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KS3 Maths · Year 7Chapter 1 · Integers, factors and powers
Year 7 · Chapter 1

Integers, factors and powers

By the end of this chapter you will be able to

  • Read and write integers and decimals, say the value of each digit, and compare them with =, ≠, <, >, ≤ and ≥
  • Add, subtract, multiply and divide positive and negative integers and decimals, using written methods
  • Use inverse operations and fact families to check an answer and find a missing number
  • Work out calculations with brackets, powers and reciprocals in the correct order (BIDMAS)
  • Find squares, cubes and roots, and tell whether a root is a whole number
  • Find factors, multiples, primes, the HCF and the LCM, and write a number as a product of prime factors
  • Recognise integers, rational numbers and real numbers, and explain why there are infinitely many integers

You should already know

  • Place value of numbers up to 10 million and negative numbers on a number line (Year 6, Chapter 1)
  • The four operations with whole numbers and decimals (Year 6, Chapter 2)
  • Factors, multiples and prime numbers (Year 6, Chapter 3)

Where this leads

  • Fractions, decimals and percentages, using the same place value (Chapter 2)
  • Standard form and rounding, using powers of 10 (Chapter 3)
  • Solving equations with inverse operations (Chapter 6)

National Curriculum for England, Key Stage 3 mathematics, Year 7. Content: N1 place value for decimals, measures and integers of any size, N2 order positive and negative integers, decimals and fractions using =, ≠, <, >, ≤ and ≥ on the number line, N3 primes, factors, multiples, HCF, LCM, prime factorisation and the unique factorisation property, N4 the four operations, including written methods, on integers and decimals, positive and negative, N5 priority of operations, including brackets, powers, roots and reciprocals, N6 relationships between operations, and inverse operations, N7 integer powers and real roots, powers of 2 to 5, exact roots and decimal approximations, N16 the infinite nature of the sets of integers, real and rational numbers.

Exercise P  Do you remember?

Non-calculator
1
FWhat is the value of the 7 in 4,732?
2
FWork out 3,456 + 2,789.
3
FList all the factors of 24.
4
FWrite the first seven multiples of 7.
5
FThe temperature is 2 °C. It falls by 5 °C. What is the new temperature?
1.1

Place value and ordering

Year 7
N1, N2
Video lesson: Place value and orderingComing soon
A football league publishes the total number of people who went to matches in one season: 2,483,910. Every digit in that number matters, and the position of each digit tells you how much it is worth.

Our number system uses place value. Each column is ten times the column on its right: ones, tens, hundreds, thousands, ten thousands, hundred thousands and millions. A digit is worth its own value times the value of its column.

Decimals continue the pattern to the right of the point, but the columns get ten times smaller: tenths, hundredths and thousandths. So in 0.3 the 3 is worth three tenths, and in 0.03 the 3 is worth three hundredths.

MillionsHundred thousandsTen thousandsThousandsHundredsTensOnes
2483910

We use these signs to compare numbers: = means equal to, ≠ means not equal to, < means less than, > means greater than, ≤ means less than or equal to, and ≥ means greater than or equal to. The open end of a sign always faces the bigger number.

Integers include negative numbers, which are less than zero. On a number line, numbers get bigger as you move to the right. So every negative number is less than zero, and −9 is less than −2, even though 9 is bigger than 2.

−10−9−8−7−6−5−4−3−2−1012345678910−7−24
On the number line the red dots are −7, −2 and 4. Going right, the numbers get bigger.

Key ideas

  • The value of a digit = the digit × the value of its column.
  • Each column is ten times the column on its right, and decimal columns are ten times smaller.
  • The open end of < or > faces the bigger number.
  • For decimals, write the same number of places before you compare: 0.5 = 0.50.
  • Every negative number is less than zero, and a bigger size means a smaller value below zero.
Worked example 1.1.1
What is the value of the 4 in 8,462,015?

Method. Find the column that holds the digit, then multiply the digit by the value of that column.

  1. Find the column of the 4. Count the columns from the right: ones, tens, hundreds and so on.
    8,462,015: the 4 is in the hundred thousands column
  2. Multiply the digit by the value of its column. A digit is worth its own value times the value of its column, so a digit in a big column is worth much more.
    4 × 100,000 = 400,000
  3. Write the answer. We want the value of the digit, not the digit itself.
    400,000

Check. The digit 4 is in the hundred thousands column, so it stands for 4 lots of 100,000. ✓

Watch out. Look at the column first. A 6 in the hundreds column is worth 600, not 6.

Worked example 1.1.2
Write these in order, smallest first: 0.6, −0.25, −2, 0.06, 1.5

Method. Sort the negative numbers first, then zero and the positive numbers. The further right a number is, the bigger it is.

  1. Find the negative numbers. With negative numbers, the bigger the size, the smaller the value.
    −2, −0.25
  2. Sort the zero and the positive numbers. Zero is less than every positive number, and smaller positive numbers come first.
    0.06, 0.6, 1.5
  3. Write the whole list from smallest to largest. Put the negative numbers first, then the others.
    −2, −0.25, 0.06, 0.6, 1.5

Check. Read the list again: each number is bigger than the one before it. ✓

Watch out. Do not sort negative numbers by their size alone: the negative number with the biggest size is the smallest value.

Worked example 1.1.3
Fill in the box with < or >: 3.4 □ 3.04

Method. Write both numbers with the same number of decimal places, then compare digit by digit from the left.

  1. Write both with the same number of decimal places. Adding a zero at the end does not change the value, but it makes the places line up.
    3.4 = 3.40 and 3.04 = 3.04
  2. Compare the digits from the left. The first place where the digits differ decides which number is bigger.
    They first differ in the tenths place
  3. Write the symbol. The open end of the sign faces the bigger number.
    3.4 > 3.04

Check. Read the statement back with the same places: 3.40 > 3.04 is true. ✓

Watch out. Do not compare the digits after the point as whole numbers: 0.45 is smaller than 0.5 because 4 tenths is less than 5 tenths.

Watch out

Watch out: do not decide that 0.45 is bigger than 0.5 because 45 is bigger than 5. Compare the tenths first: 0.45 has 4 tenths and 0.5 has 5 tenths, so 0.5 is bigger. Also, do not think that −9 is bigger than −2 because 9 is bigger than 2.

Exam tip

Exam tip. When you order a list with negative numbers, sort the negative numbers first, then zero, then the positive numbers. The number with the biggest size among the negatives is the smallest value.

Exercise 1A  Place value, comparing and ordering

Non-calculator
1
FWhat is the value of the 6 in 5,600,321?
2
FWhat is the value of the 8 in 1,482,907?
3
FWhat is the value of the 7 in 3.217?
4
FWrite three million, forty thousand and nine in figures.
5
FFill in the box with < or >: 0.305 □ 0.35
6
FWrite these in order, smallest first: 14, −9, 0, −30, 6
7
FWhich is colder, −3 °C or −7 °C? Write the colder temperature.
8
PSWrite down every integer n for which −3 < n ≤ 2.
−4−3−2−10123−22
9
PSHow many integers n satisfy −4 ≤ n < 3?
10
RTom says 0.125 is bigger than 0.3 because 125 is bigger than 3. Is Tom right? Explain your answer.
1.2

The four operations with negative numbers and decimals

Year 7
N4
Video lesson: The four operations with negative numbers and decimalsComing soon

Scuba divers and lift operators use negative numbers every day. A diver at −12 m is 12 metres below the surface. If she rises 5 metres, she moves up the number line to −7 m. When she goes down, the number moves left.

Adding a negative number means moving left, which is the same as taking away its size. Subtracting a negative number is the same as adding its size. Multiplying and dividing follow the sign rules: two numbers with the same sign give a positive answer, and two numbers with different signs give a negative answer.

For decimals, use the written method you learned in Year 6. For a multiplication, ignore the decimal points, multiply, and then put the point back so that the answer has as many decimal places as both numbers together. For a division, make the number you divide by a whole number first.

Key ideas

  • Adding a negative number = taking away its size. Subtracting a negative = adding its size.
  • Same signs give a positive answer. Different signs give a negative answer. This is true for × and ÷.
  • Line up decimal points when you add or subtract.
  • In a decimal multiplication, the answer has the total number of decimal places of both numbers.
Worked example 1.2.1
Work out −8 + 5.

Method. Adding a negative number means moving left on the number line. Adding a positive number means moving right.

  1. Write the sum, with brackets around negative numbers. Brackets stop the two signs from clashing.
    (−8) + 5
  2. Start at −8 and move 5 right. A positive number on the end of a sum moves you right.
    −8 + 5
  3. Write the answer.
    (−8) + 5 = −3

Check. −3 − 5 = −8, so the answer is right. ✓

Watch out. Adding a negative number makes the answer smaller. Do not just add the sizes.

Worked example 1.2.2
Work out (−5) × (−8).

Method. Multiply the sizes first, then decide the sign of the answer.

  1. Multiply the sizes. Ignore the signs for a moment.
    5 × 8 = 40
  2. Decide the sign. Two negatives make a positive. A negative and a positive make a negative.
    the answer is positive
  3. Write the answer.
    (−5) × (−8) = 40

Check. Divide 40 by (−8) to get back to −5. ✓

Watch out. The sign rule works for division too: (−36) ÷ (−4) is positive.

Worked example 1.2.3
Work out 3.6 × 0.4.

Method. Ignore the decimal points, multiply the whole numbers, then put the decimal point back in the right place.

  1. Ignore the decimal points. Multiply the two whole numbers exactly as you would without points.
    3.6 × 0.4 becomes 36 × 4 = 144
  2. Count the decimal places. Count the decimal places in each number and add them. The answer needs 2 decimal places.
    1 + 1 = 2
  3. Place the decimal point. Count 2 places from the right of 144 and put the point there.
    144 becomes 1.44

Check. 0.4 is less than 1, so the answer should be smaller than 3.6. 1.44 is smaller than 3.6. ✓

Watch out. The answer has the same number of decimal places as both numbers added together. Do not just copy the points.

Watch out

Watch out: 5 − (−3) is 8, not 2. Taking away a negative number makes the answer bigger, because you are adding the size. Also, for decimals, do not line up the right-hand digits: 4.6 + 2.85 needs 4.60 + 2.85.

Exam tip

Exam tip. Use a number line or a thermometer to check any answer that involves a negative number. Ask yourself whether the answer should be above or below zero.

Exercise 1B  Positive and negative integers, and decimals

Non-calculator
1
FWork out −8 + 5.
2
FWork out 6 − 11.
3
FWork out (−6) × 7.
4
FWork out (−5) × (−8).
5
FWork out 36 ÷ (−4).
6
FWork out 4.6 + 2.85.
7
FWork out 3.6 × 0.4.
8
FWork out 6.3 ÷ 0.7.
9
PSA diver is at −12 m, which means 12 m below the surface. She rises 5 m, then descends 9 m. What is her final position?
10
RSam says (−6) ÷ (−2) = −3, because a negative divided by a negative is negative. Is Sam right? Explain.
1.3

Inverse operations and fact families

Year 7
N6
Video lesson: Inverse operations and fact familiesComing soon
A bakery packs 6 cakes into each box. On Monday it packed 42 cakes and needed to know how many boxes it had filled.

Every operation has an inverse, which is the operation that undoes it. Addition and subtraction are inverses of each other. Multiplication and division are inverses of each other. If 6 × 7 = 42, then 42 ÷ 7 = 6 and 42 ÷ 6 = 7 must also be true.

A fact family is the group of four facts that uses the same three numbers. Two are multiplication facts and two are division facts. Inverse operations are useful for checking an answer, and for finding a missing number in a statement such as □ × 8 = 96.

Key ideas

  • Addition and subtraction are inverses. Multiplication and division are inverses.
  • A fact family uses three numbers to make four facts: two multiplications and two divisions.
  • To check a division, multiply the answer by the number you divided by.
  • To find a missing number, use the inverse operation on both sides.
Worked example 1.3.1
Write the four facts in the fact family for 6, 7 and 42.

Method. Find the two multiplication facts first. Then turn each one round to make a division fact.

  1. Write the multiplication facts. Multiplication works in either order.
    6 × 7 = 42 and 7 × 6 = 42
  2. Write the division fact using 6. Division is the inverse of multiplication, so 42 ÷ 6 = 7.
    42 ÷ 6 = 7
  3. Write the division fact using 7. The other division fact uses the other number.
    42 ÷ 7 = 6

Check. All four facts use 6, 7 and 42, so they form one fact family. ✓

Watch out. Check the fact family by multiplying: 7 × 6 is the same as 6 × 7.

Worked example 1.3.2
Find the missing number: □ × 8 = 96

Method. The box is a missing number. Use the inverse (opposite) operation on both sides to find it.

  1. Say what has been done to the box. Read the statement in words.
    the box has been multiplied by 8
  2. Use the inverse operation on both sides. The inverse undoes what was done, so the box is left on its own.
    □ × 8 = 96
    □ = 96 ÷ 8
  3. Work out the missing number.
    □ = 12

Check. Put 12 in the box: 12 × 8 = 96. ✓

Watch out. The inverse of × is ÷, and the inverse of + is −. Do not use the same operation again.

Worked example 1.3.3
Find the missing number: □ + 47 = 93

Method. The box is a missing number. Use the inverse (opposite) operation on both sides to find it.

  1. Say what has been done to the box. Read the statement in words.
    the box has had 47 added to it
  2. Use the inverse operation on both sides. The inverse undoes what was done, so the box is left on its own.
    □ + 47 = 93
    □ = 93 − 47
  3. Work out the missing number.
    □ = 46

Check. Put 46 in the box: 46 + 47 = 93. ✓

Watch out. The inverse of × is ÷, and the inverse of + is −. Do not use the same operation again.

Watch out

Watch out: to check 96 ÷ 8 = 12, do not just work out the division again. Multiply 12 by 8 and check that you get 96 back.

Exam tip

Exam tip. Check every missing-number answer by putting it back into the statement. If both sides match, the answer is right.

Exercise 1C  Inverse operations, fact families and missing numbers

Non-calculator
1
FWrite down the four facts in the fact family for 6, 7 and 42.
2
FFind the missing number: □ × 8 = 96
3
FFind the missing number: □ − 17 = 25
4
FFind the missing number: □ + 47 = 93
5
FFind the missing number: □ ÷ 9 = 7
6
FUse multiplication to check whether 96 ÷ 8 = 12 is correct. Write yes or no.
7
PSA bakery packs 6 cakes into each box. It packs 42 cakes. How many full boxes does it fill?
8
PSLeo has £20. He spends £7.35 on a book. How much money does he have left?
9
PSA number is multiplied by 5, and the answer is 65. Write this as a missing-number statement and find the number.
10
RSam knows that 3 × 9 = 27. He says that 9 ÷ 27 = 3. Is Sam right? Explain your answer.
1.4

Priority of operations, brackets and reciprocals

Year 7
N5
Video lesson: Priority of operations, brackets and reciprocalsComing soon
A cinema charges different prices for adults and children, and adds one booking fee to the whole order. To work out the total, the order of the calculation really matters.

When a calculation has more than one operation, we use the order of operations, which you may know as BIDMAS: Brackets first, then Indices (powers and roots), then Division and Multiplication, working from left to right, and last Addition and Subtraction, also from left to right.

The reciprocal of a number is 1 divided by that number. The reciprocal of 4 is 1/4, and the reciprocal of 0.2 is 5, because 0.2 × 5 = 1. Dividing by a number gives the same answer as multiplying by its reciprocal.

Key ideas

  • Brackets, then powers and roots, then multiplication and division (left to right), then addition and subtraction (left to right).
  • Multiplication and division have equal priority. Work from left to right.
  • The reciprocal of a number is 1 divided by it. A number times its reciprocal is 1.
  • The reciprocal of a negative number is negative.
Worked example 1.4.1
Work out 3 + 4 × 5.

Method. Multiplication comes before addition, so work out 4 × 5 first.

  1. Multiply first. BIDMAS puts multiplication before addition.
    4 × 5 = 20
  2. Add. Now the calculation is 3 + 20.
    3 + 20 = 23
  3. Write the answer.
    23

Check. Without the rule you would get 35, which is wrong. ✓

Watch out. 3 + 4 × 5 is not (3 + 4) × 5, which is 35.

Worked example 1.4.2
Work out (7 + 5) ÷ 4.

Method. Work out the brackets first, then the multiplication, then the division.

  1. Brackets first. The brackets say to work out 7 + 5 first.
    (7 + 5) = 12, so the calculation is 12 ÷ 4
  2. Divide. Now there is only one operation.
    12 ÷ 4 = 3
  3. Write the answer.
    3

Check. 3 × 4 = 12, which is the number in the brackets. ✓

Watch out. Without brackets, 7 + 5 ÷ 4 would be 8.25 and not 3.

Worked example 1.4.3
Work out 2 × 32.

Method. Use BIDMAS: powers first, then multiply and divide from left to right.

  1. Power first. Indices come before multiplication.
    32 = 9, so the calculation is 2 × 9
  2. Multiply. Multiplication is the last step here.
    2 × 9 = 18
  3. Write the answer.
    18

Check. If you worked out (2 × 3)2 instead, you would get 36, which is wrong. ✓

Watch out. 2 × 32 = 18, not 36, because the power is worked out first.

Watch out

Watch out: 24 ÷ 6 × 2 is 8, not 2. Divide and multiply have the same priority, so work from left to right: 24 ÷ 6 = 4, then 4 × 2 = 8. Never multiply first just because multiplication looks more important.

Exam tip

Exam tip. Write each step on its own line. Underline the part you will work out next, and cross it out when it is done.

Exercise 1D  Order of operations, powers and reciprocals

Non-calculator
1
FWork out 3 + 4 × 5.
2
FWork out 18 − 3 × 4.
3
FWork out (7 + 5) ÷ 4.
4
FWork out 20 − 2 × (3 + 4).
5
FWork out 2 × 32.
6
FWork out 24 ÷ 6 × 2.
7
FWork out 5 × (8 − 2)2 ÷ 9.
8
FWrite the reciprocal of each number.
(a)8
(b)0.2
(c)−4
9
PSA cinema charges £8.50 for an adult ticket and £5.20 for a child ticket, and a booking fee of £1.50 for the order. Work out the total cost of 3 adult and 2 child tickets.
10
RShow that (a + b) × c is not always the same as a + (b × c). Use a = 1, b = 2 and c = 3.
1.5

Powers, squares, cubes and roots

Year 7
N7
Video lesson: Powers, squares, cubes and rootsComing soon

A power (or index) shows repeated multiplication. In 25 the base is 2 and the index is 5, so 25 = 2 × 2 × 2 × 2 × 2 = 32. The index is not the same as multiplying the base by the index: 25 is not 2 × 5.

Squares and cubes are powers with index 2 and 3. A square number such as 49 is 72, and it can be shown as a square of dots. A root reverses a power. The square root of 49 is 7, because 7 × 7 = 49. The cube root of 27 is 3, because 3 × 3 × 3 = 27.

Here are some powers you should know: 22 = 4, 23 = 8, 24 = 16, 25 = 32; 32 = 9, 33 = 27, 34 = 81; 42 = 16, 43 = 64; 52 = 25, 53 = 125.

A square of 4 by 4 dots has 16 dots, so 42 = 16. The square root of 16 is 4.

Key ideas

  • A power 53 means three 5s multiplied together: 5 × 5 × 5 = 125.
  • The square root of a number is the positive number that multiplies by itself to make it.
  • A root is exact when the answer is a whole number. Otherwise it lies between two whole numbers.
  • Brackets matter: (−3)2 = 9, but −32 = −9.
Worked example 1.5.1
Work out 25.

Method. A power tells you to multiply the base by itself. The small raised number, the index, says how many copies to multiply.

  1. Write the power as a multiplication. The index 5 says how many copies of 2 to multiply.
    25 = 2 × 2 × 2 × 2 × 2
  2. Multiply two at a time, from left to right. Keep the running answer and multiply it again.
    2 → 4 → 8 → 16 → 32
  3. Write the answer.
    = 32

Check. Count the copies: there are 5 copies of 2 in the multiplication. ✓

Watch out. A negative base to an odd index gives a negative answer. A negative base to an even index gives a positive answer.

Worked example 1.5.2
Work out √64.

Method. A root undoes a power. To find a square root or a cube root, find the number that gives the starting number.

  1. Which number, multiplied by itself, gives this? The root is the number we multiply to get back the value inside the root sign.
    8 × 8 = 64
  2. Check the working. Multiply the number by itself the right number of times.
    8 × 8 = 64
  3. Write the answer.
    √64 = 8

Check. 8 works, so the root is exact. ✓

Watch out. The square root symbol means the positive root: √49 = 7, not −7.

Worked example 1.5.3
Work out ∛125.

Method. A root undoes a power. To find a square root or a cube root, find the number that gives the starting number.

  1. Which number, multiplied by itself three times, gives this? The root is the number we multiply to get back the value inside the root sign.
    5 × 5 × 5 = 125
  2. Check the working. Multiply the number by itself the right number of times.
    5 × 5 × 5 = 125
  3. Write the answer.
    ∛125 = 5

Check. 5 is the cube root, so the root is exact. ✓

Watch out. A cube root keeps the sign: the cube root of −8 is −2.

Watch out

Watch out: 25 is 32, not 10. Also, −32 is −9, because the index only applies to the 3; but (−3)2 is 9, because the bracket is squared.

Exam tip

Exam tip. Use your times tables. Know the square numbers up to 152 = 225 and the cubes up to 53 = 125.

Exercise 1E  Powers, squares, cubes and roots

Non-calculator
1
FWork out 25.
2
FWork out 34.
3
FWork out 103.
4
FWork out 72.
5
FWork out √64.
6
FWork out ∛125.
7
FWork out each power.
(a)(−4)2
(b)−42
(c)(−2)3
8
PSWhich is greater, 25 or 52? Write down the greater number.
9
PSA square garden has an area of 144 m2. Find the length of one side.
10
RIs √50 a whole number? Explain your answer.
1.6

Factors, multiples, primes, HCF and LCM

Year 7
N3
Video lesson: Factors, multiples, primes, HCF and LCMComing soon
Two bells ring at the same time each morning. One rings every 6 minutes and the other rings every 8 minutes. When will they ring together again?

A factor of a number divides into it exactly, with no remainder. A multiple of a number is the answer when you multiply it by a whole number. A prime number has exactly two factors: 1 and itself. The number 1 is not prime, and 2 is the only even prime.

Every whole number greater than 1 can be written as a product of prime numbers in only one way, apart from changing the order. This is the unique factorisation property. We write the answer using prime factorisation, with indices for repeated primes.

The highest common factor (HCF) is the largest factor that two numbers share. The lowest common multiple (LCM) is the smallest multiple that two numbers share.

Key ideas

  • A factor divides exactly. A multiple is a number in the times table of another number.
  • A prime number has exactly two factors: 1 and itself. 1 is not prime.
  • Prime factorisation writes a number as a product of primes, using indices for repeats.
  • HCF is the biggest common factor. LCM is the smallest common multiple.
Worked example 1.6.1
Write 360 as a product of prime factors.

Method. Keep dividing by the smallest prime number that goes in exactly, until you reach 1. Then write all the primes together.

  1. Divide by 2. Use the smallest prime that divides exactly, and keep going with the answer.
    360 ÷ 2 = 180
  2. Divide by 2. Use the smallest prime that divides exactly, and keep going with the answer.
    180 ÷ 2 = 90
  3. Divide by 2. Use the smallest prime that divides exactly, and keep going with the answer.
    90 ÷ 2 = 45
  4. Divide by 3. Use the smallest prime that divides exactly, and keep going with the answer.
    45 ÷ 3 = 15
  5. Divide by 3. Use the smallest prime that divides exactly, and keep going with the answer.
    15 ÷ 3 = 5
  6. Divide by 5. Use the smallest prime that divides exactly, and keep going with the answer.
    5 ÷ 5 = 1
  7. Write the number as a product of primes. Gather repeated primes with an index: 2 × 2 = 22.
    360 = 23 × 32 × 5

Check. Multiply the primes back together: 2 × 2 × 2 × 3 × 3 × 5 = 360. ✓

Watch out. 1 is not a prime number, so stop dividing when you reach 1, and never write 1 as a factor in the answer.

Worked example 1.6.2
Find the HCF of 36 and 60.

Method. The highest common factor (HCF) is the biggest number that divides exactly into both numbers.

  1. List the factors of 36. A factor divides exactly, with no remainder.
    36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  2. List the factors of 60. Write them in order so that they are easy to match.
    60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
  3. Find the common factors and choose the biggest. The HCF is the largest factor that is on both lists.
    common factors: 1, 2, 3, 4, 6, 12
    HCF = 12

Check. 12 divides 36 and 60 exactly. ✓

Watch out. HCF is the biggest common factor. It is not the same as LCM, which is the smallest common multiple.

Worked example 1.6.3
Find the LCM of 12 and 18.

Method. The lowest common multiple (LCM) is the smallest number that both numbers divide into exactly.

  1. List the multiples of 12. Multiples are the times-table answers for that number.
    12: 12, 24, 36
  2. List the multiples of 18. Keep going until you find a number that is on both lists.
    18: 18, 36
  3. Choose the smallest number on both lists. The first number that appears on both lists is the LCM.
    LCM = 36

Check. 36 = 12 × 3 = 18 × 2. ✓

Watch out. Do not stop at the first multiple of the bigger number. Check that it is also a multiple of the smaller one.

Watch out

Watch out: 1 is not a prime number, because it has only one factor. Also, do not confuse HCF and LCM: the HCF is always smaller than or equal to both numbers, and the LCM is always bigger than or equal to both numbers.

Exam tip

Exam tip. When you list factors, go in pairs: 1 × 36, 2 × 18, 3 × 12, and so on. Stop when the pairs repeat.

Exercise 1F  Factors, multiples, primes, HCF and LCM

Non-calculator
1
FList all the factors of 36.
2
FIs 91 a prime number? Explain.
3
FWrite 60 as a product of prime factors.
4
FWhich of these numbers are prime numbers? 2, 9, 15, 17, 21, 1
5
FFind the HCF of 12 and 30.
6
FFind the LCM of 4 and 6.
7
FFind the smallest number that is a multiple of both 8 and 12.
8
PSA bell rings every 6 minutes and another rings every 8 minutes. Both ring at 09:00. After how many minutes do they next ring together?
9
PSSam has 24 red counters and 36 blue counters. He makes as many identical bags as possible, using every counter, with the same mix in each bag. How many bags does he make?
10
RExplain why 2 is the only even prime number.
1.7

Sets of numbers: integers, rational numbers and real numbers

Year 7
N16
Video lesson: Sets of numbers: integers, rational numbers and real numbersComing soon

The integers are the whole numbers and their negatives: …, −2, −1, 0, 1, 2, …. They go on forever in both directions, so there are infinitely many integers. There is no largest integer, because you can always add 1 to get a bigger one.

A rational number can be written as a fraction p/q, where p and q are integers and q is not zero. Every integer is rational, because 5 = 5/1. A decimal that stops, such as 0.75 = 3/4, is rational. A decimal that repeats, such as 1/3 = 0.333…, is also rational.

Real numbers include every rational number and also irrational numbers, which cannot be written exactly as a fraction. Examples are √2 and π. Their decimals never stop and never repeat. Every real number can be marked on a number line.

Key ideas

  • Integers: whole numbers and their negatives. There are infinitely many of them.
  • Rational: any number that can be written as a fraction with an integer top and bottom (not 0 on the bottom).
  • Real: every number on the number line, including irrational numbers such as √2.
  • A square root of a whole number that is not a square number is irrational, for example √2 and √50.
Worked example 1.7.1
Write 0.75 as a fraction in its simplest form.

Method. Write 0.75 as a fraction with a power of ten on the bottom, then simplify it.

  1. Write 0.75 as a fraction. The 5 is in the hundredths place, so the bottom is 100.
    0.75 = 75/100
  2. Simplify. Divide the top and the bottom by their highest common factor, 25.
    75/100 = 3/4
  3. Write the answer.
    3/4

Check. 3 ÷ 4 = 0.75. ✓

Watch out. A decimal that stops is always rational.

Worked example 1.7.2
Is √9 a rational number? Explain.

Method. A rational number is a number that can be written as a fraction. Look for a way to write each number as a fraction.

  1. Write √9 as a whole number. The square root of 9 is 3, because 3 × 3 = 9.
    √9 = 3
  2. Write 3 as a fraction. Any integer can be written as itself over 1.
    3 = 3/1
  3. Decide. It is a fraction with integers on the top and bottom, so it is rational.
    So √9 is rational

Check. 3 × 3 = 9. ✓

Watch out. √9 looks irrational, but its value is a whole number.

Worked example 1.7.3
Is √2 a rational number? Explain.

Method. Think about whether √2 can be written as an exact fraction.

  1. Find the whole numbers either side. 1 × 1 = 1 and 2 × 2 = 4, so √2 lies between 1 and 2.
    1 < √2 < 2
  2. Think about its decimal. Its decimal goes on forever and never repeats.
    1.41421…
  3. Decide. A rational number has a decimal that stops or repeats, so √2 is not rational.
    √2 is irrational, so it is a real number that is not rational

Check. 1.41421 × 1.41421 is close to 2 but never exactly 2. ✓

Watch out. Not every real number is rational. Real numbers include irrational numbers such as √2 and π.

Watch out

Watch out: a decimal that looks as if it might stop can still be irrational. Also, √50 is not a whole number, so it is not an integer. Being a real number is not the same as being rational.

Exam tip

Exam tip. Use the words integer, rational and real exactly. Every integer is rational, and every rational number is real.

Exercise 1G  Integers, rational numbers and real numbers

Non-calculator
1
FWhich of these numbers are integers? −7, 0.5, 3, 5.1
2
FWrite 0.6 as a fraction in its simplest form.
3
FIs √9 a rational number? Explain.
4
FIs √2 a rational number? Explain.
5
FIs there an integer between 5.2 and 5.4? Explain.
6
FList all the integers n for which −1.5 < n ≤ 3.2.
7
PSWrite down a rational number between 0.2 and 0.3.
8
PSExplain why there is no largest integer.
9
PSIs 1/3 a rational number? Write it as a decimal and explain your answer.
10
RTom says every real number is rational because every real number can be marked on a number line. Is Tom right? Explain.

Chapter summary

SkillMethod
Place value and comparingValue = digit × column value; use < > with the open end facing the bigger number
Negative numbersAdding a negative = subtracting its size; subtracting a negative = adding its size
Sign rulesSame signs give a positive answer; different signs give a negative answer (for × and ÷)
Decimal calculationsMultiply ignoring points, then count the total decimal places; divide by making the divisor whole
Inverse operationsUndo with the opposite operation on both sides; fact families use three numbers
Order of operationsBIDMAS: brackets, indices, then × and ÷ left to right, then + and − left to right
Powers and rootsA power is repeated multiplication; a root reverses it; check roots by multiplying back
Factors and primesList factors in pairs; a prime has exactly two factors; 1 is not prime
Prime factorisationDivide by the smallest prime repeatedly; write repeats with indices
HCF and LCMHCF: largest common factor; LCM: smallest common multiple
Number setsIntegers are infinite; rational numbers are fractions (terminating or recurring decimals); real numbers include irrationals

Can you…?

  • I can read, write and compare integers and decimals, using = ≠ < > ≤ and ≥
  • I can add, subtract, multiply and divide positive and negative numbers, and decimals
  • I can use inverse operations and fact families to check and find a missing number
  • I can work out a calculation with brackets, powers and reciprocals in the right order
  • I can find squares, cubes and roots, and say when a root is not a whole number
  • I can find factors, multiples, primes, the HCF and the LCM
  • I can write a number as a product of prime factors
  • I can tell integers, rational numbers and real numbers apart, and explain why integers are infinite

Key terms

integer
a whole number, positive, negative or zero, such as −4, 0 or 7
place value
the value of a digit, which depends on its column
inequality
a statement using <, >, ≤ or ≥, showing one value is less or greater than another
inverse operation
an operation that undoes another: addition and subtraction; multiplication and division
fact family
the four related facts made from three numbers, two multiplications and two divisions
index (power)
the small number showing how many times a base is multiplied by itself, as in 25
square root
the positive number that, multiplied by itself, gives the number under the root sign
reciprocal
the number 1 divided by a given number; a number times its reciprocal is 1
BIDMAS
the order of operations: Brackets, Indices, Division and Multiplication, Addition and Subtraction
factor
a number that divides exactly into another number, with no remainder
multiple
a number that is the product of a given number and a whole number
prime number
a number with exactly two factors, 1 and itself; 1 is not prime
prime factorisation
writing a number as a product of prime numbers
HCF
the highest common factor: the largest factor that two numbers share
LCM
the lowest common multiple: the smallest multiple that two numbers share
rational number
a number that can be written as a fraction with integers on the top and bottom
irrational number
a real number that cannot be written exactly as a fraction, such as √2
real number
any number that can be marked on the number line, rational or irrational

Exercise 1Z  Check-up

Non-calculator
1
What is the value of the 7 in 4,705,812?
2
Write these in order, smallest first: 0.4, −0.8, 0, −0.35, 0.04
3
Work out −9 + 14 − (−3).
4
Work out 5.4 × 0.7.
5
Work out 12 + 3 × (9 − 4) ÷ 5.
6
Find the missing number: □ × 9 = 72.
7
Work out 34 − 23.
8
Find the HCF of 84 and 126.
9
Find the LCM of 15 and 20.
10
Write 180 as a product of prime factors.

Total 19 marks. Allow about 19 minutes.

Exercise 1X  Exam-style questions

Non-calculator
1
FWrite down the value of the 5 in 2,850,467.[1 mark]
2
FWork out −12 + 20.[1 mark]
3
FWork out 4.8 × 0.6.[2 marks]
4
FWork out (5 − 9)2 + 2 × 3.[2 marks]
5
FFind the HCF of 48 and 60.[2 marks]
6
FFind the LCM of 9 and 12.[2 marks]
7
FWrite 84 as a product of prime factors.[2 marks]
8
FSam says 3 × 42 = 144. Is he right? Show your working.[2 marks]
9
FWrite down all the prime numbers n for which −2 < n < 4.5.[2 marks]
10
FA bus leaves the station every 12 minutes. A train leaves every 18 minutes. Both leave at 09:00. At what time do they next leave together?[3 marks]
Chapter 2: Fractions, decimals and percentages →

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