Express 6.455 correct to 1 d.p, 2 s.f, and nearest whole number. Express 0.52 as a common fraction. Express 102 as a product of its prime factors. Find HCF and LCM of 30 and 24.

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Asked on April 15, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-15T11:22:14.179Z

Fresh day Nmk, let's solve. Here are the solutions to the questions you've provided. 1. (a) Express 6.455 correct to: i) 1 d.p ii) 2 s.f iii) nearest whole number i) 1 d.p (1 decimal place) The first decimal place is 4. The digit after it is 5, so we round up. 6.455 ≈ 6.5 ii) 2 s.f (2 significant figures) The first significant figure is 6, the second is 4. The digit after it is 5, so we round up. 6.455 ≈ 6.5 iii) nearest whole number The whole number is 6. The digit after it is 4, so we round down. 6.455 ≈ 6 1. (b) Express 0.52 as a common fraction in its simplest form. Step 1: Write the decimal as a fraction. 0.52 = (52)/(100) Step 2: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4. (52 ÷ 4)/(100 ÷ 4) = (13)/(25) The fraction is already in its simplest form. (13)/(25) 2. (a) Express 102 as a product of its prime factors in index form. Step 1: Find the prime factors of 102. 102 ÷ 2 = 51 51 ÷ 3 = 17 17 ÷ 17 = 1 The prime factors are 2, 3, and 17. Step 2: Write the prime factors in index form. 102 = 2^1 × 3^1 × 17^1 2 × 3 × 17 2. (b) Find the HCF and LCM of 30 and 24. Step 1: Find the prime factors of each number. 30 = 2 × 3 × 5 24 = 2 × 2 × 2 × 3 = 2^3 × 3 Step 2: Find the HCF (Highest Common Factor). The common prime factors are 2 and 3. Take the lowest power of each common factor. HCF = 2^1 × 3^1 = 2 × 3 = 6 Step 3: Find the LCM (Lowest Common Multiple). Take the highest power of all prime factors present in either number. LCM = 2^3 × 3^1 × 5^1 = 8 × 3 × 5 = 120 HCF = 6, LCM = 120 3. (a) If 2n4_6 is a number in base 6. State the largest possible value of n. In base 6, the digits used must be less than 6. The digits in 2n4_6 are 2, n, and 4. For n to be a valid digit in base 6, it must satisfy 0 n < 6. The largest possible integer value for n is 5. 5 3. (b) i) Convert 333_4 to base 10. Step 1: Write the number in expanded form using powers of the base. 333_4 = 3 × 4^2 + 3 × 4^1 + 3 × 4^0 Step 2: Calculate the value. 3 × 16 + 3 × 4 + 3 × 1 48 + 12 + 3 = 63 63_10 3. (b) ii) Convert 501_6 to base 5. Step 1: Convert 501_6 to base 10. 501_6 = 5 × 6^2 + 0 × 6^1 + 1 × 6^0 5 × 36 + 0 × 6 + 1 × 1 180 + 0 + 1 = 181_10 Step 2: Convert 181_10 to base 5. Divide 181 by 5 and record the remainders. 181 ÷ 5 = 36 remainder 1 36 ÷ 5 = 7 remainder 1 7 ÷ 5 = 1 remainder 2 1 ÷ 5 = 0 remainder 1 Read the remainders from bottom to top. 1211_5 4. Given that E = \1-10\, A = \1,4,5,7,8\, B = \2,6,8,10\ a) Represent the sets on a Venn diagram. The universal set E = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10\. A = \1,4,5,7,8\ B = \2,6,8,10\ First, find the intersection A B. A B = \8\ Next, find the elements unique to A: A B = \1,4,5,7\ Next, find the elements unique to B: B A = \2,6,10\ Finally, find the elements outside A and B (in E but not in A or B): E (A B) = \1,2,3,4,5,6,7,8,9,10\ \1,2,4,5,6,7,8,10\ = \3,9\ `mermaid graph TD subgraph E A(A) B(B) A -- 1,4,5,7 --> A B -- 2,6,10 --> B A -- 8 --> B E -- 3,9 --> E end ` (Note: The Venn diagram cannot be fully rendered in text. The description above details the placement of elements.) b) Find i) n(A B) ii) n(B') iii) List the elements in (A B)' i) n(A B) A B = \8\ The number of elements in A B is 1. n(A B) = 1 ii) n(B') B' is the complement of B, meaning all elements in the universal set E that are not in B. E = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10\ B = \2, 6, 8, 10\ B' = \1, 3, 4, 5, 7, 9\ The number of elements in B' is 6. n(B') = 6 iii) List the elements in (A B)' (A B)' is the complement of A B. A B = \8\ (A B)' = E \8\ = \1, 2, 3, 4, 5, 6, 7, 9, 10\ \1, 2, 3, 4, 5, 6, 7, 9, 10\ 5. (a) Find the radius of a circle whose area is 440 cm^2. Step 1: Use the formula for the area of a circle, A = r^2. Given A = 440 cm^2. Use ≈ (22)/(7). 440 = (22)/(7) r^2 Step 2: Solve for r^2. r^2 = (440 × 7)/(22) r^2 = 20 × 7 r^2 = 140 Step 3: Solve for r. r = sqrt(140) r ≈ 11.832 cm Rounding to 3 significant figures: 11.8 cm 5. (b) Find the volume of a cylinder with radius 14 cm and height 40 cm. Step 1: Use the formula for the volume of a cylinder, V = r^2 h. Given r = 14 cm and h = 40 cm. Use ≈ (22)/(7). V = (22)/(7) × (14 cm)^2 × 40 cm V = (22)/(7) × 196 cm^2 × 40 cm Step 2: Calculate the volume. V = 22 × 28 cm^2 × 40 cm V = 616 cm^2 × 40 cm V = 24640 cm^3 24640 cm^3 5. (c) Express 3.5 m^2 to km^2. Step 1: Recall the conversion factors. 1 km = 1000 m 1 km^2 = (1000 m)^2 = 1000000 m^2 = 10^6 m^2 Step 2: Convert 3.5 m^2 to km^2. 3.5 m^2 × 1 km^210^6 m^2 3.5 × 10^-6 km^2 0.0000035 km^2 6. (a) Expand and simplify (x-1)(x+2). Step 1: Use the FOIL method (First, Outer, Inner, Last). (x-1)(x+2) = x × x + x × 2 - 1 × x - 1 × 2 = x^2 + 2x - x - 2 Step 2: Combine like terms. = x^2 + x - 2 x^2 + x - 2 6. (b) Express (3x+5)/(3) - (x-2)/(2) as a single fraction in its simplest form. Step 1: Find a common denominator, which is 6. (2(3x+5))/(2 × 3) - (3(x-2))/(3 × 2) (2(3x+5))/(6) - (3(x-2))/(6) Step 2: Combine the numerators over the common denominator. (2(3x+5) - 3(x-2))/(6) Step 3: Expand the numerators. (6x + 10 - (3x - 6))/(6) Step 4: Distribute the negative sign and combine like terms. (6x + 10 - 3x + 6)/(6) (3x + 16)/(6) (3x+16)/(6) 7. Factorise completely: a) am - an + bm - bn Step 1: Group terms with common factors. (am - an) + (bm - bn) Step 2: Factor out the common factor from each group. a(m - n) + b(m - n) Step 3: Factor out the common binomial factor (m-n). (m - n)(a + b) (m - n)(a + b) b) 49 - y^2 This is a difference of squares, a^2 - b^2 = (a-b)(a+b). Here, a^2 = 49 a = 7 and b^2 = y^2 b = y. (7 - y)(7 + y) (7 - y)(7 + y) c) x^2 - 10x - 24 Step 1: Find two numbers that multiply to -24 and add to -10. The numbers are -12 and 2. -12 × 2 = -24 -12 + 2 = -10 Step 2: Write the quadratic expression in factored form. (x - 12)(x + 2) (x - 12)(x + 2) 8. 210 is shared in the ratio 1:2:4, find the difference between the largest and smallest share. Step 1: Find the total number of parts in the ratio. 1 + 2 + 4 = 7 parts Step 2: Calculate the value of one part. (210)/(7) = 30 Step 3: Calculate the smallest share (1 part) and the largest share (4 parts). Smallest share: 1 × 30 = 30 Largest share: 4 × 30 = 120 Step 4: Find the difference between the largest and smallest share. 120 - 30 = 90 90 9. A map is drawn to scale 1:35000 i) Express the scale in the form 1 cm : n km Step 1: Convert 35000 cm to km. 1 km = 1000 m 1 m = 100 cm So, 1 km = 1000 × 100 cm = 100000 cm. Step 2: Divide 35000 by 100000 to convert to km. 35000 cm = (35000)/(100000) km = 0.35 km So the scale is 1 cm : 0.35 km. 1 cm : 0.35 km ii) Hence or otherwise find the actual area, in km^2, represented by 20 cm^2 on the map. Step 1: Determine the linear scale factor. From part (i), 1 cm on the map represents 0.35 km in actual distance. Step 2: Determine the area scale factor. If 1 cm represents 0.35 km, then 1 cm^2 represents (0.35 km)^2. (0.35 km)^2 = 0.1225 km^2 So, 1 cm^2 on the map represents 0.1225 km^2 in actual area. Step 3: Calculate the actual area represented by 20 cm^2$ on the map. 20 cm^2 × 0.1225 km^2cm^2 = 2.45 km^2 2.45 km^2 What's next? Send 'em!

Accepted Answer · 2026-04-15T11:22:14.179Z

Fresh day Nmk, let's solve. Here are the solutions to the questions you've provided. 1. (a) Express 6.455 correct to: i) 1 d.p ii) 2 s.f iii) nearest whole number i) 1 d.p (1 decimal place) The first decimal place is 4. The digit after it is 5, so we round up. 6.455 ≈ 6.5 ii) 2 s.f (2 significant figures) The first significant figure is 6, the second is 4. The digit after it is 5, so we round up. 6.455 ≈ 6.5 iii) nearest whole number The whole number is 6. The digit after it is 4, so we round down. 6.455 ≈ 6 1. (b) Express 0.52 as a common fraction in its simplest form. Step 1: Write the decimal as a fraction. 0.52 = (52)/(100) Step 2: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4. (52 ÷ 4)/(100 ÷ 4) = (13)/(25) The fraction is already in its simplest form. (13)/(25) 2. (a) Express 102 as a product of its prime factors in index form. Step 1: Find the prime factors of 102. 102 ÷ 2 = 51 51 ÷ 3 = 17 17 ÷ 17 = 1 The prime factors are 2, 3, and 17. Step 2: Write the prime factors in index form. 102 = 2^1 × 3^1 × 17^1 2 × 3 × 17 2. (b) Find the HCF and LCM of 30 and 24. Step 1: Find the prime factors of each number. 30 = 2 × 3 × 5 24 = 2 × 2 × 2 × 3 = 2^3 × 3 Step 2: Find the HCF (Highest Common Factor). The common prime factors are 2 and 3. Take the lowest power of each common factor. HCF = 2^1 × 3^1 = 2 × 3 = 6 Step 3: Find the LCM (Lowest Common Multiple). Take the highest power of all prime factors present in either number. LCM = 2^3 × 3^1 × 5^1 = 8 × 3 × 5 = 120 HCF = 6, LCM = 120 3. (a) If 2n4_6 is a number in base 6. State the largest possible value of n. In base 6, the digits used must be less than 6. The digits in 2n4_6 are 2, n, and 4. For n to be a valid digit in base 6, it must satisfy 0 n < 6. The largest possible integer value for n is 5. 5 3. (b) i) Convert 333_4 to base 10. Step 1: Write the number in expanded form using powers of the base. 333_4 = 3 × 4^2 + 3 × 4^1 + 3 × 4^0 Step 2: Calculate the value. 3 × 16 + 3 × 4 + 3 × 1 48 + 12 + 3 = 63 63_10 3. (b) ii) Convert 501_6 to base 5. Step 1: Convert 501_6 to base 10. 501_6 = 5 × 6^2 + 0 × 6^1 + 1 × 6^0 5 × 36 + 0 × 6 + 1 × 1 180 + 0 + 1 = 181_10 Step 2: Convert 181_10 to base 5. Divide 181 by 5 and record the remainders. 181 ÷ 5 = 36 remainder 1 36 ÷ 5 = 7 remainder 1 7 ÷ 5 = 1 remainder 2 1 ÷ 5 = 0 remainder 1 Read the remainders from bottom to top. 1211_5 4. Given that E = \1-10\, A = \1,4,5,7,8\, B = \2,6,8,10\ a) Represent the sets on a Venn diagram. The universal set E = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10\. A = \1,4,5,7,8\ B = \2,6,8,10\ First, find the intersection A B. A B = \8\ Next, find the elements unique to A: A B = \1,4,5,7\ Next, find the elements unique to B: B A = \2,6,10\ Finally, find the elements outside A and B (in E but not in A or B): E (A B) = \1,2,3,4,5,6,7,8,9,10\ \1,2,4,5,6,7,8,10\ = \3,9\ `mermaid graph TD subgraph E A(A) B(B) A -- 1,4,5,7 --> A B -- 2,6,10 --> B A -- 8 --> B E -- 3,9 --> E end ` (Note: The Venn diagram cannot be fully rendered in text. The description above details the placement of elements.) b) Find i) n(A B) ii) n(B') iii) List the elements in (A B)' i) n(A B) A B = \8\ The number of elements in A B is 1. n(A B) = 1 ii) n(B') B' is the complement of B, meaning all elements in the universal set E that are not in B. E = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10\ B = \2, 6, 8, 10\ B' = \1, 3, 4, 5, 7, 9\ The number of elements in B' is 6. n(B') = 6 iii) List the elements in (A B)' (A B)' is the complement of A B. A B = \8\ (A B)' = E \8\ = \1, 2, 3, 4, 5, 6, 7, 9, 10\ \1, 2, 3, 4, 5, 6, 7, 9, 10\ 5. (a) Find the radius of a circle whose area is 440 cm^2. Step 1: Use the formula for the area of a circle, A = r^2. Given A = 440 cm^2. Use ≈ (22)/(7). 440 = (22)/(7) r^2 Step 2: Solve for r^2. r^2 = (440 × 7)/(22) r^2 = 20 × 7 r^2 = 140 Step 3: Solve for r. r = sqrt(140) r ≈ 11.832 cm Rounding to 3 significant figures: 11.8 cm 5. (b) Find the volume of a cylinder with radius 14 cm and height 40 cm. Step 1: Use the formula for the volume of a cylinder, V = r^2 h. Given r = 14 cm and h = 40 cm. Use ≈ (22)/(7). V = (22)/(7) × (14 cm)^2 × 40 cm V = (22)/(7) × 196 cm^2 × 40 cm Step 2: Calculate the volume. V = 22 × 28 cm^2 × 40 cm V = 616 cm^2 × 40 cm V = 24640 cm^3 24640 cm^3 5. (c) Express 3.5 m^2 to km^2. Step 1: Recall the conversion factors. 1 km = 1000 m 1 km^2 = (1000 m)^2 = 1000000 m^2 = 10^6 m^2 Step 2: Convert 3.5 m^2 to km^2. 3.5 m^2 × 1 km^210^6 m^2 3.5 × 10^-6 km^2 0.0000035 km^2 6. (a) Expand and simplify (x-1)(x+2). Step 1: Use the FOIL method (First, Outer, Inner, Last). (x-1)(x+2) = x × x + x × 2 - 1 × x - 1 × 2 = x^2 + 2x - x - 2 Step 2: Combine like terms. = x^2 + x - 2 x^2 + x - 2 6. (b) Express (3x+5)/(3) - (x-2)/(2) as a single fraction in its simplest form. Step 1: Find a common denominator, which is 6. (2(3x+5))/(2 × 3) - (3(x-2))/(3 × 2) (2(3x+5))/(6) - (3(x-2))/(6) Step 2: Combine the numerators over the common denominator. (2(3x+5) - 3(x-2))/(6) Step 3: Expand the numerators. (6x + 10 - (3x - 6))/(6) Step 4: Distribute the negative sign and combine like terms. (6x + 10 - 3x + 6)/(6) (3x + 16)/(6) (3x+16)/(6) 7. Factorise completely: a) am - an + bm - bn Step 1: Group terms with common factors. (am - an) + (bm - bn) Step 2: Factor out the common factor from each group. a(m - n) + b(m - n) Step 3: Factor out the common binomial factor (m-n). (m - n)(a + b) (m - n)(a + b) b) 49 - y^2 This is a difference of squares, a^2 - b^2 = (a-b)(a+b). Here, a^2 = 49 a = 7 and b^2 = y^2 b = y. (7 - y)(7 + y) (7 - y)(7 + y) c) x^2 - 10x - 24 Step 1: Find two numbers that multiply to -24 and add to -10. The numbers are -12 and 2. -12 × 2 = -24 -12 + 2 = -10 Step 2: Write the quadratic expression in factored form. (x - 12)(x + 2) (x - 12)(x + 2) 8. 210 is shared in the ratio 1:2:4, find the difference between the largest and smallest share. Step 1: Find the total number of parts in the ratio. 1 + 2 + 4 = 7 parts Step 2: Calculate the value of one part. (210)/(7) = 30 Step 3: Calculate the smallest share (1 part) and the largest share (4 parts). Smallest share: 1 × 30 = 30 Largest share: 4 × 30 = 120 Step 4: Find the difference between the largest and smallest share. 120 - 30 = 90 90 9. A map is drawn to scale 1:35000 i) Express the scale in the form 1 cm : n km Step 1: Convert 35000 cm to km. 1 km = 1000 m 1 m = 100 cm So, 1 km = 1000 × 100 cm = 100000 cm. Step 2: Divide 35000 by 100000 to convert to km. 35000 cm = (35000)/(100000) km = 0.35 km So the scale is 1 cm : 0.35 km. 1 cm : 0.35 km ii) Hence or otherwise find the actual area, in km^2, represented by 20 cm^2 on the map. Step 1: Determine the linear scale factor. From part (i), 1 cm on the map represents 0.35 km in actual distance. Step 2: Determine the area scale factor. If 1 cm represents 0.35 km, then 1 cm^2 represents (0.35 km)^2. (0.35 km)^2 = 0.1225 km^2 So, 1 cm^2 on the map represents 0.1225 km^2 in actual area. Step 3: Calculate the actual area represented by 20 cm^2$ on the map. 20 cm^2 × 0.1225 km^2cm^2 = 2.45 km^2 2.45 km^2 What's next? Send 'em!

Express 6.455 correct to 1 d.p, 2 s.f, and nearest whole number. Express 0.52 as a common fraction. Express 102 as a product of its prime factors. Find HCF and LCM of 30 and 24.

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Express 6.455 correct to 1 d.p, 2 s.f, and nearest whole number. Express 0.52 as a common fraction. Express 102 as a product of its prime factors. Find HCF and LCM of 30 and 24.

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