# KS LearningMaths Exercises

### Maths Exercises

The Maths exercises on this page are for secondary school up to and including year 11. They are provided free of charge by KS Learning for anyone teaching pupils at home or in school. They may not be used for commericial purposes.

• Factors and Multiples

• 1) Explain each term.
1. multiple
2. factor
3. common multiple
4. common factor

• 2) List the first 5 multiples of
1. 3
2. 8
3. 7
4. 12

• 3) List three common multiples
1. 8 and 10
2. 20 and 15
3. 2, 5, and 7
4. 16, 12, and 20

• 4) Make a list of
1. multiples of 8 between 20 and 50
2. multiples of 3 less than 40 that are odd numbers
3. even numbers between 10 and 60 that are also multiples of 5
4. the multiples of 12 less than 150 whose digits sum to less than 10

• 5) List the factor pairs of
1. 12
2. 16
3. 30
4. 100

• 6) List three common factors of
1. 16 and 22
2. 20, 100, and 40
3. 88, 132, and 110
4. 32, 128, and 256

• 7) Make a list of
1. factors of 144 from 10 to 30
2. even numbers that are also factors of 72
3. factors of 120 that are also integers
4. numbers between 0 and 100 that are factors of 200 and multiples of 5

• 8) Write
1. the first three cube numbers
2. the smallest square number that is also a cube number

1. Define a prime number
2. How many even prime numbers are there?
3. List the prime numbers between 10 and 50
4. Is 149 a prime number? Explain.

• 10) Write the following numbers as products of their primes
1. 18
2. 150
3. 840
4. 264

• 11) Explain the terms
1. LCM
2. HCF

• 12) Find the HCF of the pairs of numbers
1. 24 and 16
2. 42 and 48
3. 120 and 144
4. 96 and 72

• 13) Find the HCF of the groups of numbers
1. 24, 42, and 72
2. 30, 75, and 100
3. 54, 117, and 216

• 14) Find the LCM of the pairs of numbers
1. 48 and 108
2. 54 and 72
3. 90 and 120
4. 100 and 125

• 15) Determine both the HCF and LCM of the following pairs of numbers
1. 84 and 60
2. 72 and 108
3. 180 and 200
• Percentages

• 1) Write each percentage (i) as a decimal, and (ii) as a fraction
1. 48%
2. 15%
3. 75%
4. 27%

• 2) Write each fraction (i) as a decimal, and (ii) as a percentage
1. $$\frac{4}{20}$$
2. $$\frac{12}{18}$$
3. $$\frac{10}{35}$$
4. $$\frac{11}{15}$$

• 3) Find
1. $$\frac{4}{20}$$ of 210kg
2. 28% of £295
3. 82% of 0.32 ml
4. $$1\frac{4}{20}$$ of 32m

• 4) Increase
1. £34 by 12%
2. 270km by 48%
3. 54.5g by 21.5%
4. 4m by 135%

• 5) Decrease
1. £29.50 by 22%
2. 1kg by 9.75%
3. £1.12 by 72%
4. 95m by 12.5%

• 6) A pair of shoes is selling for £54 at George's Emporium.
1. They are reduced by £5. What is the percentage decrease?
2. George bought them for £18. Find the percentage profit on the original price?
3. The shoes eventually sell for £38. What is George's percentage profit?

• 7) Ami makes 5l of daal at a cost of £7.20 which she sells for £18.50.
1. What does it cost to make a litre of daal?
2. What Ami's percentage profit?
3. She sells 10l at an 8% discount. What does she charge?
4. How much profit does she make on the sale of 10l of daal?

• 8) Anjali sets up a store. Express her profit and loss for each item. Anjali sets up a store. Express her profit and loss for each item.
1. an old black shoe with a hole in it that bought for £5 & sold at £3
2. half a chocolate bar that she found under her bed, bought for 85p and sold for 15p
3. three old samosas bought for 30p each & sold for a total of £1

• 9) Hrishita invests £55 and must choose between 3.5% compound interest or 4.2% simple interest. Which is better if she leaves the investment untouched for
1. 2 years
2. 8 years
3. 15 years

• 10) The number of crows in Bushy Park is due to fall by 12% per year. In 2020, there were 4000 crows.
1. How many will there be in 2021?
2. How many will there be in 2030?
3. When will there be 1000?

• 11) Oscar puts £120 in a savings account. He gets 5% for the first 3 years, and the interest rate falls to 3.5%, both compound interest
1. 3 years
2. 6 years
3. 10 years
4. 25 years

• 12) Alisha puts £120 in a savings account at 5% simple interest at the same time as Oscar. At each interval in the previous question, who has more money?

• 13) A watch was increased by 5% to £145. What was its original price?

• 14) A laptop is reduced by 21% to £428. What was its original price?

• 15) VAT is added to the price of goods at a rate of 20%. Given the prices of the following items, how much of each price is VAT?
1. bicycle £120
2. shirt £35
3. calculator £6.40
4. phone £78
• Fractions

• 1) Write Sam's results as fractions
1. 10 out of 20 for English
2. 15 out of 25 for Maths
3. 18 out of 30 for French
4. 8 out of 24 for History

• 2) Find the value of $$x$$ in the equivalent fractions
1. $$\frac{x}{5} = \frac{4}{20}$$
2. $$\frac{2}{3} = \frac{8}{x}$$
3. $$\frac{4}{x} = \frac{12}{18}$$
4. $$\frac{2}{7} = \frac{x}{35}$$

• 3) Write as an improper fraction
1. $$2\frac{3}{5}$$
2. $$5\frac{1}{4}$$
3. $$-1\frac{3}{8}$$
4. $$-3\frac{2}{3}$$

• 4) Write as a mixed fraction
1. $$\frac{15}{4}$$
2. $$\frac{121}{9}$$
3. $$\frac{23}{5}$$
4. $$\frac{84}{6}$$

• 5) Work out the answer
1. $$\frac{2}{3} + \frac{1}{4}$$
2. $$\frac{5}{6} + \frac{1}{2}$$
3. $$\frac{3}{5} + \frac{2}{3}$$
4. $$\frac{3}{2} + \frac{2}{5}$$

• 6) Determine the result
1. $$\frac{3}{8} - \frac{1}{4}$$
2. $$\frac{5}{6} - \frac{2}{3}$$
3. $$\frac{4}{5} - \frac{2}{3}$$
4. $$\frac{7}{10} - \frac{2}{5}$$

• 7) xxx
1. $$\frac{3}{8} \times \frac{1}{4}$$
2. $$\frac{5}{6} \times \frac{2}{3}$$
3. $$\frac{7}{10} \times \frac{2}{5}$$
4. $$\frac{8}{9} \times \frac{3}{4}$$

• 8) xxx
1. $$\frac{2}{5} \div \frac{1}{4}$$
2. $$\frac{3}{5} \div \frac{2}{3}$$
3. $$\frac{1}{2} \div \frac{3}{8}$$
4. $$\frac{5}{12} \div \frac{3}{10}$$

• 9) Convert each decimal to a fraction
1. 0.35
2. 0.64
3. 0.08
4. 0.13

• 10) Convert each fraction to a decimal
1. $$\frac{3}{8}$$
2. $$\frac{2}{5}$$
3. $$\frac{7}{10}$$
4. $$\frac{3}{4}$$

• 11) Convert each fraction to a decimal
1. $$\frac{2}{9}$$
2. $$\frac{1}{11}$$
3. $$\frac{3}{9}$$
4. $$\frac{5}{11}$$

• 12) Write from smallest to largest
1. $$0.25, \frac{3}{8}, \frac{2}{9}, 0.3, \frac{5}{10}$$
2. $$\frac{5}{8}, 0.5, \frac{4}{7}, 0.55, \frac{4}{6}$$
3. $$0.41, 0.4\dot{1}, \frac{3}{7}, \frac{5}{11}, \frac{2}{5}$$
4. $$\frac{7}{8}, \frac{8}{9}, 0.9, \frac{6}{7}, \frac{9}{11}$$
• Bidmas

• 1) Work our
1. $$1 - 5 + 4 - 2$$
2. $$1 - (5 + 4) - 2$$
3. $$1 - 5 + (4 - 2)$$
4. $$(1 - 5) + 4 - 2$$
5. $$1 - (5 + 4 - 2)$$
6. $$(1 + 5 - 4) - 2$$

• 2) Determine
1. $$4 \times 3 - 5 + 3$$
2. $$4 \times (3 - 5) + 3$$
3. $$(4 \times 3) - 5 + 3$$
4. $$4 \times 3 - (5 + 3)$$
5. $$(4 \times 3 - 5) + 3$$
6. $$4 \times 3 - (5 + 3)$$

• 3) Calculate
1. $$8 - (2 + 7) - 5 + 3$$
2. $$8 - 2 + 7 - 5 + 3$$
3. $$(8 - 2 + 7) - 5 + 3$$
4. $$8 - (2 + 7) - (5 + 3)$$
5. $$8 - (2 + 7 - 5) + 3$$
6. $$(8 - 2) + 7 - (5 + 3)$$

• 4) Determine
1. $$-(2 + 5 - 7) + 2 - 4$$
2. $$-(2 + 5 - (7 + 2)) - 4$$
3. $$(-2 + 5) - (7 + 2 - 4)$$
4. $$-(2 + (5 - (7 + 2 - 4)))$$
5. $$-(2 + 5 - ((7 + 2) - 4))$$
6. $$-(2 + (5 - 7)) + 2 - 4$$

• 5) Work out
1. $$5 - 7 \times 2 - 6 \div 2$$
2. $$(5 - 7) \times 2 - 6 \div 2$$
3. $$5 - 7 \times (2 - 6) \div 2$$
4. $$5 - (7 \times 2 - 6) \div 2$$
5. $$(5 - 7) \times (2 - 6 ) \div 2$$
6. $$(5 - 7 \times (2 - 6)) \div 2$$

• 6) List the following numbers
1. the first 5 square numbers
2. the first 5 cube numbers
3. the prime numbers from 50 to 80
4. all the factors of 144
5. three multiples of 4 greater than 200
6. a number that is a multiple of 6 and has a factor of 5

1. $$( ( - 7 ) - ( - 8 ) ) - 6$$
2. $$3 + ( ( - 2 ) - ( - 9 ) )$$
3. $$6 + ( 1 + ( - 5 ) )$$
4. $$( - 3 ) - (1 - ( - 2 ))$$
5. $$( ( - 6 ) - 4 ) - ( - 8 )$$
6. $$( 3 - ( - 3 ) ) - 5$$
7. $$( - 7 ) - ( ( - 1 ) + 7 )$$
8. $$( - 4 ) - ( ( - 7 ) + ( - 8 ) )$$

• 8 Work out the result
1. $$(-2) \times (3 + -5)$$
2. $$(-2) \times (3) + -5$$
3. $$-(2 \times 3) + -5$$
4. $$-(2 \times 3 + - 5)$$
5. $$(-3 - - 5 \times (-4))$$
6. $$-3 - (- 5 \times -4)$$
7. $$(-3 - -5) \times (-4)$$
8. $$-3 - 5 \times -4$$

• 9) Find the result
1. $$-20 \div 5 + 2$$
2. $$2 - 4 \div (-3 - 1)$$
3. $$3 + -4 \div 16$$
4. $$-(3 \times -2 \div 3 + -5)$$
5. $$-2(-3 - -5) \div -4$$
6. $$-2 \div -2 \div -2 + 6$$
7. $$-(-2 + 7) \div (-3 \times -2)$$
8. $$6 - (10 \div - 5) \times -2$$

• 10) Find the missing number
1. $$-2 +$$    $$= 5 + 2$$
2. $$-2 +$$    $$= -5$$
3. $$- 3 +$$    $$= 0$$
4. $$-( 3 -$$    $$) = 4$$
5. $$4 \times$$    $$= -12$$
6. $$-8 = 3 +$$
7. $$-3 + -4 =$$    $$+ 5$$
8. $$2 \times-3 = 1 -$$
9.    $$\times -2 = 10$$
• Positive and Negative Numbers

• 1) Work out
1. 5 - 2
2. -5 + 2
3. -5 - 2
4. 5 + 2
5. 3 + 8
6. -3 + 8
7. 3 - 8
8. -3 - 8

• 2) Determine
1. 3 - 2
2. 1 + 3
3. 4 - 8
4. 7 + 2
5. 5 + 4
6. 2 - 5
7. 4 - 3
8. 1 - 5

• 3) Calculate
1. 2 + 7 + 3
2. 6 + 3 - 1
3. - 1 + 5 - 2
4. 0 - 4 + 1
5. 6 + 9 - 15
6. 10 - 5 - 2
7. 2 - 5 - 2
8. -3 + 4 - 3

• 4) Determine
1. -2 + 5 - 7 + 2 - 4
2. 16 - 11 - 7 + 5 + 3
3. - 6 - 4 - 9 + 5 - 7
4. 10 - 14 + 8 + 9 - 3
5. 9 - 6 + 3 - 2 - 5
6. -11 - 5 - 2 - 9 - 6
7. 7 - 5 + 11 - 2 + 3
8. 8 - 3 + 9 - 4 - 3

• 5) Work out
1. $$-20 \div 5 + 2$$
2. $$-7 - 4$$
3. $$7 \times -4$$
4. $$- 7 \times 4$$
5. $$7 - 4$$
6. $$-7 + -4$$
7. $$7 \times 4$$
8. $$4 - 7$$
9. $$-7 \times -4$$

• 6) Work out
1. (-8) + (-10)
2. 20 - (-9)
3. -6 + (+24)
4. (-8) - (-5)
5. -5 - 2 + (-1)
6. +5 - (+5) - 7
7. -2 - -7 - +4
8. -(-11) + 5 - 4

1. ( ( - 7 ) - ( - 8 ) ) - 6
2. 3 + ( ( - 2 ) - ( - 9 ) )
3. 6 + ( 1 + ( - 5 ) )
4. ( - 3 ) - (1 - ( - 2 ))
5. ( ( - 6 ) - 4 ) - ( - 8 )
6. ( 3 - ( - 3 ) ) - 5
7. ( - 7 ) - ( ( - 1 ) + 7 )
8. ( - 4 ) - ( ( - 7 ) + ( - 8 ) )

• 8) Work out the result
1. $$(-2) \times (3 + -5)$$
2. $$(-2) \times (3) + -5$$
3. $$-(2 \times 3) + -5$$
4. $$-(2 \times 3 + -5)$$
5. $$(-3 - - 5 \times (-4))$$
6. $$-3 - (- 5 \times -4)$$
7. $$(-3 - -5) \times (-4)$$
8. $$-3 - 5 \times -4$$

• 9) Find the result
1. $$-20 \div 5 + 2$$
2. $$2 - 4 \div (-3 - 1)$$
3. $$3 + -4 \div 16$$
4. $$-(3 \times -2 \div 3 + -5)$$
5. $$-2(-3 - -5) \div -4$$
6. $$-2 \div -2 \div -2 + 6$$
7. $$-(-2 + 7) \div (-3 \times -2)$$
8. $$6 - (10 \div - 5) \times -2$$

• 10) Find the missing number
1. $$-2 +$$    $$= -5$$
2. $$- 3 +$$    $$= 0$$
3. $$-( 3 -$$    $$) = 4$$
4. $$4 \times$$    $$= -12$$
5. $$-8 = 3 +$$    
6. $$-3 + -4 =$$    $$+ 5$$
7. $$2 \times -3 = 1 -$$    
8.     $$\times -2 = 10$$
• Surds

• 1) Work out the surds
1. $$\sqrt{2} \times \sqrt{3}$$
2. $$\sqrt{5} \times \sqrt{5}$$
3. $$\sqrt{1} + \sqrt{16} + \sqrt{4}$$
4. $$-2\sqrt{3} \times 2\sqrt{5}$$
5. $$-2\sqrt{6} \times - 3\sqrt{7}$$
6. $$3\sqrt{4} - 5$$

• 2) Find the answers as surds
1. $$3\sqrt{2} + 2\sqrt{2} - \sqrt{2}$$
2. $$4\sqrt{7} - 2\sqrt{7} + 5\sqrt{7}$$
3. $$3\sqrt{11} + 8\sqrt{11} - 4\sqrt{11}$$
4. $$7\sqrt{3} - 4\sqrt{3} - 3\sqrt{3}$$
5. $$2\sqrt{6} - - 4\sqrt{6} + 2\sqrt{6}$$
6. $$3\sqrt{3} - (\sqrt{3} - 6\sqrt{3})$$
7. $$8\sqrt{5} - 2(5\sqrt{5} - 3\sqrt{5})$$

• 3) Determine the answers as surds
1. $$\sqrt{32}$$
2. $$\sqrt{20}$$
3. $$- 3\sqrt{27}$$
4. $$2\sqrt{64}$$
5. $$- 7\sqrt{90}$$
6. $$\sqrt{288}$$
7. $$- 4\sqrt{125}$$

• 4) Words out as surds
1. $$\sqrt{80} + \sqrt{45} - \sqrt{20}$$
2. $$-\sqrt{50} + \sqrt{32} + \sqrt{18}$$
3. $$2\sqrt{50} + \sqrt{48} - 5\sqrt{12}$$
4. $$-3\sqrt{50} - 5\sqrt{28} - 2\sqrt{63}$$
5. $$-5\sqrt{50} + 3\sqrt{99} - 6\sqrt{44}$$
6. $$2\sqrt{50} - \sqrt{125} + 5\sqrt{20}$$
7. $$-2\sqrt{50} - 3\sqrt{75} + \sqrt{108}$$

• 5) Find the answers as surds
1. $$2\sqrt{3} \times \sqrt{6}$$
2. $$- 5\sqrt{10} \times \sqrt{5}$$
3. $$5\sqrt{8} \times \sqrt{6}$$
4. $$- 3\sqrt{20} \times \sqrt{10}$$
5. $$- 3\sqrt{15} \times \sqrt{10} \times 2\sqrt{3}$$

1. $$3\sqrt{6} \times 2\sqrt{2} + 5\sqrt{27}$$
2. $$5\sqrt{28} - 3\sqrt{14} \times \sqrt{2}$$
3. $$3\sqrt{10} \times \sqrt{30} + \sqrt{48} + \sqrt{32}$$
4. $$10\sqrt{27} - 5\sqrt{15} \times 8\sqrt{2}$$
5. $$5\sqrt{72} - 2\sqrt{25} \times 2\sqrt{2}$$

• 7) Work out as surds
1. $$\sqrt{3}(1 + \sqrt{2})$$
2. $$-\sqrt{5}(\sqrt{10} - \sqrt{5})$$
3. $$\sqrt{4}(\sqrt{11} - 3)$$
4. $$3(\sqrt{3} + \sqrt{2})$$
5. $$\sqrt{5}(\sqrt{12} + \sqrt{10})$$

• 8) Give answers in exact form
1. $$(\sqrt{5} + \sqrt{3})(1 - \sqrt{2})$$
2. $$(3 - \sqrt{2})(\sqrt{2} - 5)$$
3. $$(\sqrt{20} - \sqrt{2})(5 - \sqrt{2})$$
4. $$(\sqrt{6} - 1)(\sqrt{15} + \sqrt{8})$$
5. $$(\sqrt{5} - \sqrt{15})(3 - \sqrt{10})$$

• 9) Rationalise giving answers as surds
1. $$\frac{2}{\sqrt{5}}$$
2. $$- \frac{1}{\sqrt{2}}$$
3. $$\frac{3}{\sqrt{11}}$$
4. $$\frac{2\sqrt{3}}{\sqrt{5}}$$
5. $$- \frac{2\sqrt{6}}{5\sqrt{3}}$$

• 10) Write the exact answer after rationalising
1. $$\frac{3}{3 - \sqrt{11}}$$
2. $$\frac{2}{\sqrt{3}+1}$$
3. $$\frac{\sqrt{3}}{\sqrt{5} - \sqrt{3}}$$
4. $$- \frac{5\sqrt{2}}{2 - \sqrt{5}}$$
5. $$\frac{3\sqrt{2}}{\sqrt{8}-\sqrt{32}}$$
6. $$\frac{5\sqrt{5}}{\sqrt{50}-\sqrt{5}}$$
7. $$\frac{12}{\sqrt{27}-\sqrt{45}}$$
• Indices 01

• 1) Write in index form
1. $$3 \times 3 \times 3 \times 3 \times 3$$
2. $$7 \times 7 \times 7$$
3. $$2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$$
4. $$11 \times 11$$
5. $$15 \times 15 \times 15 \times 15 \times 15 \times 15$$

• 2) Write in expanded form
1. $$4 ^ 5$$
2. $$5 ^ 4$$
3. $$3 ^ 7$$
4. $$2 ^ 3$$
5. $$11 ^ 6$$

• 3) Work out without using a calculator
1. $$2 ^ 3$$
2. $$1 ^ 6$$
3. $$3 ^ 4$$
4. $$6 ^ 0$$
5. $$4 ^ 5$$

• 4) Work out by using a calculator
1. $$12 ^ 4$$
2. $$8 ^ 5$$
3. $$101 ^ 3$$
4. $$15 ^ 2$$
5. $$4 ^ {14}$$

• 5) Write the answer in index form
1. $$3 ^ 5 \times 3 ^ 2$$
2. $$5 ^ 2 \times 5 ^ 0$$
3. $$8 ^ 1 \times 8 ^ 2$$
4. $$7 ^ 5 \times 5 ^ 7$$
5. $$9 ^ 7 \times 9 ^ 3$$

• 6) Write the answer in index form
1. $$7 ^ 8 \div 7 ^ 2$$
2. $$4 ^ {12} \div 4 ^ 7$$
3. $$8 ^ {10} \div 8 ^ 3$$
4. $$2 ^ 6 \div 2$$
5. $$6 ^ {11} \div 6 ^ 8$$

• 7) Determine without a calculator
1. $$5 ^ {-2}$$
2. $$2 ^ {-5}$$
3. $$5 ^ 3$$
4. $$3 ^ {-4}$$
5. $$12 ^ 2$$

• 8) Find the answer without a calculator
1. $$\left( \frac{2}{3} \right) ^ 5$$
2. $$\left( \frac{1}{5} \right) ^ 2$$
3. $$\left( 1 \frac{1}{4} \right) ^ 4$$
4. $$\left( \frac{2}{7} \right) ^ 3$$
5. $$\left( 2 \frac{1}{2} \right) ^ 7$$

• 9) Determine without a calculator
1. $$\left( \frac{1}{3} \right) ^ {-6}$$
2. $$\left( 1 \frac{1}{2} \right) ^ {-4}$$
3. $$\left( \frac{5}{6} \right) ^ {-2}$$
4. $$\left( 2 \frac{3}{4} \right) ^ {-3}$$
5. $$\left( \frac{8}{11} \right) ^ {-0}$$

• 10) Work out not using a calculator
1. $$3 ^ 2$$
2. $$5 ^ {-2}$$
3. $$2 ^ 5$$
4. $$-3 ^ 2$$
5. $$(-2) ^ 4$$

• 11) Find the answer in index form
1. $$8 ^ {5} \times 8 ^ {-3}$$
2. $$5 ^ {2} \div 5 ^ {5}$$
3. $$10 ^ {1} \div 10 ^ {-3}$$
4. $$7 ^ {-5} \times 7 ^ {-2}$$
5. $$4 ^ {-7} \div 4 ^ {3}$$
• Indices 02

• 1) Work out the answer
1. $$(36) ^ {\frac{1}{2}}$$
2. $$(8) ^ {\frac{1}{3}}$$
3. $$(32) ^ {\frac{1}{5}}$$
4. $$(\frac{1}{9}) ^ {\frac{1}{2}}$$
5. $$(3\frac{3}{8}) ^ {\frac{1}{3}}$$

• 2) Determine
1. $$(-243) ^ {\frac{1}{5}}$$
2. $$(-27) ^ {\frac{1}{3}}$$
3. $$(1 \frac{7}{9}) ^ {\frac{1}{2}}$$
4. $$(- {\frac{1}{64}}) ^ {\frac{1}{3}}$$
5. $$(- {\frac{1024}{3125}}) ^ {\frac{1}{5}}$$

• 3) Work out
1. $$27 ^ {\frac{4}{3}}$$
2. $$(-512) ^ {\frac{2}{3}}$$
3. $$({\frac{9}{4}}) ^ {\frac{3}{2}}$$
4. $$(5 \frac{1}{16}) ^ {\frac{3}{4}}$$
5. $$(- {\frac{8}{27}}) ^ {\frac{5}{3}}$$

• 4) Work out by using a calculator
1. $$(8) ^ {-\frac{1}{3}}$$
2. $$(125) ^ {-\frac{2}{3}}$$
3. $$( {\frac{49}{144}}) ^ {-\frac{1}{2}}$$
4. $$(7 {\frac{19}{32}}) ^ {-\frac{2}{5}}$$
5. $$(- {\frac{64}{216}}) ^ {-1\frac{1}{3}}$$

• 5) Calculate
1. $$\sqrt[3]{27}$$
2. $$\sqrt[5]{-32}$$
3. $$\sqrt{(4)^3}$$
4. $$(\sqrt{16})^3$$
5. $$-2 \times (\sqrt{64}) ^ {-1\frac{1}{3}}$$

1. $$(2 ^ {-3}) ^ 2$$
2. $$(-2 \times 4^2) ^ \frac{1}{5}$$
3. $$((3 \times 9) ^ \frac{1}{3}) ^ 2$$
4. $$((729) ^ \frac{1}{3}) ^ {-\frac{1}{2}}$$
5. $$(2 ^ {-4} \times 4 ^ 2) ^ \frac{1}{2}$$

• 7) Rewrite to the given base
1. 36 to base 6
2. 8 to base 2
3. 1 to base 5
4. 81 to base 3
5. 10 000 to base 10

• 8) Work out
1. $$(2 \frac{1}{4}) ^ {-\frac{1}{2}} \times (3\frac{3}{8}) ^ {\frac{1}{3}}$$
2. $$(1.5) ^ {-2} \times 16 ^ {-\frac{1}{2}} \times (81) ^ {\frac{3}{4}}$$
3. $$(1.25) ^ 3 \times (5) ^ {-2} \times 7 ^ 0$$
4. $$-3 ^ 5 \div (-1 \frac{1}{2}) \times (3) ^ {-2}$$
5. $$-(4) ^ {-2} \times (-2) ^ 5 \times {-8} ^ 2$$

• 9) Determine
1. $$5 ^ {-2} \times 5 ^ 2 \times 5$$
2. $$2 ^ {-3} \times (\frac{1}{8}) ^ 2 \times 4 ^ 3$$
3. $$3 ^ 2 \times (-3) ^ 3 \times 3 ^ {-3}$$
4. $$16 \times 2 ^ 3 \div 2 ^ 2$$
5. $$12 ^ 2 \times 2 ^ {-2} \times \frac{2}{9}$$
• Standard Form

• 1) Express the following in ordinary notation
1. $$3.2 \times 10 ^ 3$$
2. $$1.85 \times 10 ^ 5$$
3. $$14.8 \times 10 ^ 4$$
4. $$2.07 \times 10 ^ 11$$
5. $$540 \times 10 ^ 2$$
6. $$0.252 \times 10 ^ 8$$
7. $$0.034 \times 10 ^ 3$$
8. $$0.0000145 \times 10 ^ 6$$

• 2) Express the following in ordinary notation
1. $$2.7 \times 10 ^ {-3}$$
2. $$63 \times 10 ^ {-7}$$
3. $$152 \times 10 ^ {-4}$$
4. $$0.4 \times 10 ^ {-2}$$
5. $$0.0000069 \times 10 ^ {-5}$$
6. $$10.0005 \times 10 ^ {-6}$$
7. $$25.2 \times 10 ^ {-8}$$
8. $$0.034 \times 10 ^ {-3}$$

• 3} Write the following numbers in standard form
1. $$350$$
2. $$2 \text{ } 365$$
3. $$985 \text{ } 001$$
4. $$34 \text{ } 700 \text{ } 000$$
5. $$87 \text{ } 960 \text{ } 000 \text{ } 000 \text{ } 000 \text{ } 000 \text{ } 000$$
6. $$27$$ million
7. $$721 \text{ } 462$$
8. $$8$$

• 4} Express each in standard form
1. $$1.234$$
2. $$23.672$$
3. $$1 \text{ } 200.4$$
4. $$0.714$$
5. $$435$$
6. $$0.000 \text{ } 000 \text{ } 000 \text{ } 008 \text{ } 9$$
7. $$0.056 \text{ } 000 \text{ } 000$$
8. $$0.000 \text{ } 001 \text{ } 288$$

• 5) The area of the surface of the earth is 510 000 000 km2.
1. Convert the surface area of the earth to m2.
2. Express this in standard form.

• 6. The speed of light is 3 x 108m/s and the speed of sound is 340 m/s.
1. Express the speed of light in normal form
2. Write the speed of sound in standard form
3. How many times larger is the speed of light than the speed of sound. Express the answer in standard form.

1. $$8 \div 1000$$
2. $$20 \times 100$$
3. $$6 \div 24$$
4. $$80 \times 200 \times 30$$
5. $$20 \div 1 000$$
6. $$55 000 \div 11$$
7. $$0.02 \times 0.008$$
8. $$20 \times 500 \div 10$$

• 8) Determine the answer and write in standard form
1. $$(6 \times 10^5) \times (7 \times 10^2)$$
2. $$(2 \times 10^3) \times (20 \times 10^-5)$$
3. $$(0.4 \times 10^{-4}) \times (5 \times 10^{-3})$$
4. $$(320 \times 10^8) \times (3 \times 10^{-10})$$
5. $$(11 \times 10^5) \times (40 \times 10^{2})$$
6. $$(0.12 \times 10^5) \times (0.02 \times 10^{-2})$$

• 9) Work out the result and write in standard form
1. $$(9 \times 10^6) \div (3 \times 10^2)$$
2. $$(325 \times 10^{-4}) \div (25 \times 10^5)$$
3. $$(40 \times 10^2) \div (0.8 \times 10^{-8})$$
4. $$(0.5 \times 10^3) \div (0.01 \times 10^{-1})$$
5. $$(180 \times 10^{-9}) \div (30 \times 10^{-3})$$
6. $$(0.2 \times 10^3) \div (0.1 \times 10^{-4})$$

• 10) Write from largest to smallest
1. $$0.002 435 \times 10^9$$
2. $$24 350$$
3. $$2.435 \times 10^{-2}$$
4. $$243.5 \times 10^1$$
5. $$0.2435 \times 10^6$$
6. $$2 435 000 \times 10^{-4}$$
• Units

• 1) Convert to the given units
1. 15 cm to m
2. 23 kg to mg
3. 1.2 l to cl
4. 0.003kN to dN
5. 650 Hm to Dm
6. 0.0003 Mg to cg

• 2) Convert to the given units
1. 65cm2 to m2
2. 25km2 to mm2
3. 1.25dm2 to Dm2
4. 0.005Hm2 to cm2
5. 0. 0305m2 to km2
6. 12 000mm2 to dm2

• 3) Convert to the given units
1. 2.5m3 to cm3
2. 500Hm3 to mm3
3. 4.2Dm3 to m3
4. 10100km3 to m3
5. 0. 28mm3 to dm3
6. 0.00042cm3 to km3

• 4) It takes 3mg of salt to make a chocolate cookie. How many cookies can be made with 0.5kg of salt?

• 5) A book is 2.4cm wide. How many books will fit on a 2.08m shelf?

• 6) It takes 10dl of concentrated lemon syrup to make 2 l of lemonade. How many litres of lemonade can be made with 24kl of lemon syrup?

• 7) 150cm3 of vinegar are added to every 4.5m3 of cleaning solution. How many dm3 of cleaning solution can be made with 0.3dm3 of vinegar?

• 8) A teacher wants to cut squares with 8mm sides from a sheet of paper that is 28cm long and 21cm wide. How many squares can the teacher cut from the piece of paper?

• 9) Carl jogs 5km when David jogs 6km
1. Write the distances they jog in the form of 1:n
2. How many km does Carl jog, if David jogs 20km?

• 10) The ratio of men to women is 2:3 when there are 40 people in the room. What is the new ratio if 5 men enter the room?

• 11) A herd of 52 cows has 12 white cows and the rest brown. What is the simplest ratio of white to brown cows?

• 12) A pattern has 5 red triangles and 30 blue triangles. Write the red to blue triangle ratio in a simplest form.

• 13) A group of children at a nursery as 63 girls and 27 boys. Write the number of boys to children as a ratio in its simplest form.

• 14) 6kg of cement powder is mixed with 10kg of sand and 5l of water.
1. Write the ratio of cement to sand to water as a ratio in its simplest form.
2. How much cement and sand are mixed with 24l of water?

• 15) A baker uses 20g of yeast to make 9 bread rolls. How many bread rolls can be made with 65g of yeast?
• Number Types

• 1) Given the following set of numbers
{ $$12, \frac{2}{3}, 11, -4, 0, 9, -\frac{2}{5}, -6, 8, 3, 17$$},
select all that are:
1. integers
2. fractions
3. even numbers
4. odd numbers
5. natural numbers
6. real numbers
7. surds

• 2) Define with at least two examples
1. even numbers
2. natural numbers
3. integers
4. odd numbers
5. real numbers
6. prime numbers
7. surds
8. counting numbers
9. square numbers
10. composite numbers

• 3) Mark on a number line from -6 to +6
1. all even numbers with an X
2. three consecutive natural numbers with an S
3. two odd numbers that add up to 8 with an M
4. the position of a half with a Y
5. three numbers that are not even with a W
6. three real numbers with a Z that are not whole numbers
7. all the prime numbers with a P
8. a square number with Q

• 4) Write using maths symbols
1. six is greater than five
2. eight is equal to two times four
3. six is greater than or equal to two
4. x is between three and five
5. seven is not equal to one
6. pi is approximately equal to 3.14
7. the sum of three and four is lessthan the product of three and four

• 5) List the following numbers
1. all the odd numbers between 6 and 15
2. eight real numbers between 9 and 10
3. five even numbers greater than twenty
4. the next 5 natural numbers that follow right after 11
5. the first 10 positive prime numbers
6. all even prime numbers
7. 4 surds
8. all the integers from -5 to +5
9. five consecutive natural numbers between 100 and 110
10. all the square numbers from 4 to 60
11. a number that is both a square and a cube number

• 6) Use a number line to illustrate
1. $$2 \le x \lt 6$$ for real numbers
2. $$x \ge -3$$ for integers
3. $$x = -5, -2, 1, 2, 3$$
4. $$-3 \le x \le 2$$ for real numbers
5. $$x \gt 5$$ and $$x \lt 2$$ for integers

• 7) Sort in ascending order
• 2, 13, 8, 25, 21, 3
• -5, 11, -8, 7, -6, -3
• -321, 241, -54, 333, -21, 231

• 8) Sort in descending order
1. $$23, \frac{1}{4}, 3, -8, 1\frac{2}{3}, 12$$
2. $$-51, 9, -\frac{8}{9}, 17, -2, \frac{2}{3}$$
3. $$-32, 2241, -854, 3433, -2311, 227$$

• 9) Use a calculator to find
1. $$17^2$$
2. $$\sqrt[3]{729}$$
3. $$23^4$$
4. $$\sqrt{324}$$
5. $$\sqrt[5]{16807}$$
6. $$(-3)^7$$
• xx

• 1) xx
1. $$xx$$
2. $$xx$$

• 2) xx
1. $$xx$$
2. $$xx$$

• 3) xx
1. $$xx$$
2. $$xx$$

• 4) xx
1. $$xx$$
2. $$xx$$

• 5) xx
1. $$xx$$
2. $$xx$$

• 6) xx
1. $$xx$$
2. $$xx$$

• 7) xx
1. $$xx$$
2. $$xx$$

• 8) xx
1. $$xx$$
2. $$xx$$

• 9) xx
1. $$xx$$
2. $$xx$$

• 10) xx
1. $$xx$$
2. $$xx$$
Sites of Interest

### Confidence

A good tutor can build the confidence of a learner enabling subject success

### Skills

A private tutor can improve the skills a pupil needs to master a subject

### Progress

Regular tutoring can drive progress and better results in school subjects

### Support

Support can help students and parents make the right academic decisions