Answers – Practice 1

1. Value of 7 in 570 296: 70,000

   In 570 296, the digit 7 is in the ten thousands place, so its value is 7 × 10,000 = 70,000

2. Fifty-three thousand and thirty-five: 53,035

3. 8379 to the nearest hundred: 8400

   Since 79 is less than 50, we round down to 8400

4. Ordering from smallest to largest:

   First, convert all to decimal form:

   • ¹³/₂₀₁ = 0.0647…
   • 5.6% = 0.056
   • 0.065 = 0.065
   • ⁵/₈₉ = 0.0562…

   Therefore, in order from smallest to largest: 5.6%, ⁵/₈₉, 0.065, ¹³/₂₀₁

1. Number of students studying Music: 8 + 3 + 2 + 4 = 17

2. Number of students studying Drama: 8 + 3 + 2 + 9 = 22

3. Number of students studying Geography: 12 + 4 + 2 + 9 = 27

4. Number of students studying exactly 2 subjects: 3 + 4 + 9 = 16

   (This includes: Music & Drama (3), Music & Geography (4), Drama & Geography (9))

5. Number of students studying only one subject: 8 + 8 + 12 = 28

   (This includes: Music only (8), Drama only (8), Geography only (12))

6. Number of students studying Music and Geography: 4 + 2 = 6

   (This includes students studying just Music & Geography (4), and all three subjects (2))

7. Number of students studying Drama and Geography: 9 + 2 = 11

   (This includes students studying just Drama & Geography (9), and all three subjects (2))

8. Number of students studying Drama and Music: 3 + 2 = 5

   (This includes students studying just Music & Drama (3), and all three subjects (2))

9. Total number of students: 8 + 8 + 12 + 3 + 4 + 9 + 2 = 46

1. Elements of set F (factors of 14):

   F = {1, 2, 7, 14}

2. Elements of set P (prime numbers less than 14):

   P = {2, 3, 5, 7, 11, 13}

3. The elements in set (F ∩ P)’ (elements not in the intersection of F and P):

   First, find F ∩ P = {2, 7} (prime numbers that are also factors of 14)

   Therefore (F ∩ P)’ = {1, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14}

   These are all the elements in ξ that are not both factors of 14 and prime numbers.

4. F ∩ P = {2, 7}

   These are the numbers that are both factors of 14 and prime numbers.

To write 195 as a product of its prime factors:

Step 1: Find the smallest prime number that divides 195.

195 ÷ 3 = 65 (3 is the smallest prime that divides 195)

Step 2: Continue the process with 65.

65 ÷ 5 = 13 (5 is the smallest prime that divides 65)

Step 3: 13 is a prime number.

Therefore: 195 = 3 × 5 × 13

1. 3¹/₃ – 2¹/₃

   Convert to improper fractions:

   3¹/₃ = (3 × 3 + 1)/3 = ¹⁰/₃

   2¹/₃ = (2 × 3 + 1)/3 = ⁷/₃

   Subtraction: ¹⁰/₃ – ⁷/₃ = ³/₃ = 1

   Therefore, 3¹/₃ – 2¹/₃ = 1

2. ¹⁵/₂₈ ÷ ⁴/₇

   To divide by a fraction, multiply by its reciprocal:

   ¹⁵/₂₈ ÷ ⁴/₇ = ¹⁵/₂₈ × ⁷/₄

   = ^{(15 × 7)}/_{(28 × 4)}

   = ¹⁰⁵/₁₁₂

   To simplify, find the GCD of 105 and 112:

   GCD of 105 and 112 is 7

   ¹⁰⁵/₁₁₂ = ^{(105 ÷ 7)}/_{(112 ÷ 7)} = ¹⁵/₁₆

   Therefore, ¹⁵/₂₈ ÷ ⁴/₇ = ¹⁵/₁₆

1. Factors of 56:

   First, find all pairs of numbers that multiply to give 56:

   1 × 56 = 56

   2 × 28 = 56

   4 × 14 = 56

   7 × 8 = 56

   Therefore, the factors of 56 are: 1, 2, 4, 7, 8, 14, 28, 56

2. Write 108 as a product of its prime factors:

   Step 1: Find the smallest prime number that divides 108.

   108 ÷ 2 = 54

   Step 2: Continue the process.

   54 ÷ 2 = 27

   27 ÷ 3 = 9

   9 ÷ 3 = 3

   3 ÷ 3 = 1

   Therefore: 108 = 2² × 3³ = 2 × 2 × 3 × 3 × 3

3. Find the LCM of 84 and 60:

   Method: Find the prime factorization of each number, then multiply the highest powers of each prime factor.

   84 = 2² × 3 × 7 = 2 × 2 × 3 × 7

   60 = 2² × 3 × 5 = 2 × 2 × 3 × 5

   LCM = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420

4. Find the HCF of 90 and 144 using the division method:

   Step 1: Divide the larger number by the smaller.

   144 ÷ 90 = 1 remainder 54

   Step 2: Divide the divisor by the remainder.

   90 ÷ 54 = 1 remainder 36

   Step 3: Continue until there is no remainder.

   54 ÷ 36 = 1 remainder 18

   36 ÷ 18 = 2 remainder 0

   Since the remainder is 0, the HCF is 18.