# KS LearningMaths Bronze Certificate Answers

### Maths Bronze Certificate Worksheets Answers

The Maths Bronze Certificate is the first in a series of four certificates produced jointly by KS Learning and Porridge and Rice. The four certificates - Bronze, Silver, Gold, and Platinum - cover the AQA IGCSE Maths syllabus in four convenient sections and are designed to be taken in order before the IGCSE.

• Syllabus

• B01 Fractions, decimals and rounding

• B02 Ratio and percentages

• B03 Powers and roots

• B04 Working with algebra

• B05 Algebraic Equations
• Syllabus

• B06 Graphs of straight lines

• B07 Simultaneous equations

• B08 Revision Worksheet 1

• B09 Revision Worksheet 2

• B10 Revision Worksheet 3
• B01 Fractions, decimals and rounding

• Content
• B01.1 Equivalent fractions
• B01.2 Multiplying and dividing with fractions
• B01.3 Decimals and fractions
• B01.4 Rounding and approximation
• B01.6 Upper and lower bounds

• 1) Write each fraction in its simplest form.
1. $$\frac{12}{48}$$ = $$\frac{1}{4}$$
2. $$\frac{36}{84}$$ = $$\frac{3}{7}$$
3. $$\frac{28}{63}$$ = $$\frac{4}{9}$$
4. $$\frac{8}{36}$$ = $$\frac{2}{9}$$

• 2) Write 2 equivalent fractions for each fraction.
1. $$\frac{2}{5}$$ = $$\frac{4}{10}, \frac{6}{15}$$
2. $$\frac{4}{8}$$ = $$\frac{1}{2}, \frac{2}{4}$$
3. $$\frac{2}{3}$$ = $$\frac{4}{6}, \frac{6}{9}$$
4. $$\frac{8}{9}$$ = $$\frac{16}{18}, \frac{24}{27}$$

• 3) Write each set of fractions from smallest to largest.
1. $$\frac{2}{5},\ \frac{1}{3},\ \frac{3}{7},\ \frac{4}{9}$$
= $$\frac{1}{3},\ \frac{2}{5},\ \frac{3}{7},\ \frac{4}{9}$$
2. $$\frac{11}{20},\ \frac{1}{2},\ \frac{3}{5},\ \frac{5}{8}$$
= $$\frac{1}{2},\ \frac{11}{20},\ \frac{3}{5},\ \frac{5}{8}$$
3. $$\frac{13}{15},\ \frac{5}{6},\ \frac{2}{3},\ \frac{4}{5}$$
= $$\frac{2}{3},\ \frac{4}{5},\ \frac{5}{6},\ \frac{13}{15}$$

• 4) Write as top heavy fractions
1. $$2\frac{1}{4}$$ = $$\frac{9}{4}$$
2. $$3\frac{5}{8}$$ = $$\frac{29}{8}$$
3. $$5\frac{2}{3}$$ = $$\frac{17}{3}$$
4. $$2\frac{4}{9}$$ = $$\frac{22}{9}$$

• 5) Work out
1. $$\frac{2}{5} +\ \frac{1}{3}$$ = $$\frac{11}{15}$$
2. $$3\frac{5}{8} +\ 1\frac{2}{3}$$ = $$5\frac{7}{24}$$
3. $$\frac{5}{9} -\ \frac{1}{5}$$ = $$\frac{16}{45}$$
4. $$2\frac{3}{5} -\ 1\frac{7}{8}$$ = $$\frac{29}{40}$$

• 6) Work out
1. $$\frac{2}{5} \times \frac{3}{4}$$ = $$\frac{3}{10}$$
2. $$\frac{9}{14} \div \frac{1}{3}$$ = $$1\frac{13}{14}$$
3. $$3\frac{3}{5} \times 2\frac{5}{6}$$ = $$10\frac{1}{5}$$
4. $$2\frac{9}{14} \div 1\frac{1}{3}$$ = $$1\frac{55}{56}$$
5. $$5 \times 1\frac{2}{5}$$ = $$7$$
6. $$\frac{4}{7} \div 6$$ = $$\frac{2}{21}$$

• 7) Write the decimals as exact fractions
1. 0.24 = $$\frac{6}{25}$$
2. 0.0625 = $$\frac{1}{16}$$
3. 0.$$\dot{6}$$ = $$\frac{2}{3}$$
4. 0.$$\dot{2}\dot{6}$$ = $$\frac{26}{99}$$
5. 0.0$$\dot{5}\dot{4}$$ = $$\frac{3}{55}$$
6. 1.2$$\dot{7}$$ = $$1\frac{5}{18}$$
7. 0.$$\dot{6}{3}\dot{9}$$ = $$\frac{71}{111}$$

• 8) Convert the decimals to fractions to find out
1. $$0.\dot{2}\dot{7} = \frac{3}{11}\ = \frac{75}{275}$$ and 0.28 is $$\frac{7}{25}\ = \frac{77}{275}$$ so 0.28 is greater than $$0.\dot{2}\dot{7}$$
2. if 0.$$\dot{7} = \frac{7}{9} = \frac{77}{99}$$
0.$$\dot{3}\dot{5} = \frac{35}{99}$$ and $$2 \times \frac{35}{99}\ = \frac{70}{99}$$
0.$$\dot{7}$$ is thus not double 0.$$\dot{3}\dot{5}$$

• 9) Write the fractions as decimals
1. $$\frac{5}{8}$$
= $$\ 0.625$$
2. $$\frac{4}{9}$$
= $$0.\dot{4}$$
3. $$3\frac{2}{11}$$
= $$3.\dot{1}\dot{8}$$

• 10) Round the numbers as indicated
1. 2.3162 (2dp)
= 2.32 (2dp)
2. 1.31671 (2sf)
= 1.3 (2sf)
3. 321.376 (1dp)
= 321.4 (1dp)
4. 38.627 (3sf)
= 38.6 (3sf)
5. 0.000376 (1sf)
= 0.0004 (1sf)
6. 0.000288 (3dp)
= 0.000 (3dp)

• 11) Approximate
1. $$\frac{2.13 \times 5.31}{0.789}$$
= $$\frac{2 \times 5}{1}$$ = 10
2. $$\frac{38.9 \times 107}{168 -\ 29}$$
= $$\frac{40 \times 100}{200 -\ 30} = \frac{4000}{170}$$ = 20
3. $$\frac{7.8 \times (9.81 -\ 6.35)}{5.5 -\ 2.1}$$
= $$\frac{8 \times (10 -\ 6)}{6 -\ 2} = \frac{8 \times 4}{4} = 8$$

• 12) Work out the largest and smallest value in each case.
1. The perimeter of a square with a side of 8cm to the nearest cm
smallest is 7.5 x 4 = 30cm
largest is 8.5 x 4 = 34cm
2. The area of rectangle with sides of 133mm and 24mm to the nearest mm
smallest is 132.5 x 23.5 = 3113.75mm2
largest is 133.5 x 24.5 = 3270.75mm2
3. The circumference of a circle with a radius of 12cm to the nearest cm.
23π or 69
25π or 75
4. The difference between the attendees at a game where 24 400 attend one week and 25 000 the next week, correct to the nearest hundred.
$$24 \text{ } 950 -\ 24 \text{ } 450 = 500$$
$$25 \text{ } 050 -\ 24 \text{ } 350 = 700$$

• 13) Use a calculator to work out providing answers to 2sf.
1. $$(16.3 -\ 3.6 \times 2.1) \times 3.4$$ = 30
2. $$\frac{2.4 +\ 3 \times 2.34}{11.1 -\ 2.1^2}$$ = 1.4
3. $$\frac{\sqrt{7.4 +\ 3.34^2}}{4.2^2 -\ 3.2^2}$$ = 0.58

• 14) A rectangle has a length of 8cm and 5cm to the nearest centimetre.
1. Calculate the upper and lower bound for the perimeter of the rectangle.
upper bound $$2 \times 8.5 +\ 2 \times 5.5$$ = 28cm
lower bound $$2 \times 7.5 +\ 2 \times 4.5$$ = 24cm
2. Calculate the upper and lower bound for the area of the rectangle.
upper bound $$8.5 \times 5.5$$ = 46.75cm2
lower bound $$7.5 \times 4.5$$ = 33.75cm2

• 15) A car travels for 1200m for 1.3 hours to 2 significant figures.
1. Work out the longest and shortest possible distance travelled.
longest distance = 1250m
shortest distance = 1150m
2. Calculate the upper and lower bound for the speed of the car.
upper speed = $$\frac{1250}{1.25}$$ = 1000 m.h-1 = 0.28 m.s-1
lower speed = $$\frac{1150}{1.35}$$ = 851.851 m.h-1 = 0.24 m.s-1
• B02 Ratio and percentages

• Content
• B02.1 Working with ratios
• B02.2 Simple percentages
• B02.3 Percentage increase and decrease
• B02.4 Reverse percentage problems
• B02.5 Compound interest

• 1) Express each of the ratios in their simplist form.
1. 27 : 63
3:7
2. 16 : 20 : 28
4:5:7
3. 144 : 48
3:1
4. 24 : 8 : 44
6:2:11

• 2) Share in the ratio given
1. 160 metres in 1 : 5
26.7m : 133.3m
2. £450 in 3 : 2 : 4
£150: £100: £200
3. $144 in 3 : 1 : 4$54 : $18 :$72
4. 560kg in 4 : 5 : 1
224kg: 280kg : 56kg

• 3) Lou, Sean and Aaron win a prize of £540 which they share in the ratio of 5:1:3.
1. How much does each person get?
Lou: £300 Sean: £60 Aaron: £180
2. What percentage of the prize do Lou and Sean get combined?
Lou & Sean have £360
$$\frac{360}{540}$$ = 66.$$\dot{6}$$%
3. Express Sean's share of the prize as a fraction?
$$\frac{60}{540}$$ = $$\frac{1}{9}$$
4. Express Sean and Aaron's combined share of the prize as a fraction?
Sam & Aaron have £240
$$\frac{240}{540}$$ = $$\frac{4}{9}$$

• 4)Below is the recipe for making 10 scones. How much of each ingredient is needed for 24 scones?
• 225g self raising flour
• 1/4 teaspoon of salt
• 55g butter
• 25g caster sugar
• 150ml milk

• 540g self raising flour
• 3/5 teaspoon of salt
• 132g butter
• 60g caster sugar
• 360ml milk

• 5) Write the percentages as fractions, in their lowest terms.
1. 35%
$$\frac{7}{20}$$
2. 56%
$$\frac{14}{25}$$
3. 15%
$$\frac{3}{20}$$
4. 84%
$$\frac{21}{25}$$

• 6) Write the fractions as percentages to an appropriate number of decimal places.
1. $$\frac{5}{8}$$
62.5%
2. $$\frac{2}{5}$$
40%
3. $$\frac{3}{40}$$
7.5%
4. $$\frac{12}{80}$$
15%

• 7) Lactny High School has 820 pupils.
1. Girls are 45% of the school. How many girls are there?
369 girls
2. 18 years old are 5% of the school. How many 18 year olds are there?
41 18 year-olds
3. What percentage of the school is absent when 11 pupils miss school?
1.34%

• 8) Thomas buys 10 mugs for 80p each and 24 teaspoons for 12p each. He sells the mugs for £1.50 each and the teaspoons for 50p each.
1. How much money does he make in total?
£16.12
2. What is his percentage profit on the sale of the teaspoons?
317% profit
3. What fraction of the sale price is profit on the mugs?
$$\frac{7}{15}$$

• 9) Work out the new prices when a store adjusts the price of its stock as below.
1. a t-shirt for £7.85 is increased by 22%.
£9.58 (3sf)
2. a tie for £11.95 is reduced by 30%.
£8.37 (3sf)
3. a pair of socks for £3.50 is increased by 12%.
£3.92
4. a pair of shorts for £16.50 is reduced by 8%.
£15.18

• 10) The value of a car is reduced in price by 9.5% each year. If the initial value of the car is £16,900, what is the value of the car after
1. 1 year
£15,294.50
2. 2 years
£13,841.52
3. 5 years
£10,259.58
4. 9 years
£6,882.15

• 11) If the items in a camera store below were increased or decreased as shown, find the original price.
1. a UV lens is £4.50 after being reduced by 20%.
£5.63
2. a disposable camera is £7.20 after being increased by 18%.
£6.10
3. a camera bag is £38 after being increased by 12%.
£33.93
4. a zoom lens is £124.12 after being decreased by 16%.
£147.76

• 12) Joseph invests £120 at a rate of 3% compound per annum. How much money does he have after
1. 2 years?
£127.31
2. 3 years?
£131.13
3. 8 years?
£152.01

• 13) Noah has a choice of two savings accounts, one which offers 3% simple interest and the other 2.6% compound. Which is the better choice if he invests £400 for the following number of years?
1. 1 year
simple £412.00 and compound £410.40
simple interest is the better choice
2. 5 years
simple £460.00 and compound £452.56
simple interest is the better choice
3. 12 years
simple £544.00 and compound £544.29
compound interest is the better choice
4. 25 years
simple £700.00 and compound £759.88
compound interest is the better choice

• 14) Daniel invested £240 at a compound interest rate of 5% per annum until he has £306.31 in his investment.
1. How long did Daniel leave the money invested?
5 Years
2. How much will he have if he leaves his money invested for 8 years?
£354.59
• B03 Powers and roots

• Content
• B03.1 Basic powers and roots
• B03.2 Higher powers and roots
• B03.3 Fractional (rational) indices
• B03.4 Negative powers
• B03.5 The laws of indices
• B03.6 Standard index forms
• B03.7 Calculating with numbers in index form
• B03.8 Factors, multiples and primes
• B03.9 Highest Common Factor, HCF
• B03.10 Lowest Common Multiple, LCM

• 1) Without a calculator, work out
1. $$2^5$$ = 32
2. $$-4^2$$ = $$-16$$
3. $$(-5)^2$$ = 25
4. $$(-2)^3$$ = -8
5. $$\sqrt{36}$$ = +6 and -6

• 2) Work out
1. $$16 ^ \frac{1}{4}$$ = 2 and -2
2. $$8 ^ \frac{2}{3}$$ = 4
3. $$81 ^ \frac{3}{4}$$ = 27
4. $$6 ^ {-\ 2}$$ = $$\frac{1}{36}$$
5. $$\left( \frac{9}{64} \right) ^ {-\ \frac{1}{2}}$$ = $$\frac{8}{3}$$
6. $$\left( \frac{8}{125} \right) ^ {-\ \frac{2}{3}}$$ = $$\frac{25}{4}$$

• 3) Write the answer as a number to a single power
1. $$2^5 \times 2^3$$ = $$2^8$$
2. $$3^7 \div 3^3$$ = $$3^4$$
3. $$3^4 \times 3^{-\ 2}$$ = $$3^2$$
4. $$(5^3)^{2}$$ = $$5^6$$
5. $$5^\frac{1}{4} \times 5^\frac{2}{3}$$ = $$5^\frac{11}{12}$$
6. $$4^ {-\ \frac{2}{3}} \times 4^\frac{1}{3}$$ = $$4^ {-\ \frac{1}{3}}$$
7. $$7^\frac{2}{5} \div 7^\frac{1}{3}$$ = $$7^ { \frac{1}{15}}$$
8. $$(5^3)^{\frac{2}{3}}$$ = $$5^2$$
9. $$(6^0)^3 \times 6 ^ {\frac{3}{4}}$$ = $$6 ^ {\frac{3}{4}}$$
10. $$(((5^ {-\ 3}) ^2) ^ \frac{1}{4})$$ = $$5^ { \frac{-\ 3}{2}}$$

• 4) Write the following numbers in standard form
1. $$540 \,000$$ = $$5.4 \times 10^5$$
2. $$841$$ = $$8.41 \times 10^2$$
3. $$12.7$$ = $$1.27 \times 10^1$$
4. $$0.0037$$ = $$3.7 \times 10^{-\ 3}$$
5. $$0.00000000805$$ = $$8.05 \times 10^{-\ 9}$$

• 5) Write the numbers as ordinary numbers
1. $$4.5 \times 10^7$$ = 45,000,000
2. $$2.15 \times 10^{-\ 4}$$ = 0.000215
3. $$1.003 \times 10^5$$ = 100,300
4. $$3 700 \times 10^{-\ 8}$$ = 0.000037

• 6) Express answers in standard form
1. $$2.7 \times 10^6 +\ 3.9 \times 10^7$$
= $$4.17 \times 10^7$$
2. $$1.8 \times 10^4 -\ 9.7 \times 10^3$$
= $$8.3 \times 10^3$$
3. $$(3.4 \times 10^8) \times (2.1 \times 10^2)$$
= $$7.14 \times 10^{10}$$
4. $$5.3 \times 10^5 -\ 4.7 \times 10^6$$
= $$-4.17 \times 10^6$$
5. $$(6.4 \times 10^3) \div (1.6 \times 10^9)$$
= $$4.0 \times 10^{-6}$$
6. $$4.1 \times 10^4 +\ 5.2 \times 10^3$$
= $$4.62 \times 10^4$$
7. $$(9.6 \times 10^7) \div (6.0 \times 10^{-\ 4})$$
= $$1.6 \times 10^{11}$$
8. $$(8.1 \times 10^{-\ 9}) \times (8.3 \times 10^8)$$
= $$6.723 \times 10^{0}$$

• 7) Write the numbers as the product of their primes
1. 180
= $$2^2 \times 3^2 \times 5$$
2. 720
= $$2^4 \times 3^2 \times 5^1$$
3. 252
= $$2^2 \times 3^2 \times 7$$
4. 1485
= $$5 \times 3^3 \times 11$$

• 8) Find the HCF for each pair of numbers
1. 72 and 108
36
2. 90 and 140
10
3. 120 and 195
15
4. 48 and 360
24

• 9) Find the HCF and LCM for each pair of numbers
1. 28 and 42
HCF = 14 and LCM = 84
2. 33 and 55
HCF = 11 and LCM = 165
3. 25 and 40
HCF = 5 and LCM = 200
4. 240 and 96
HCF = 48 and LCM = 480

• 10) Write in the form $$(2^a \times 3^b \times 5^c)$$
1. 48
= $$2^4 \times 3^1 \times 5^0$$
2. 90
= $$2^1 \times 3^2 \times 5^1$$
3. 96
= $$2^5 \times 3^1 \times 5^0$$

• 11) Use prime factors to find the following
1. $$\sqrt{576}$$
= $$576 = 2^6 \times 3^2$$
thus $$\sqrt{576} = 2^3 \times 3^1 = 24$$
2. $$\sqrt[3]{512}$$
= $$512 = 2^9$$
thus $$\sqrt[3]{512} = 2^3 = 8$$
3. $$\sqrt[5]{7776}$$
= $$7776 = 2^5 \times 3^5$$
thus $$\sqrt[5]{7776} = 2^1 \times 3^1 = 6$$
4. $$\sqrt[3]{27000}$$
= $$27000 = 2^3 \times 3^3 \times 5^3$$
thus $$\sqrt[3]{27000} = 2^1 \times 3^1 \times 5^1 = 30$$
5. $$\sqrt{1024}$$
= $$1024 = 2^{10}$$
thus $$\sqrt{1024} = 2^5 = 32$$

• 12) Given that $$y^2 = \frac {(a^3 \times b)}{(a +\ b)}$$, find y when
1. $$a = 3 \times 10^8$$ and $$b = 2 \times 10^7$$
= $$1.3 \times 10^{12}$$
2. $$a = 2 \times 10^5$$ and $$b = 4 \times 10^{-3}$$
= $$1.26 \times 10^{4}$$
3. $$a = 1.6 \times 10^6$$ and $$b = 1.8 \times 10^4$$
= $$2.13 \times 10^{8}$$

• 13) Jack decides to call his parents every 14 days and his sister every 10 days. If he spoke to both today, how many days before he speaks to both again.
= 70 days

• 14) Tim, Tom, and Tam exercise together. Tim repeats his routine every 24 minutes, Tom repeats his routine every 36 minutes, and Tam repeats his routine every 45 minutes. All three start their first routine at 1pm and continue until 8pm.
1. How many times do Tim and Tam start their routines together?
= Tim and Tam start at the same time every 360 minutes or 6 hours so only twice between 1pm and 8pm, at 1pm and 7pm
2. What is the latest time that Tim, Tom, and Tam start their routines together?
= Tim, Tom and Tam start their routines together every 360 minutes or 6 hours so only twice, so 7pm
3. What are the times that Tom and Tam start their routines at the same time?
= Tom and Tam start their routines together every 180 minutes or 3 hours so only twice, so 1pm, 4pm and 7pm
• B04 Working with algebra

• Content
• B04.1 Substituting numbers into formulae and expressions
• B04.2 Working with indices
• B04.3 Expanding brackets
• B04.4 Multiplying brackets together
• B04.5 Factorising - common factors
• B04.6 Factorising - quadratic expressions
• B04.7 Factorising - harder quadratic expressions
• B04.8 Factorising - difference of two squares
• B04.9 Generating formulae
• B04.10 Changing the subject of a formula

• 1) If $$a = 5$$ and $$b = - 3$$, find the value of
1. $$3a +\ 2b$$
= $$3 \times 5 +\ 2 \times (-3) = 9$$
2. $$a -\ 5b$$
= $$5 -\ 5 \times (-3) = 20$$
3. $$a^2 -\ 2ab$$
= $$5^2 -\ 2 \times (5) \times (-3) = 55$$
4. $$(-\ a +\ 4b)^2$$
= $$(-\ 5 +\ 4 \times (-3))^2 = 289$$
5. $$( ab -\ 3b)(-2a)^2$$
= $$( 5 \times (-3) -\ 3 \times (-3))(-2 \times 5)^2 = - 600$$
6. $$\frac{2a + b}{a^3 -\ b}$$
= $$\frac{2 \times 5 + (-3)}{5^3 -\ (-3)} = \frac{7}{128}$$
7. $$(5 -\ b)\sqrt{5a}$$
= $$(5 -\ (-3))\sqrt{5 \times 5} = 40$$

• 2) If $$x = 4$$, $$y = 6$$ and $$z = - 3$$, find the value of
1. $$2x +\ 5y -\ 3z$$
= $$47$$
2. $$x -\ y^2 \times z$$
= $$112$$
3. $$xy -\ 5z -\ yz$$
= $$57$$
4. $$z^3 -\ 2(xy +\ 11)$$
= $$-97$$
5. $$(y -\ y^2) \times z \div x$$
= $$22 { \frac{1}{2}}$$
6. $$\frac{1}{2}(-\ z +\ 3xy)$$
= $$37 { \frac{1}{2}}$$
7. $$z (-\ xy)\times \sqrt{xy}$$
= $$144\sqrt{6}$$

• 3) Simplify the expression
1. $$3x^0$$
2. $$2x^2 \times 5x^3$$
3. $$4x^{-2} \div 20x^7$$
4. $$5x^4 \times x^3 \times 8x^{-2}$$
5. $$(3x^4)^3$$
6. $$4x \sqrt{x^3} \times x^{-2}$$
7. $$( \frac{1}{2} x^{-2}) ^{-5}$$
8. $$9x^7 \times {-7}x^6$$

• 4) Expand
1. $$x(3x +\ 2)$$
2. $$2x(x^2 -\ 5x +\ 3)$$
3. $$-3(6x -\ 1) -\ 5(-x -\ 3)$$
4. $$4x(2x +\ 3) +\ 5x(5x -\ 7)$$
5. $$(6x +\ 1)(5x -\ 3)$$
6. $$(x -\ 2)(3x -\ 1)(2x +\ 3)$$
7. $$(4x -\ 1)^2(2x +\ 3)$$
8. $$5(3x -\ 4)^3$$

• 5) Factorise
1. $$ab^2 +\ ab$$
2. $$pq^2 -\ p^3q -\ p^3q^2$$
3. $$6xy^2 +\ 18$$
4. $$x^2 -\ x -\ 2$$
5. $$8y^3 -\ 6x^2$$
6. $$x^2 +\ 3x -\ 4$$
7. $$x^2 +\ x -\ 6$$
8. $$4x^2 -\ 1$$

• 6) Factorise
1. $$2x^2 +\ 3x +\ 1$$
2. $$5x^2 -\ 9x -\ 2$$
3. $$2x^2 -\ 9x +\ 9$$
4. $$2x^2 +\ 11x -\ 6$$
5. $$3x^2 -\ 8x +\ 4$$
6. $$7y^2 -\ 700$$
7. $$4x^2 +\ 10x -\ 6$$
8. $$2a^2 +\ a -\ 10$$

• 7) Each week, Toby pays £x for cereal and £y for milk.
1. Write a formula for the total amount T that he spends each week.
2. Write a formula for the amount T5 that he spends over 5 weeks.
3. If cereal costs 3 times the amount of milk, write a formula for T in terms of x only.

• 8) Anna makes earrings from a sheet of silver metal. She cuts two circles with radius r1 for the earrings, and from each circle she cuts out a smaller circle with radius r2.
1. Write a formula for the area of silver cut out for the holes in the earrings.
2. Write a formula for the area of silver used to make the earrings.
3. If r1 is three times the size of r2, write a formula for the area of silver used in terms of r1.
4. If 24cm2 of silver is used, find r1 and r2.

• 9) Tom is 5 years older than Tam but 1 year younger than Tim.
1. Write a formula for each of Tom, Tam, and Tim's ages if Tom's age is n.
2. If the sum of the boys' ages is 35, find each of their ages.

• 10) Change the subject of the formula
1. given $$y = mx -\ c$$, rearrange for $$x$$
2. given $$y = mx -\ c$$, rearrange for $$m$$
3. given $$v = u +\ at$$, rearrange for $$u$$
4. given $$v = u +\ at$$, rearrange for $$a$$
5. given $$p = 3x +\ xt$$, rearrange for $$t$$
6. given $$p = 3x +\ xt$$, rearrange for $$x$$

• 11) Change the subject of the formula
1. given $$r = mx^2 -\ ab$$, rearrange for $$b$$
2. given $$r = mx^2 -\ ab$$, rearrange for $$x$$
3. given $$y = \frac{lm}{a \sqrt{t}}$$, rearrange for $$l$$
4. given $$y = \frac{lm}{a \sqrt{t}}$$, rearrange for $$t$$
5. given $$v = ut +\ \frac{1}{2} at^2$$, rearrange for $$a$$
6. given $$v = ut +\ \frac{1}{2} at^2$$, rearrange for $$t$$
• B05 Algebraic Equations

• Content
• B05.1 Expressions, equations and identities
• B05.2 Simple equations
• B05.3 Harder linear equations
• B05.4 Equations with brackets
• B05.5 Equations with fractional coefficients

• 1) Solve each equation for $$x$$
1. $$3x =\ 12$$
2. $$2x -\ 5=\ 9$$
3. $$\frac{x}{3} =\ 5$$
4. $$2x +\ 4 =\ 10$$
5. $$\frac{1}{4}x =\ 6$$
6. $$-\ 2x -\ 3 =\ 7$$
7. $$\frac{x}{5} =\ \frac{3}{4}$$
8. $$8 =\ \frac{3x}{4} +\ 2$$

• 2) Solve each of the following algebraic equations
1. $$2x +\ 6 =\ x +\ 5$$
2. $$-x + 5 = -11x$$
3. $$9y - 3 =\ 6 -\ 3y$$
4. $$7 +\ 2t =\ 11 -\ 6t$$
5. $$-x +\ 5 =\ 3x -\ 1$$
6. $$3a -\ 7 = a +\ 1$$
7. $$3 -\ 3t = 3t -\ 6$$
8. $$2x +\ 3 -\ 5x = 20$$

• 3) Solve for the variable in each equation
1. $$3(p -\ 2) =\ 12p +\ 1$$
2. $$2(4 - x) =\ 3(2x -\ 5)$$
3. $$-\ (3y -\ 2) =\ 2(8 - y)$$
4. $$5(2z -\ 1) =\ 3(1 -\ 4z)$$
5. $$8z +\ 3 = -\ (2 -\ 3z)$$
6. $$2(3x +\ 2) -1 = (4 - 4x)$$
7. $$6 + 2(3m - 1) = 5(1 -\ m)$$
8. $$\frac{3}{4} (8 +\ x) = (x -\ 5)$$

• 4) Find the value of $$x$$ for each equation
1. $$3(2x + 1) -\ (1 -\ 2x) =\ 3$$
2. $$(2x -\ 3) -\ 1 =\ 2(1 -\ 2x) +\ (2x +\ 7)$$
3. $$4(3z +\ 2) -\ 3(1 -\ z) =\ 3(1 +\ z)$$
4. $$3(7x -\ 3) +\ 4(2 +\ 2x) =\ -\ 6$$
5. $$2(5 +\ 2y) +\ 5(y -\ 2) =\ -\ 3( y +\ 1 )$$
6. $$2(2t + 8) =\ 9 (t -\ 4) +\ (t -\ 2)$$
7. $$-\ 3(2x + 3) -\ 5(2 -\ 3x) =\ -\ x$$
8. $$6(7 -\ 2y) -\ 4 =\ (1 -\ 2x) +\ 10(3x - 4)$$

• 5)
1. $$2(\frac{1}{4} -\ x) = (x +\ 2)$$
2. $$\frac{2}{5}(5 +\ 3x) = (\frac{x}{5} -\ 1)$$
3. $$-\ 1(\frac{2x}{3} -\ 2) = (x +\ 2)$$
4. $$3(2x -\ \frac{1}{3}) = \frac{1}{2}( 3 -\ x)$$
5. $$(\frac{1}{3} -\ x) = (3 -\ 2x)$$
6. $$5(2 +\ \frac{x}{2}) = (x +\ 4)$$
7. $$-(\frac{x}{4} -\ 6)= 2\frac{1}{3}(4x +\ 3)$$
8. $$\frac{3}{2}(\frac{5}{6} -\ x) = (4 -\ \frac{x}{3})$$

• 6) Write an equation for each problem then solve.
1. Pree is twice the age of Pru, and together their ages add up to 21. Find their ages.
2. Harry ate half his sweets on Monday, then two more on Tuesday after which he had 6 left. Find out how many sweets he had.
3. Tag and Tug like to play pick-up-sticks after school. Tag wins two and a half times as many as Tug out of the 14 games they play. Work out how many games each boy wins.
4. Lim Ho has 5 times as many sweets as Suzie D and double the sum of their sweets is 144. Find how many sweets does each girl have.
5. Max and George own the same number of books. When Max buys three more books and George gives away three books, Max has four times as many books as George. Find out how many books they each had to start with.
6. Mark has three times as many trophies as his two friends, Mick and Mac, combined. Mick has half as many as Mac. They have 24 trophies between them. Find out how many trophy each boy has.
7. After Tag lost £2 and Fat Sami lost £3, Fat Sami still had twice as much money as Tag. How much money did each have to start with?
8. There are red, green, and yellow boxes on a shelf. The product of the number of red and green boxes is 10 more than double the number of yellow boxes, and there are 2 more red boxes than green. Find out how many boxes there are of each colour.
• B06 Graphs of straight lines

• Content
• S06.1 Coordinates in all four quadrants
• S06.2 Graphs of linear functions
• S06.3 Gradient and intercept of linear functions
• S06.4 Equations and graphs
• S06.5 Parallel and perpendicular lnes

• 1) On cartesian axes, plot the following points -
1. points A(1,2) and B(-3,6)
2. points D(-3,-5) and E(5,-2)
3. the midpoint of AB labelling it C
4. the midpoint of CD labelling it F
5. the point G which has the same x-coordinate as A and the same y-cordinate as B
6. the point H which has the same x-coordinate as E but double the y-coordinate
7. the midpoint of GB labelling it I
8. the point J whose x-cordinate is half that of C and whose y-cordinate is double the difference of the y-cordinates of D and B

• 2) Calculate three points using the equation between -5 and 5, then plot the equation.
1. $$y = 2x -\ 3$$
2. $$3x + 2y = 1$$
3. $$-2x -\ 3 = y$$
4. $$\frac{1}{4}x -\ 3 +\ 2y = 0$$
5. $$x + \frac{2}{3}y -\ 1 = 0$$
6. $$3x - 5y = \frac{1}{2}x +1$$
7. $$\frac{x}{5} +\ \frac{y}{3} = \frac{3}{4}$$
8. $$-3x + \frac{1}{2}y = 2$$

• 3) Write the equation for the straight line with
1. gradient m = 2 and y-intercept c = 0
2. gradient m = - 3 and y-intercept c = $$- \frac{2}{3}$$
3. gradient m = 1 and y-intercept c = -6
4. gradient m = $$\frac{1}{2}$$ and y-intercept c = 2

• 4) Find the equation of the line that passes through the given points.
1. P(1,3) and Q(-5,4)
2. P(-4,6) and O(-2,4)
3. R(-1,2) and S(-3,0)
4. B1(-5,-4) and B2(5,5)

• 5) Find the equation of the line through the point given and with the gradient specified.
1. m = 2 and (2,-1)
2. m = -4 and (3,2)
3. m = 1 and (5,1)
4. m = 3 and (8,-2)

• 6) Write the gradient, y-intercept, and x-intercept for each equation.
1. $$y = 3x -\ 1$$
2. $$-4x -\ 5 = 2y$$
3. $$2x + \frac{1}{3}y -\ 2 = 0$$
4. $$x - 2y = \frac{3}{2}x -\ 2$$

• 7) Rewrite each equation in the form $$ax +\ by +\ c = 0$$
1. $$5x + 2y = -\ 5$$
2. $$\frac{1}{4}x -\ 3 +\ 2y = 0$$
3. $$\frac{1}{4}x -\ 3 + \frac{2}{3}y = -\ 1$$
4. $$\frac{2x}{5} - 5y = \frac{1}{2}x +1$$

• 8) Given the equation $$y = 3x -\ 4$$
1. find the equation of the line parallel with y-intercept of 2
2. find the equation of the line perpendicular with y-intercept of 1
3. find the equation of the line parallel through the point (-1,8)
4. find the equation of the line perpendicular through the point (3,2)

• 8) Given the equation $$\frac{2}{3}x -\ 4 +\ 2y = 0$$
1. find the equation of the line parallel with y-intercept of -1
2. find the equation of the line perpendicular with y-intercept of 5
3. find the equation of the parallel line through the point (5,1)
4. find the equation of the line perpendicular through the point (2,-6)
• B07 Simultaneous equations

• Content
• S07.1 Solving simultaneous equations by inspection
• S07.2 Solving simultaneous equations by algebraic elimination
• S07.3 Solving simultaneous equations by a graphical method
• S07.4 Setting up and solving problems using simultaneous equations

• 1) Solve the simultaneous equations by inspection
1. $$2x -\ y = 1$$
$$x -\ y = 0$$
2. $$x +\ 4y = 10$$
$$x +\ 5y = 12$$
3. $$3x +\ 3y = 12$$
$$3x +\ 4y = 13$$
4. $$6x -\ y = 9$$
$$5x -\ y = 7$$
5. $$2x -\ y = 7$$
$$4x -\ y = 13$$
6. $$2x +\ 3y = 12$$
$$4x +\ 3y = 18$$
7. $$5x +\ 4y = 14$$
$$4x +\ 5y = 13$$
8. $$2x -\ 4y = 2$$
$$2x -\ 3y = 4$$

• 2) Solve the simultaneous equations by elimination
1. $$4x +\ 2y = 22$$
$$3x -\ 2y = 6$$
2. $$2x +\ 3y = 13$$
$$x +\ 2y = 8$$
3. $$3x +\ 2y = 5$$
$$5x -\ 4y = 1$$
4. $$6x -\ y = \frac{2}{5}$$
$$2x -\ 3y = 28$$
5. $$3x -\ 4y = 3$$
$$x +\ 6y = 12$$
6. $$3x -\ 2y = 33$$
$$2x +\ 3y = -4$$
7. $$2x +\ \frac{1}{2} y = 10$$
$$x +\ 11y = 5$$
8. $$5x +\ 3y = 1$$
$$7x +\ 5y = 1$$

• 3) Solve the simultaneous equations graphically
1. $$x +\ 2y = 2$$
$$3x +\ y = 3$$
2. $$3x -\ 2y = 1$$
$$3x +\ y = 7$$
3. $$- x -\ y = 1$$
$$- 3x +\ 3y = 2$$
4. $$4x -\ 2y = 2$$
$$3x -\ y = 3$$
5. $$- 5x +\ 2y = 1$$
$$3x +\ 2y = 9$$
6. $$\frac{1}{2} x +\ 2y = 1$$
$$-2x +\ y = -4$$
7. $$2x +\ 5y = \frac{2}{3}$$
$$3x -\ 2y = 1$$
8. $$\frac{3}{4}x -\ 2y = \frac{1}{2}$$
$$-5x -\ y = -4$$

• 4) Solve the simultaneous equations by substitution
1. $$4x -\ y = 2$$
$$-x -\ y = -1$$
2. $$4x +\ 2y = 1$$
$$5x -\ 3y = 1$$
3. $$3x -\ 4y = 6$$
$$-2x +\ 5y = 10$$
4. $$3x +\ 2y = \frac{1}{2}$$
$$2x -\ y = 3$$
5. $$2x +\ 2y = 1$$
$$-x +\ 3y = -2$$
6. $$- \frac{1}{4} x +\ 2y = 3$$
$$2x -\ \frac{1}{2} y = 1$$
7. $$-5x -\ 3y = 4$$
$$3x +\ 2y = -4$$
8. $$-x +\ 4y = \frac{2}{3}$$
$$3x -\ 2y = -1$$

• 6) Ben and Bas shop for stationary. Ben buys 4 pens and 5 pencils, while Bas buys 2 pens and 6 pencils. Ben spends £9.05 and Bas spends £7.50.
1. Write an equation to represent what Ben spends.
2. Write an equation to represent what Bas spends.
3. Find the cost of pens and pencils.

• 7) Jessica sold 9 books and Jennifer sold only 7. The girls then sold CDs with Jessica selling 4 and Jennifer selling 8. The girls charged the same prices to their customers. Jessica made £17.90 and Jennifer made £19.30 after the sales.
1. Write an equation for how much money Jessica made.
2. Write an equation that represents the money that Jennifer made.
3. Using the two equations written, work out the price of books and CDs the girls used.

• 8) Two numbers X and Y are chosen at random. The sum of 10% of X and 10% of Y produce 38.2 and the sum of 25% of X and 4% of Y produce 57.7.
1. Find the numbers.
2. Work out the product of 5% of X and 10% of Y.
• B08 Revision Worksheet 1

• 1) Write the fractions from largest to smallest
$$0.\dot{6},\ \frac{5}{7},\ \frac{2}{3},\ 0.68,\ \frac{9}{13},\ \ 0.6\dot{7} ,\ 0.\dot{6}\dot{7},\ \frac{7}{9}$$

• 2) Work out
1. $$2\frac{2}{3} -\ 1\frac{4}{5}$$
2. $$1\frac{1}{4} \div 2\frac{3}{8}$$

• 3) Consider $$\frac{6.19 \times (5.92 -\ 2.69)}{3.51 -\ 1.91}$$ and
2. using a calculator find an answer to 2sf

• 4) Round the following numbers to 1dp and 2sf.
1. 0.0002784
2. 1461.01882

• 5) Ha, He, and Ho win a prize of £240 which they share on the ratio of 3:1:4. How much money does each person get.

• 6) Treehouse Store reduces it products by 5% in June, and a further 8% in July. If the following products are selling for the prices below, find their prices before both discounts.
1. shirt £21.40
2. trousers £70
3. jumper £110

• 7) Find the HCF and LCM for each pair of numbers
1. 144 and 270
2. 252 and 324

• 8) Write the answers as exact numbers
1. $$(128)^\frac{1}{2}$$
2. $$(6)^\frac{1}{2} \times (24) ^\frac{1}{2}$$
3. $$(124)^0 \times 321$$

• 9) Write the numbers in standard form
1. 120.65
2. 0.000831
3. -19.0009
4. 235 000 001

• 10) Expand
1. $$2a(x -\ 5a)$$
2. $$-3x(2x +\ 6) +\ 4x(3x -\ 5)$$

• 11) Factorise and solve
1. $$x^2 -\ 3x -\ 10 = 0$$
2. $$12x^2 +\ 7x -\ 10 = 0$$
3. $$2x^4 +\ 15x^2 -\ 8 = 0$$

• 12) Solve
1. $$(3x -\ 5) =\ 2(1 +\ x)$$
2. $$5a +\ 2 -\ 3(5 -\ a) =\ 7 -\ 4(2 +\ 3a)$$
3. $$2(x -\ \frac{1}{2}) = \frac{2}{3}( 3 -\ 6x)$$

• 13) Joseph and Carl work for their father planting potatoes earning £1 between them. Joseph works three and half times as many hours as Carl.
1. Write an equation for the total money earned by both boys.
2. Work out how much each boy earns.

• 14) Given $$s = \frac{2}{3}b -\ dt^2$$, rearrange for
1. $$b$$
2. $$t$$

• 15) Given A(1,-2) and B(6,2)
1. find the equation of the straight line between A and B in the form $$y = ax +\ b$$
2. work out the mid point for AB
3. work out the intercepts for the line

• 16) Given the equation $$2y -\ 3x +\ 2 = 0$$
1. find the equation of the line parallel with y-intercept of 12
2. find the equation of the line perpendicular through the point (1,4) in the form $$y = mx + c$$
3. sketch both equations on the same axis

• 17) Write the equations in the form $$ax +\ bx +\ c = 0$$
1. $$2x -\ \frac{1}{2}y = 2$$
2. $$-\ 2x +\ \frac{2}{3}y = 11$$

• 18) Solve the simultaneous equations
1. $$4x +\ 1y = 14$$
$$3x -\ 2y = 5$$
2. $$3x +\ \frac{1}{2}y = 8$$
$$5x -\ 4y = 23$$

• 19) Mary and Honey play a card game that requires finding two specific combinations, each with its own score. Mary scores 132 and Honey scores 215 after Mary obtains the first card combination 5 times and Honey obtains it 8 times, and Mary obtains the second combination 3 times and Honey obtains it 5 times. Find the scores gained by obtaining each combination.

• 20) Thomas decides to deposit £500 in a savings account for 8 years. He has three options.
1. 6% simple interest for 8 years,
2. 4% compound interest for 5 years dropping to 3% for the remaining 3 years, or
3. 3% compound interest for 8 years.
Which option will earn him the most money?
• B09 Revision Worksheet 2

• 1) Consider the fractions $$\frac{3}{5},\ \frac{5}{7},\ \frac{2}{3},\ \frac{3}{4},\ \frac{5}{6}$$
1. write the second fraction as a decimal
2. find the sum of the first three fractions
3. find the product of the last two fractions
4. write two equivalent fractions for the first fraction
5. find the difference between the first and last fraction
6. write the fractions from largest to smallest

• 2) Given the number 0.$$\dot{6}\dot{3}$$
1. write the first 10 digits of the number
2. round the number to 3dp
3. convert it to a proper fraction
4. approximate $$\dot{6}\dot{3}^2 -\ \dot{6}\dot{3}$$

• 3) Ayaan drives, illegally because he doesn't have a licence yet, at 12 km/h for 3.25 hours.
1. convert the speed to m/s
2. find the distance travelled
3. state the upper and lower bound of the distance travelled

• 4) Below is the recipe for making 50 kg concrete.
• 10 buckets cement
• 2 bucket water
• 4 buckets stones
• 2 bucket river sand
1. write the ratio for the ingredients in simplest form
2. rewrite the recipe for 120kg of cement

• 5) Lavinia wants to invest her birthday money for 8 years. She has the three options below. Which option is best?
1. simple interest of 4 $$\frac{1}{2}$$ percent per year
2. 3 $$\frac{3}{4}$$ percent compound interest per year
3. 3 per cent compound interest per year for the first 6 years, and 5 per cent simple for the remaining two years

• 6) Savanah Store reduces it stock for a sale first by 5% and then by 7 $$\frac{1}{2}$$ per cent giving the prices below. Find the original price of each item.
1. shirt £11.55
2. tie £8.32
3. suit £59.85

• 7) Express answers in standard form
1. $$7.0 \times 10^6 +\ 5 \times 10^7$$
2. $$7.2 \times 10^6 -\ 2.7 \times 10^{-\ 4}$$
3. $$(2.4 \times 10^4) \div (7.2 \times 10^3)$$
4. $$(3.5 \times 10^{-\ 2}) \times (5.0 \times 10^{-\ 5})$$

• 8) Given 240 and 216,
1. write 240 as a product of its primes
2. write 216 in the form $$(2^a \times 3^b)$$
3. find the HCF for the pair of numbers
4. find the LCM for the two numbers

• 9) Given $$2x +\ 3yx -\ 3z = 2$$
1. find $$y$$ when $$x = 4$$ and $$z = -2$$
2. find $$z$$ when $$x = -1$$ and $$y = \frac{1}{2}$$
3. rearrange the subject of the formula for $$x$$

• 10) Expand $$(-2x +\ 3)^2(3x -\ 5)$$

• 11) Factorise
1. $$3x^2y -\ 6xy^2$$
2. $$x^2 +\ 3x -\ 18$$
3. $$6x^2 +\ 5x -\ 4$$

• 12) Given the equation $$3x + \frac{3}{4}y -\ 9 = 0$$,
1. find the $$y$$ and $$x$$intercepts
3. find the equation parallel with y-intercept 1
4. work out the equation of the straight line perpendicular to the line through (2,-3)

• 13) Solve the simultaneous equations by substitution
1. $$2x -\ 6y = 3$$
$$3x +\ 4y = 11$$
2. $$4x -\ 1y = 11$$
$$2\frac{1}{2}x +\ 4y = -7$$

• 14) Ed paid £7.13 for 3 books and 2 chocolate bars while his friend Eve paid £7.10 for 2 books and 5 chocolate bars.
1. Write an equation for each person to represent their purchase.
2. Find the cost of books and chocolate bars.

• 4)
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2. 222
3. 333
4. 444

• 5)
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2. 222
3. 333
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• 6)
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2. 222
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• 7)
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• 8)
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• 9)
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• 10)
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• 11)
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• 12)
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• 13)
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• 14)
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• B10 Revision Worksheet 3

• 1) Given the fractions $$\frac{2}{5},\ \frac{3}{9},\ \frac{3}{7},\ \frac{4}{5},\ \frac{7}{9},\ \frac{2}{6}$$
1. identify the equivalent fractions
2. write the fractions from largest to smallest
3. write the largest fraction as a decimal
4. find the sum of all the fractions

• 2) Work out
1. $$1\frac{2}{5} \times \frac{2}{3}$$
2. $$2\frac{4}{7} \div 3\frac{5}{6}$$
3. $$\frac{5}{6} +\ 2\frac{3}{5}$$
4. $$1\frac{1}{5} -\ 1\frac{3}{4}$$

• 6) Write the decimals as exact fractions
1. 0.24
2. 0.0625
3. 0.$$\dot{2}\dot{6}$$
4. 0.0$$\dot{5}\dot{4}$$
5. 1.2$$\dot{7}$$
6. 0.$$\dot{6}{3}\dot{9}$$

• 7) Convert the decimals to fractions to find out
1. whether 0.$$\dot{2}\dot{7}$$ or 0.28 is larger.
2. if 0.$$\dot{7}$$ is double 0.$$\dot{3}\dot{5}$$.

• 9) Round the numbers as indicated
1. 2.3162 (2dp)
2. 1.31671 (2sf)
3. 321.376 (1dp)
4. 38.627 (3sf)
5. 0.000376 (1sf)
6. 0.000288 (3dp)

• 10) Approximate
1. $$\frac{2.13 \times 5.31}{0.789}$$
2. $$\frac{38.9 \times 107}{168 -\ 29}$$
3. $$\frac{7.8 \times (9.81 -\ 6.35)}{5.5 -\ 2.1}$$

• 11) Work out the largest and smallest value in each case.
1. The perimeter of a square with a side of 8cm to the nearest cm.
2. The area of rectangle with sides of 133mm and 24mm to the nearest mm.
3. The circumference of a circle with a radius of 12cm to the nearest cm.
4. The difference between the attendees at a game where 24 400 attend one week and 25 000 the next week, correct to the nearest hundred.

• 12) Use a calculator to work out providing answers to 2sf.
1. $$(16.3 -\ 3.6 \times 2.1) \times 3.4$$
2. $$\frac{2.4 +\ 3 \times 2.34}{11.1 -\ 2.1^2}$$
3. $$\frac{\sqrt{7.4 +\ 3.34^2}}{4.2^2 -\ 3.2^2}$$

• 13) A rectangle has a length of 8cm and 5cm to the nearest centimetre.
1. Calculate the upper and lower bound for the perimeter of the rectangle.
2. Calculate the upper and lower bound for the area of the rectangle.

• 14) A car travels for 1200m for 1.25 hours to 2 significant figures.
1. Work out the longest and shortest possible distance travelled.
2. Calculate the upper and lower bound for the speed of the car.

• 4)
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2. 222
3. 333
4. 444

• 5)
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2. 222
3. 333
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• 6)
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2. 222
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• 7)
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• 8)
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• 9)
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• 10)
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• 11)
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• 12)
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• 13)
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• 14)
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