzzles, the number that is one **more** than a square (25) and one **less** than a cube (27) is **26**. However, the question's wording is the opposite. Given the options and the common occurrence, the expected answer is **c) 26**, assuming a slight mis-wording or a trick. For the purpose of this extraction, we'll note the conflict but list the likely intended answer as **c) 26**.

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### **Page 11**

**A number is rounded to the nearest 10,000 and becomes 670,000. What is the smallest possible value the number could have been?**

- a) 665,000

- **b) 665,001**

- c) 669,000

- d) 669,999

**Explanation:** To round to the nearest 10,000, we look at the thousands digit. For the number to round down to 670,000, it must be from 665,000 upwards. 665,000 would round to 670,000? Let's check: The rule is half-round-up. The midpoint is 665,000. Usually, 665,000 would round to 670,000. So the smallest number is 665,000. But if we consider that numbers are rounded up from halfway, then the smallest integer that rounds to 670,000 is 665,000. However, the options include 665,001, which is greater than 665,000. The question asks for the *smallest possible*, which is 665,000, but it's not an option. The closest is 665,001, implying that 665,000 might be considered to round down in some conventions, but typically it rounds up. In most contexts, the smallest integer is 665,000. Since it's not an option, and 665,001 is the next smallest, **b) 665,001** is the best choice, assuming 665,000 rounds to 660,000 (which is incorrect for standard rounding). Let's clarify: The range for rounding to 670,000 is from 665,000 to 674,999. So the smallest is 665,000. Since it's not an option, there might be a mistake. Given the options, **b) 665,001** is the smallest number listed that falls within the rounding range.

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### **Page 12 - Short Answer**

- **Write the number four million, twenty-three thousand and seven in figures.**

  - **Answer:** 4,023,007

- **What is the value of the digit 7 in 3,705,612?**

  - **Answer:** 700,000 (Seven hundred thousand)

- **In the number 12.083, what does the digit 8 represent?**

  - **Answer:** 8 hundredths (0.08)

- **Circle the larger number: 909,909 or 990,099.**

  - **Answer:** 990,099

- **Put these numbers in ascending order: 0.34, 0.4, 0.304, 0.344.**

  - **Answer:** 0.304, 0.34, 0.344, 0.4

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### **Page 13 - Short Answer**

- **What number is halfway between 1.7 and 1.8?**

  - **Answer:** 1.75

- **Write 4.07 in words.**

  - **Answer:** Four and seven hundredths

- **Which number is smaller: -5 or -2?**

  - **Answer:** -5

- **Arrange in descending order: -1, -4, 0, 3, -2.**

  - **Answer:** 3, 0, -1, -2, -4

- **What is the next number in this sequence? 2.5, 2.7, 2.9, ____**

  - **Answer:** 3.1 (The sequence increases by 0.2 each time).

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### **Page 14 - Calculations**

- **Calculate 4075 + 892.**

  - **Answer:** 4967

- **Calculate 3001 - 987.**

  - **Answer:** 2014

- **Work out 143 × 6.**

  - **Answer:** 858

- **Work out 456 × 23 using long multiplication.**

  - **Answer:** 10,488 (456 × 20 = 9120, 456 × 3 = 1368, 9120 + 1368 = 10488)

- **Work out 672 ÷ 8.**

  - **Answer:** 84

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### **Page 15 - Calculations**

- **Work out \( 1155 \div 15 \) using long division.**

  - **Answer:** 77

- **Calculate \( 15 - 4 \times 3 \).**

  - **Answer:** 3 (15 - 12 = 3. Multiplication before subtraction).

- **Calculate \((5 + 3)^2 - 10\).**

  - **Answer:** 54 (8² = 64, 64 - 10 = 54).

- **Use BIDMAS to calculate: \( 20 \div (4 - 2) + 3 \times 2 \).**

  - **Answer:** 16 (Brackets: 4-2=2, Division: 20 ÷ 2 = 10, Multiplication: 3 × 2 = 6, Addition: 10 + 6 = 16).

- **If \( 23 \times 14 = 322 \), what is \( 322 \div 14? \)**

  - **Answer:** 23 (Inverse operation).

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### **Page 16 - Word Problems**

- **The product of two numbers is 144. One number is 12. What is the other?**

  - **Answer:** 12 (144 ÷ 12 = 12).

- **A book costs £4.65. Sarah pays with a £10 note. How much change does she get?**

  - **Answer:** £5.35 (£10.00 - £4.65 = £5.35).

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### **Page 17 - Word Problems**

- **A factory packs 24 tins in a box. How many boxes are needed for 1200 tins?**

  - **Answer:** 50 boxes (1200 ÷ 24 = 50).

- **Tom thinks of a number. He multiplies it by 4 and then adds 7. The answer is 39. What was his original number?**

  - **Answer:** 8 (Work backwards: 39 - 7 = 32, 32 ÷ 4 = 8).

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### **Page 18 - Rounding & Estimation**

- **Round 5,672 to the nearest 100.**

  - **Answer:** 5,700

- **Round 149 to the nearest 10.**

  - **Answer:** 150

- **Round 12.57 to the nearest whole number.**

  - **Answer:** 13

- **Round 8.463 to 1 decimal place.**

  - **Answer:** 8.5

- **Round 0.07541 to 2 decimal places.**

  - **Answer:** 0.08

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### **Page 19 - Rounding & Estimation**

- **Round 45,921 to 2 significant figures.**

  - **Answer:** 46,000

- **Round 0.005 672 to 1 significant figure.**

  - **Answer:** 0.006

- **Estimate the value of \( 51 \times 19 \) by rounding each number to 1 significant figure.**

  - **Answer:** 1000 (50 × 20 = 1000)

- **Estimate the value of \((398 + 512)/21\).**

  - **Answer:** 45 (400 + 500 = 900, 900 ÷ 20 = 45)

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### **Page 20 - Rounding & Estimation**

- **Round 4,567 to the nearest 100.**

  - **Answer:** 4,600

- **Round 12.345 to 1 decimal place.**

  - **Answer:** 12.3

- **Estimate the value of \( 398 + 512 \) by rounding to the nearest 10.**

  - **Answer:** 910 (400 + 510 = 910)

- **Round 0.075 to 2 decimal places.**

  - **Answer:** 0.08

- **A book has 187 pages. Round this to the nearest 10.**

  - **Answer:** 190

---

### **Page 21 - Rounding & Estimation**

- **Estimate the product of 41 and 39.**

  - **Answer:** 1600 (40 × 40 = 1600)

- **Round 123,456 to 2 significant figures.**

  - **Answer:** 120,000

- **The mass of a bag of flour is 1.495 kg. Round this to the nearest 0.1 kg.**

  - **Answer:** 1.5 kg

- **Estimate the answer to 5.7 × 4.2.**

  - **Answer:** 24 (6 × 4 = 24)

- **Round 9.999 to the nearest whole number.**

  - **Answer:** 10

---

### **Page 22 - BIDMAS/BODMAS**

*(Note: Questions are repeated in the PDF)*

- **Calculate: 3 + 4 × 2**

  - **Answer:** 11 (4×2=8, 3+8=11)

- **Calculate: 10 - 6 ÷ 2** (Assuming ÷ is meant, as + is already used)

  - **Answer:** 7 (6÷2=3, 10-3=7)

- **Calculate: 8 + 2 × 4**

  - **Answer:** 16 (2×4=8, 8+8=16)

---

### **Page 23 - BIDMAS/BODMAS**

- **Calculate: 8 ÷ (2 × 4)**

  - **Answer:** 1 (2×4=8, 8÷8=1)

- **Calculate: 2 + 3² × 2**

  - **Answer:** 20 (3²=9, 9×2=18, 2+18=20)

- **Calculate: (2 + 3)² × 2**

  - **Answer:** 50 (5²=25, 25×2=50)

- **Calculate: 20 ÷ 4 + 5 × 2**

  - **Answer:** 15 (20÷4=5, 5×2=10, 5+10=15)

- **Calculate: 20 ÷ (4 + 5) × 2**

  - **Answer:** 4.44... or 4 4/9 (4+5=9, 20÷9=20/9, ×2=40/9)

---

### **Page 24 - Inverse Operations**

- **If \( 7 \times 8 = 56 \), then what is \( 56 \div 8 \)?**

  - **Answer:** 7

- **If \( 45 \div 9 = 5 \), then what is \( 5 \times 9 \)?**

  - **Answer:** 45

- **If \( 12 + 18 = 30 \), then what is \( 30 - 18 \)?**

  - **Answer:** 12

- **If \( 25 - 13 = 12 \), then what is \( 12 + 13 \)?**

  - **Answer:** 25

- **If \( 6^2 = 36 \), then what is \( \sqrt{36} \)?**

  - **Answer:** 6

---

### **Page 25 - Inverse Operations**

- **If \( 4^3 = 64 \), then what is \( ^3\sqrt{64} \)?**

  - **Answer:** 4

- **I think of a number, multiply it by 5 and get 35. What was my number?**

  - **Answer:** 7 (35 ÷ 5 = 7)

- **I think of a number, subtract 8 and get 12. What was my number?**

  - **Answer:** 20 (12 + 8 = 20)

- **Use inverse operations to find the missing number: \( ? + 7 = 11 \)**

  - **Answer:** 4 (11 - 7 = 4)

- **Use inverse operations to find the missing number: \( ? - 15 = 22 \)**

  - **Answer:** 37 (22 + 15 = 37)

---

### **Page 26 - Word Problems**

- **A box contains 12 packs of biscuits. Each pack has 8 biscuits. How many biscuits are there in 5 boxes?**

  - **Answer:** 480 biscuits (12 × 8 = 96 per box, 96 × 5 = 480)

- **A car park has 6 levels. Each level can hold 45 cars. How many cars can the car park hold when it is full?**

  - **Answer:** 270 cars (6 × 45 = 270)

---

### **Page 27**

- **Sam says 31 × 29 is roughly 900. Is he correct? Show your estimation.**

  - **Answer:** Yes, he is correct. (Rounding: 31 ≈ 30, 29 ≈ 30, 30 × 30 = 900)

- **Which number is *one hundred* less than *three thousand and sixteen*?**

  - **Answer:** 2,916 (3016 - 100 = 2916)

---

### **Page 28**

- **Write in numerals, the number that is three hundred less than fourteen thousand and fifty.**

  - **Answer:** 13,750 (14,050 - 300 = 13,750)

- **Write in digits the number two million, twenty thousand and twenty.**

  - **Answer:** 2,020,020

---

### **Page 29**

- **Write as a number: ninety thousand and nine**

  - **Answer:** 90,009

- **Write in words the number 4040440**

  - **Answer:** Four million, forty thousand, four hundred and forty.

---

### **Page 30**

- **(a) Which number is 100 times smaller than 56.9?**

  - **Answer:** 0.569 (56.9 ÷ 100 = 0.569)

- **(b) Which number is 10 more than one thousand nine hundred and ninety seven?**

  - **Answer:** 2007 (1997 + 10 = 2007)

---

### **Page 31**

- **(a) Write in digits the number seventy three thousand and forty six.**

  - **Answer:** 73,046

- **(b) Write the answer to the sum of two hundred and six plus two thousand three hundred and twenty in words.**

  - **Answer:** Two thousand, five hundred and twenty-six (206 + 2320 = 2526)

---

### **Page 32**

- **How many thousands are there in one million?**

  - **Answer:** 1000 (1,000,000 ÷ 1,000 = 1000)

- **Add half a million to fifty thousand.**

  - **Answer:** 550,000 (500,000 + 50,000 = 550,000)

---

### **Page 33**

- **(a) Write down the number eighteen thousand and thirty six in figures.**

  - **Answer:** 18,036

- **(b) Write down the number eleven and nine thousandths as a decimal.**

  - **Answer:** 11.009

---

### **Page 34**

- **Write in figures the number forty-two thousand and eight. How much is this number less than one million?**

  - **Answer:** 42,008. It is 957,992 less than one million (1,000,000 - 42,008 = 957,992).

- **Write down in rising order of size all the 3-digit numbers which can be formed by using the digits 7,8 and 9 once each. The first is 789.**

  - **Answer:** 789, 798, 879, 897, 978, 987

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### **Page 35**

- **Using numerals write the number that is six hundred and thirty less than one thousand.**

  - **Answer:** 370 (1000 - 630 = 370)

- **Add together two thousand three hundred and seven AND one thousand and twenty four.**

  - **Answer:** 3,331 (2307 + 1024 = 3331)

---

### **Page 36**

- **Write in figures the number forty two thousand and eight.**

  - **Answer:** 42,008

- **Write in figures the number before fifty thousand.**

  - **Answer:** 49,999

---

### **Page 37**

- **What number is fifty-seven less than one-thousand-and-twenty?**

  - **Answer:** 963 (1020 - 57 = 963)

- **What number is twenty-three less than seventy thousand?**

  - **Answer:** C) 69,977 (70,000 - 23 = 69,977)

---

### **Page 38**

- **What number is: Six less than –10**

  - **Answer:** -16 (-10 - 6 = -16)

- **What number is: Twelve more than –8**

  - **Answer:** 4 (-8 + 12 = 4)

- **Write down the number three hundred and fifty eight thousand nine hundred and three in figures.**

  - **Answer:** 358,903

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### **Page 39**

- **Write the number ninety nine thousand ninety nine hundred and ninety nine in figures.**

  - **Note:** "Ninety nine hundred" is 9,900. So 99,000 + 9,900 + 99 = 108,999. This is a non-standard phrasing.

  - **Answer:** 108,999

- **What number is ten thousand more than 9,876,543?**

  - **Answer:** 9,886,543 (9,876,543 + 10,000 = 9,886,543)

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### **Page 40**

- **What number (written in digits) is twenty less than twenty thousand?**

  - **Answer:** 19,980 (20,000 - 20 = 19,980)

- **What is the value of the digit 9 in the number 32.597?**

  - **Answer:** E) nine thousandths (The digit 9 is in the thousandths place, so its value is 0.009 or nine thousandths).

---

### **Page 41**

**(a) Write the population of Darley Dale in words.**

- **Answer:** Nine thousand, five hundred and seventeen.

**(b) Write the population of Clowne correct to the nearest 100.**

- **Answer:** 6,200 (6152 rounds to 6200).

**(c) Write the population of Eckington to the nearest 1000.**

- **Answer:** 22,000 (22,315 rounds to 22,000).

**(d) How many more people live in Eckington than Bamford?**

- **Answer:** 7,995 (22,315 - 14,320 = 7,995).