Express 6.455 correct to 1 d.p, 2 s.f, and nearest whole number. Express 0.52 as a common fraction. Express…

Question

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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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 \approx 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 \approx 6.5

iii) nearest whole number
The whole number is 6. The digit after it is 4, so we round down.

6.455 \approx 6

1. (b) Express 0.52 as a common fraction in its simplest form.

Write the decimal as a fraction.
$0.52 = \frac{52}{100}$
Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4.
$\frac{52 \div 4}{100 \div 4} = \frac{13}{25}$
The fraction is already in its simplest form.
$\boxed{\frac{13}{25}}$
2. (a) Express 102 as a product of its prime factors in index form.
Find the prime factors of 102.
$102 \div 2 = 51$ $51 \div 3 = 17$ $17 \div 17 = 1$
The prime factors are 2, 3, and 17.
Write the prime factors in index form.
$102 = 2^1 \times 3^1 \times 17^1$ $\boxed{2 \times 3 \times 17}$
2. (b) Find the HCF and LCM of 30 and 24.
Find the prime factors of each number.
$30 = 2 \times 3 \times 5$ $24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3$
Find the HCF (Highest Common Factor).
The common prime factors are 2 and 3. Take the lowest power of each common factor.
$HCF = 2^1 \times 3^1 = 2 \times 3 = 6$
Find the LCM (Lowest Common Multiple).
Take the highest power of all prime factors present in either number.
$LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120$ $\boxed{HCF = 6, LCM = 120}$
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 $0 \le n < 6$ .
The largest possible integer value for n is 5.
$\boxed{5}$
3. (b) i) Convert $333_4$ to base 10.
Write the number in expanded form using powers of the base.
$333_4 = 3 \times 4^2 + 3 \times 4^1 + 3 \times 4^0$
Calculate the value.
$3 \times 16 + 3 \times 4 + 3 \times 1$ $48 + 12 + 3 = 63$ $\boxed{63_{10}}$
3. (b) ii) Convert $501_6$ to base 5.
Convert $501_6$ to base 10.
$501_6 = 5 \times 6^2 + 0 \times 6^1 + 1 \times 6^0$ $5 \times 36 + 0 \times 6 + 1 \times 1$ $180 + 0 + 1 = 181_{10}$
Convert $181_{10}$ to base 5.
Divide 181 by 5 and record the remainders.
$181 \div 5 = 36 \text{ remainder } 1$ $36 \div 5 = 7 \text{ remainder } 1$ $7 \div 5 = 1 \text{ remainder } 2$ $1 \div 5 = 0 \text{ remainder } 1$
Read the remainders from bottom to top.
$\boxed{1211_5}$
4. Given that $\mathcal{E} = \{1-10\}$ , $A = \{1,4,5,7,8\}$ , $B = \{2,6,8,10\}$

a) Represent the sets on a Venn diagram.
The universal set $\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ .
$A = \{1,4,5,7,8\}$
$B = \{2,6,8,10\}$
First, find the intersection $A \cap B$ .
$A \cap B = \{8\}$
Next, find the elements unique to A: $A \setminus B = \{1,4,5,7\}$
Next, find the elements unique to B: $B \setminus A = \{2,6,10\}$
Finally, find the elements outside A and B (in $\mathcal{E}$ but not in A or B):
$\mathcal{E} \setminus (A \cup B) = \{1,2,3,4,5,6,7,8,9,10\} \setminus \{1,2,4,5,6,7,8,10\} = \{3,9\}$
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) $n(A \cap B)$ ii) $n(B')$ iii) List the elements in $(A \cap B)'$

i) $n(A \cap B)$
$A \cap B = \{8\}$
The number of elements in $A \cap B$ is 1.
$\boxed{n(A \cap B) = 1}$
ii) $n(B')$
$B'$ is the complement of B, meaning all elements in the universal set $\mathcal{E}$ that are not in B.
$\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$
$B = \{2, 6, 8, 10\}$
$B' = \{1, 3, 4, 5, 7, 9\}$
The number of elements in $B'$ is 6.
$\boxed{n(B') = 6}$
iii) List the elements in $(A \cap B)'$
$(A \cap B)'$ is the complement of $A \cap B$ .
$A \cap B = \{8\}$
$(A \cap B)' = \mathcal{E} \setminus \{8\} = \{1, 2, 3, 4, 5, 6, 7, 9, 10\}$
$\boxed{\{1, 2, 3, 4, 5, 6, 7, 9, 10\}}$
5. (a) Find the radius of a circle whose area is $440 \text{ cm}^2$ .
Use the formula for the area of a circle, $A = \pi r^2$ .
Given $A = 440 \text{ cm}^2$ . Use $\pi \approx \frac{22}{7}$ .
$440 = \frac{22}{7} r^2$
Solve for $r^2$ .
$r^2 = \frac{440 \times 7}{22}$ $r^2 = 20 \times 7$ $r^2 = 140$
Solve for $r$ .
$r = \sqrt{140}$ $r \approx 11.832 \text{ cm}$
Rounding to 3 significant figures:
$\boxed{11.8 \text{ cm}}$
5. (b) Find the volume of a cylinder with radius 14 cm and height 40 cm.
Use the formula for the volume of a cylinder, $V = \pi r^2 h$ .
Given $r = 14 \text{ cm}$ and $h = 40 \text{ cm}$ . Use $\pi \approx \frac{22}{7}$ .
$V = \frac{22}{7} \times (14 \text{ cm})^2 \times 40 \text{ cm}$ $V = \frac{22}{7} \times 196 \text{ cm}^2 \times 40 \text{ cm}$
Calculate the volume.
$V = 22 \times 28 \text{ cm}^2 \times 40 \text{ cm}$ $V = 616 \text{ cm}^2 \times 40 \text{ cm}$ $V = 24640 \text{ cm}^3$ $\boxed{24640 \text{ cm}^3}$
5. (c) Express $3.5 \text{ m}^2$ to $\text{km}^2$ .
Recall the conversion factors.
$1 \text{ km} = 1000 \text{ m}$
$1 \text{ km}^2 = (1000 \text{ m})^2 = 1000000 \text{ m}^2 = 10^6 \text{ m}^2$
Convert $3.5 \text{ m}^2$ to $\text{km}^2$ .
$3.5 \text{ m}^2 \times \frac{1 \text{ km}^2}{10^6 \text{ m}^2}$ $3.5 \times 10^{-6} \text{ km}^2$ $\boxed{0.0000035 \text{ km}^2}$
6. (a) Expand and simplify $(x-1)(x+2)$ .
Use the FOIL method (First, Outer, Inner, Last).
$(x-1)(x+2) = x \times x + x \times 2 - 1 \times x - 1 \times 2$ $= x^2 + 2x - x - 2$
Combine like terms.
$= x^2 + x - 2$ $\boxed{x^2 + x - 2}$
6. (b) Express $\frac{3x+5}{3} - \frac{x-2}{2}$ as a single fraction in its simplest form.
Find a common denominator, which is 6.
$\frac{2(3x+5)}{2 \times 3} - \frac{3(x-2)}{3 \times 2}$ $\frac{2(3x+5)}{6} - \frac{3(x-2)}{6}$
Combine the numerators over the common denominator.
$\frac{2(3x+5) - 3(x-2)}{6}$
Expand the numerators.
$\frac{6x + 10 - (3x - 6)}{6}$
Distribute the negative sign and combine like terms.
$\frac{6x + 10 - 3x + 6}{6}$ $\frac{3x + 16}{6}$ $\boxed{\frac{3x+16}{6}}$
7. Factorise completely:

a) $am - an + bm - bn$
Group terms with common factors.
$(am - an) + (bm - bn)$
Factor out the common factor from each group.
$a(m - n) + b(m - n)$
Factor out the common binomial factor $(m-n)$ .
$(m - n)(a + b)$ $\boxed{(m - n)(a + b)}$
b) $49 - y^2$
This is a difference of squares, $a^2 - b^2 = (a-b)(a+b)$ .
Here, $a^2 = 49 \implies a = 7$ and $b^2 = y^2 \implies b = y$ .
$(7 - y)(7 + y)$ $\boxed{(7 - y)(7 + y)}$
c) $x^2 - 10x - 24$
Find two numbers that multiply to -24 and add to -10.
The numbers are -12 and 2.
$-12 \times 2 = -24$ $-12 + 2 = -10$
Write the quadratic expression in factored form.
$(x - 12)(x + 2)$ $\boxed{(x - 12)(x + 2)}$
8. $210 is shared in the ratio 1:2:4, find the difference between the largest and smallest share.
Find the total number of parts in the ratio.
$1 + 2 + 4 = 7 \text{ parts}$
Calculate the value of one part.
$\frac{210}{7} = 30$
Calculate the smallest share (1 part) and the largest share (4 parts).
Smallest share: $1 \times 30 = 30$
Largest share: $4 \times 30 = 120$
Find the difference between the largest and smallest share.
$120 - 30 = 90$ $\boxed{90}$
9. A map is drawn to scale 1:35000

i) Express the scale in the form 1 cm : n km
Convert 35000 cm to km.
$1 \text{ km} = 1000 \text{ m}$
$1 \text{ m} = 100 \text{ cm}$
So, $1 \text{ km} = 1000 \times 100 \text{ cm} = 100000 \text{ cm}$ .
Divide 35000 by 100000 to convert to km.
$35000 \text{ cm} = \frac{35000}{100000} \text{ km} = 0.35 \text{ km}$
So the scale is $1 \text{ cm} : 0.35 \text{ km}$ .
$\boxed{1 \text{ cm} : 0.35 \text{ km}}$
ii) Hence or otherwise find the actual area, in $\text{km}^2$ , represented by $20 \text{ cm}^2$ on the map.
Determine the linear scale factor.
From part (i), $1 \text{ cm}$ on the map represents $0.35 \text{ km}$ in actual distance.
Determine the area scale factor.
If $1 \text{ cm}$ represents $0.35 \text{ km}$ , then $1 \text{ cm}^2$ represents $(0.35 \text{ km})^2$ .
$(0.35 \text{ km})^2 = 0.1225 \text{ km}^2$
So, $1 \text{ cm}^2$ on the map represents $0.1225 \text{ km}^2$ in actual area.
Calculate the actual area represented by $20 \text{ cm}^2$ on the map.
$20 \text{ cm}^2 \times 0.1225 \frac{\text{km}^2}{\text{cm}^2} = 2.45 \text{ km}^2$ $\boxed{2.45 \text{ km}^2}$

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.

Step-by-step solution

Write the decimal as a fraction.

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4.

Find the prime factors of 102.

Write the prime factors in index form.

Find the prime factors of each number.

Find the HCF (Highest Common Factor).

Find the LCM (Lowest Common Multiple).

Write the number in expanded form using powers of the base.

Calculate the value.

Convert 5016501_65016​ to base 10.

Convert 18110181_{10}18110​ to base 5.

Use the formula for the area of a circle, A=πr2A = \pi r^2A=πr2.

Solve for r2r^2r2.

Solve for rrr.

Use the formula for the volume of a cylinder, V=πr2hV = \pi r^2 hV=πr2h.

Calculate the volume.

Recall the conversion factors.

Convert 3.5 m23.5 \text{ m}^23.5 m2 to km2\text{km}^2km2.

Use the FOIL method (First, Outer, Inner, Last).

Combine like terms.

Find a common denominator, which is 6.

Combine the numerators over the common denominator.

Expand the numerators.

Distribute the negative sign and combine like terms.

Group terms with common factors.

Factor out the common factor from each group.

Factor out the common binomial factor (m−n)(m-n)(m−n).

Find two numbers that multiply to -24 and add to -10.

Write the quadratic expression in factored form.

Find the total number of parts in the ratio.

Calculate the value of one part.

Calculate the smallest share (1 part) and the largest share (4 parts).

Find the difference between the largest and smallest share.

Convert 35000 cm to km.

Divide 35000 by 100000 to convert to km.

Determine the linear scale factor.

Determine the area scale factor.

Calculate the actual area represented by 20 cm220 \text{ cm}^220 cm2 on the map.

Still stuck on this one?

More Mathematics Questions

Convert $501_6$ to base 10.

Convert $181_{10}$ to base 5.

Use the formula for the area of a circle, $A = \pi r^2$ .

Solve for $r^2$ .

Solve for $r$ .

Use the formula for the volume of a cylinder, $V = \pi r^2 h$ .

Convert $3.5 \text{ m}^2$ to $\text{km}^2$ .

Factor out the common binomial factor $(m-n)$ .

Calculate the actual area represented by $20 \text{ cm}^2$ on the map.