Number System | ICSE Class 7 Maths Notes

This note covers integers and their operations, number properties, factors and multiples, fractions as operators, reciprocals, rational numbers, decimal calculations, natural-number exponents, and sets.

How do integers extend the numbers used for counting?

Natural numbers are the counting numbers beginning with 1. Whole numbers include these numbers and zero. Integers include zero, the positive counting numbers and their negative counterparts. Positive numbers are greater than zero; negative numbers are less than zero.

The symbols +, −, × and ÷ mean addition, subtraction, multiplication and division. A sign before a number indicates whether it is positive or negative. The sum is the result of addition, and the difference is the result of subtraction. The symbol = means “is equal to”; brackets group parts of a calculation.

How does a number line show position?

A number line is a straight line on which numbers are marked at a consistent scale. Zero is the reference point. Positive integers lie to its right and negative integers to its left. Numbers increase as we move to the right.

The magnitude of an integer is its distance from zero, without its sign. A signed movement can therefore describe both distance and direction. If rightward movement is positive, leftward movement is negative. State this convention before calculating a final position.

Mark zero and equally spaced integers on both sides. Start at zero, move 5 units right, then 7 units left. The final position is −2, two units left of zero.

Additive inverses are numbers whose sum is zero. Subtracting an integer has the same effect as adding its additive inverse. This connects subtraction to the addition rules already used with signed numbers.

Worked example 1. Find 7 − 18 and rewrite the subtraction as addition.

Answer: The additive inverse of 18 is −18. Therefore, 7 − 18 = 7 + (−18) = −11. The result is negative because the amount subtracted exceeds the starting number.

Keeping the order of the numbers matters in subtraction: exchanging the starting number and the number subtracted changes the calculation.

How are integers multiplied and divided?

A product is the result of multiplication. In a multiplication, the numbers being multiplied are called factors. Multiply their magnitudes first, then use their signs to decide the sign of the product. A product involving zero is zero.

Why does multiplying two negatives give a positive?

Look at the pattern 2 × (−3) = −6, 1 × (−3) = −3 and 0 × (−3) = 0. As the first factor decreases by one, the product increases by three. Continuing gives (−1) × (−3) = 3 and (−2) × (−3) = 6.

In division, the dividend is the number being divided, the divisor is the number by which it is divided, and the quotient is the result. Division asks which number, multiplied by the divisor, gives the dividend.

Worked example 2. Calculate (−100) ÷ 25 and (−100) ÷ (−4).

Answer: Since 25 × (−4) = −100, the first quotient is −4. Since (−4) × 25 = −100, the second quotient is 25. Multiplication checks both answers.

For non-zero integers, like signs give a positive quotient and unlike signs give a negative quotient. The quotient need not be an integer. A fraction expresses a quantity using equal parts of a unit. For example, 5 ÷ 8 gives the fraction 5/8.

In 5/8, the slash is a fraction bar. The numerator, 5, counts the parts, while the denominator, 8, tells how many equal parts form a unit. The denominator cannot be zero.

Note: Division by zero is not defined. Zero divided by a non-zero integer is zero. These statements concern different positions of zero in the calculation.

In a word problem, attach signs to the quantities before operating. Gains and upward movements can be represented positively; losses and downward movements can be represented negatively. Interpret the final sign using the convention stated in the question.

Which properties help us calculate with integers?

A property is a rule describing how an operation behaves. In the rules below, the letters a, b and c stand for any integers. A letter allows one statement to describe many numerical calculations.

Property: Commutativity

An operation is commutative when exchanging the order of its two numbers leaves the result unchanged. Integer addition and multiplication have this property: a + b = b + a and a × b = b × a.

Subtraction and division are not commutative. A rule about exchanging factors must not be transferred to a subtraction or division merely because the same numbers appear. Identify the operation before rearranging it.

Property: Associativity

An operation is associative when changing the grouping of three numbers leaves the result unchanged. For integers, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.

Associativity changes the brackets; commutativity changes the order. Both can help with addition and multiplication. Subtraction and division are not associative, so their brackets cannot be moved using these rules.

Worked example 3. Evaluate 5 × (−3) × 4 by two groupings.

Answer: (5 × (−3)) × 4 = (−15) × 4 = −60. Also, 5 × ((−3) × 4) = 5 × (−12) = −60. The groupings give the same product.

Property: Distributivity

Distributivity links multiplication to addition: a × (b + c) = a × b + a × c. The outside factor multiplies every term inside the brackets. A term here means one of the quantities being added.

For example, 5 × (4 + (−2)) = 5 × 4 + 5 × (−2). Evaluating either side gives 10. Distributing the factor includes multiplying the negative term with its sign.

What do identity, inverse and closure mean?

An identity leaves a number unchanged under an operation. Zero is the additive identity; one is the multiplicative identity. Thus, a + 0 = a and a × 1 = a. The additive inverse of a is −a.

Closure means that operating on numbers from a collection keeps the result in that collection. Integers are closed under addition, subtraction and multiplication, but not division. A multiplicative inverse gives product one; an integer's multiplicative inverse need not itself be an integer.

How do divisibility, factors and multiples fit together?

A positive integer is a factor of another positive integer if it divides that number exactly. A multiple is obtained by multiplying a number by a counting number. Exact divisibility means that division leaves no remainder, or amount left over.

Divisibility rules help recognise factors without completing the division. A number is divisible by 3 when its digit sum is divisible by 3. A number is divisible by 5 when its ones digit is 5 or zero. These rules can be explored through multiplication patterns.

How do common factors lead to the HCF?

A common factor divides each of the given numbers exactly. The highest common factor, abbreviated HCF, is the greatest of these common factors. It helps when a quantity must be divided into the largest equal parts that fit exactly.

A prime number is an integer greater than one whose only positive factors are one and itself. Prime factorisation writes a number as a product of primes. For the HCF, retain only common prime factors, using the smaller number of occurrences of each.

Worked example 4. Find the HCF of 12 and 16.

Answer: Write 12 = 2 × 2 × 3 and 16 = 2 × 2 × 2 × 2. Both contain two factors of 2. Their product, 2 × 2 = 4, is the HCF.

How is the LCM different?

A common multiple is a multiple of each given number. The lowest common multiple, abbreviated LCM, is the smallest positive common multiple. It supplies a common denominator, the same denominator for each fraction, when fractions must be expressed using equal-sized parts.

To find an LCM by prime factorisation, include every prime appearing in either number, with the greater number of occurrences needed. For 14 = 2 × 7 and 35 = 5 × 7, the LCM is 2 × 5 × 7 = 70.

The HCF uses factors shared by the numbers. The LCM contains enough prime factors to include each whole number as a factor. Choosing between them depends on whether the problem asks for equal divisions or a shared multiple.

How can a fraction operate on a quantity?

A fraction expresses a quantity using equal parts of a unit. In the notation 1/3, the slash is a fraction bar: the numerator is the upper number, 1, and the denominator is the lower number, 3. The denominator cannot be zero.

A fraction can also act as an operator, meaning an instruction to act on a quantity. Finding a fraction “of” a quantity means multiplying by that fraction. Divide by the denominator and multiply by the numerator to find the required part.

How does a fractional time affect distance?

Worked example 5. Aaron walks 3 kilometres in one hour. At this rate, how far does he walk in 2/5 of an hour?

Answer: One fifth of the hourly distance is 3/5 kilometre. Two fifths is twice this distance: (2/5) × 3 = 6/5 kilometres. The fraction operates on the distance covered in one hour.

The abbreviation km means kilometre. The calculation uses the same walking rate throughout the stated time. The denominator first divides the hourly distance into equal parts; the numerator then selects how many of those parts are needed.

How are two fractions multiplied?

Multiply the numerators to obtain the new numerator and multiply the denominators to obtain the new denominator. For instance, one third of one half is (1/3) × (1/2) = 1/6. Here 1/2 refers to the original whole, and 1/3 operates on that half to give 1/6 of the original whole.

Divide a rectangle into two equal horizontal parts and shade the upper half. Divide it into three equal columns. Shade one third of the upper half again. The double-shaded region is one of six equal parts.

Equivalent fractions have the same value. Multiplying or dividing numerator and denominator by the same non-zero number preserves that value. Cancelling common factors before multiplying fractions can make the arithmetic shorter. Cancellation divides a numerator and a denominator by their common factor.

How do reciprocals help us divide fractions?

The reciprocal of a non-zero number is the number that gives a product of one when multiplied by it. For a fraction, interchange its numerator and denominator. This makes the reciprocal its multiplicative inverse.

For example, the reciprocal of 3/5 is 5/3 because (3/5) × (5/3) = 1. Zero has no reciprocal: a product with zero cannot become one. The condition that a divisor must be non-zero applies to fractions as well as integers.

How is fraction division performed?

1. Identify the dividend and divisor in the order given.
2. Check that the divisor is not zero.
3. Replace division by multiplication by the reciprocal of the divisor.
4. Multiply, cancel common factors where possible, and simplify the result.

Worked example 6. Calculate (2/3) ÷ (3/5).

Answer: The divisor is 3/5 and its reciprocal is 5/3. Therefore, (2/3) ÷ (3/5) = (2/3) × (5/3) = 10/9. The dividend remains 2/3 throughout the conversion.

Division can also count how many groups fit. The calculation (1/2) ÷ (1/4) asks how many quarters fit into one half. Two quarters fit, so the quotient is two. Dividing by a positive fraction smaller than one can increase a positive quantity.

How are mixed fractions used?

A mixed fraction combines a whole-number part with a proper fraction. A proper fraction has a positive numerator smaller than its positive denominator. An improper fraction has a positive numerator at least as large as its positive denominator.

Convert a mixed fraction to an improper fraction before multiplying or dividing. Multiply its whole-number part by the denominator, add the numerator, and keep the denominator. The conversion changes the form but preserves the quantity.

Worked example 7. Internet time costs 8 rupees per hour. Find the cost of 1 1/4 hours at this rate.

Answer: Convert 1 1/4 to 5/4. The cost is (5/4) × 8 = 10 rupees. The symbol ₹ denotes rupees, so the answer can be written ₹10.

What are rational numbers and how are they located?

Definition: A rational number can be written as p/q, where p and q are integers and q is not zero. Here p is the numerator and q is the denominator. The symbol ≠ means “is not equal to”.

Rational numbers include positive and negative fractions, integers and zero. Every integer can be written with denominator one. This extends our number collection so that division by a non-zero integer can give an answer even when the quotient is not an integer.

For example, 5 ÷ 8 = 5/8 is a rational number. The numbers −2/3, 6/7 and −9/5 are also rational. The negative sign applies to the value of the whole fraction, rather than changing its size into a positive value.

How does the denominator guide a number-line construction?

To represent a fraction with positive denominator on a number line, divide each unit interval, the distance between neighbouring integers, into the number of equal parts named by the denominator. Count the required parts from zero, using the numerator. Positive fractions lie right of zero; negative fractions lie left.

To locate −2/3, divide the interval between zero and −1 into three equal parts. Count two of these parts to the left from zero. The resulting point lies between −1 and zero, because its magnitude is less than one.

What stays unchanged in equivalent forms?

Equivalent forms describe one number and therefore one position on the number line. Changing the size and number of parts together can preserve the value. A negative rational number can be written with a positive denominator and a negative numerator.

A fraction is in lowest terms when its numerator and denominator have no common positive factor other than one. Reducing to lowest terms makes a result easier to compare and interpret; it does not move the number's point on the line.

How are the four operations performed on rational numbers?

Rational-number calculations combine the sign rules for integers with the methods for fractions. For addition and subtraction, express the numbers with a common denominator, meaning the same denominator. Their parts then have equal sizes and their numerators can be combined.

How are addition and subtraction organised?

1. Keep track of each number's sign and the operation between the numbers.
2. Choose a common denominator, often the LCM of the denominators.
3. Rewrite each fraction in an equivalent form with that denominator.
4. Add or subtract the numerators, retain the denominator, and simplify.

Worked example 8. Find 3/8 + (−5/7).

Answer: Use the common denominator 56. Then 3/8 = 21/56 and −5/7 = −40/56. Their sum is (21 + (−40))/56 = −19/56. The denominator remains 56 while the signed numerators are added.

For subtraction, either subtract after making denominators equal or add the additive inverse of the second number. For example, (−5/7) − 2/3 = (−15 − 14)/21 = −29/21. The minus sign before the second fraction must remain part of the calculation.

How are multiplication and division organised?

Multiply numerators and denominators, using the integer sign rules. Thus, (−2/3) × (4/5) = −8/15. There is no need to make the denominators equal for multiplication. The denominator of the product comes from multiplying the original denominators.

For division, retain the dividend and multiply by the reciprocal of the non-zero divisor. For example, (−5/3) ÷ (2/5) = (−5/3) × (5/2) = −25/6. A positive divisor leaves the negative sign of this dividend in the quotient.

How do these operations solve a reading problem?

Mira's novel has 400 pages. She reads 1/5 of its pages on one day and 3/10 on the next. The total fraction read is 2/10 + 3/10 = 5/10 = 1/2. The remaining fraction is also 1/2, so 200 pages remain.

The fractions here refer to the whole novel, not to different remaining quantities. First identify the whole to which each fraction refers. Then select the operations needed to combine the parts and find the quantity still required.

How are fractions and decimals connected?

Place value is the value a digit has because of its position in a number. Decimal notation extends place value to parts smaller than one. The decimal point separates whole-number places from fractional places. Immediately to its right are tenths, hundredths and thousandths, representing parts with denominators 10, 100 and 1000.

A decimal fraction has a denominator such as 10, 100 or 1000. In 27.53, the digits represent two tens, seven ones, five tenths and three hundredths. A digit's position determines its contribution to the value.

How can an equivalent fraction become a decimal?

When possible, make the denominator a power of ten, a repeated product of tens, by multiplying numerator and denominator by the same number. For example, 1/2 = 5/10 = 0.5. To convert a finite decimal back to a fraction, use its place value and simplify.

The decimal 0.254 equals 254/1000. Its value is also 2/10 + 5/100 + 4/1000. The zero before the decimal point shows that there are no whole units. Zeros inside a decimal must be kept when they hold a place.

Does every decimal expansion end?

A terminating decimal ends after a finite number of decimal places. Long division can produce a terminating answer, as in 237 ÷ 8 = 29.625. When the remainder becomes zero, no further non-zero decimal digits are needed.

Other divisions continue. In 10 ÷ 3, each division step leaves a remainder of one, so the decimal is 3.333… . The dots mean that the pattern continues. A recurring decimal has a digit or block of digits that repeats indefinitely.

For 1 ÷ 7, the repeating block is 142857. Keep the distinction between an exact recurring value and a finite approximation, a value close to the exact number. Stopping the written digits without indicating continuation does not give the complete decimal expansion.

How are decimals multiplied and divided accurately?

Decimal multiplication follows from multiplication of decimal fractions. First multiply as if the decimal points were absent. Then place the decimal point using the total number of decimal places in the two factors. Place value explains this procedure.

Why do decimal places combine in a product?

Worked example 9. A car travels 12.5 kilometres per litre of petrol. Find the distance it can travel using 7.5 litres at this rate.

Answer: The distance is 12.5 × 7.5 = (125/10) × (75/10) = 9375/100 = 93.75 kilometres. Each factor contributes one decimal place, giving two in the product.

When multiplying 5.8 by 1.24, first calculate 58 × 124 = 7192. The two factors have three decimal places altogether, so the product is 7.192. The positions of the decimal points determine the scale of the answer.

Multiplication does not necessarily increase a positive number. For example, 0.25 × 8 = 2, while 0.25 × 0.8 = 0.2. Multiplying by a positive number between zero and one takes a part of the other positive number.

What changes when the divisor is a decimal?

Multiply both the dividend and divisor by the same suitable power of ten so that the divisor becomes a whole number. Then divide. Scaling both numbers together preserves the quotient; changing just one changes the question.

Worked example 10. Calculate 4.68 ÷ 0.13.

Answer: Multiply both numbers by 100. Then 4.68 ÷ 0.13 = 468 ÷ 13 = 36. Check by multiplication: 36 × 0.13 = 4.68, the original dividend.

Dividing by 10, 100 or 1000 moves the decimal point left by one, two or three places respectively. For example, 3.9 ÷ 10 = 0.39. This divides the value by ten; it does not merely change the appearance of the number.

In calculations involving quantities, keep the units. A length divided into equal pieces gives a length per piece. Distance divided by travel time gives average speed, meaning distance travelled per unit of time.

How do natural-number exponents shorten repeated multiplication?

An exponent tells how many times a base appears as a factor in repeated multiplication. The base is the repeated factor. In 5⁴ = 5 × 5 × 5 × 5 = 625, the base is 5 and the exponent is 4.

The expression 5⁴ is read as “five raised to the power four”. Exponential form is this shorter way of writing repeated multiplication. It must be distinguished from repeated addition: 4 + 4 + 4 = 12, whereas 4³ = 64.

Which laws apply to powers?

In the following rules, m and n are natural-number exponents, while a and b represent bases. The notation aᵐ means m copies of a multiplied together. A product such as mn means m × n.

These laws follow by expanding the powers into factors and regrouping or cancelling. The conditions matter: adding exponents is a rule for multiplication with the same base, not for adding powers.

Worked example 11. Evaluate 3⁴ × 3³ using the law for a common base.

Answer: Combine four factors of 3 with three more factors of 3. Thus, 3⁴ × 3³ = 3⁷ = 81 × 27 = 2187. The exponents add because the numbers of repeated factors add.

How can powers describe repeated choices?

A lock with five digit positions has ten choices, from zero to nine, at each position. If digits may repeat and the first digit may be zero, the number of possible passwords is 10 × 10 × 10 × 10 × 10 = 10⁵ = 1,00,000.

This counts repeated choices using the same factor. Each extra position multiplies the number of possibilities by ten. The permission to begin with zero matters because passwords are strings of digits, rather than ordinary five-digit counting numbers.

How are sets represented and compared?

A set is a well-defined collection of objects. “Well-defined” means that membership can be decided clearly. An element, also called a member, is an object belonging to the set. Sets are usually denoted by capital letters.

In roster form, elements are listed inside braces, the symbols { and }. Let V name the set of English vowels: V = {a, e, i, o, u}. Here the small letters are vowel elements, not numerical variables.

The symbol ∈ means “belongs to”, and ∉ means “does not belong to”. Thus a ∈ V and b ∉ V. In this statement b denotes the letter b. In roster form, an element is not generally repeated; listed elements are taken as distinct.

How does cardinality describe a finite set?

A finite set has a definite finite number of elements, including the possibility of none. An infinite set is not finite. An empty set contains no elements and can be written { }. The set of natural numbers is infinite.

Cardinality is the number of distinct elements in a finite set. The notation n(V) means the cardinality of V, so n(V) = 5. Repeated writing of an element does not create an additional distinct member.

What is the difference between equal and equivalent sets?

Equal sets contain exactly the same elements. If A and B name two sets, A = B states that they are equal. Order of listing does not matter: {1, 2, 3, 4} and {3, 1, 4, 2} are equal.

Equivalent finite sets have the same cardinality. Their elements need not be identical. The vowel set and the set {1, 3, 5, 7, 9} each have five elements, so they are equivalent but not equal. Equal finite sets are necessarily equivalent.

Worked example 12. Write the set of letters in SCHOOL and find its cardinality.

Answer: The set is {S, C, H, O, L}. The repeated O contributes one distinct element. There are five distinct letters, so the cardinality is 5.

What makes a set universal?

A universal set contains all the elements under consideration in a particular context. It is usually denoted by U. When discussing natural numbers and collections such as the primes or even natural numbers, the natural numbers can serve as U.

The choice of universal set depends on the discussion. It is not a single fixed collection for every problem. Specify the context before deciding which elements belong to the universal set and to the collections being studied within it.

Glossary

• Integer — A number belonging to the collection of positive counting numbers, zero and negative counting numbers.
• Magnitude — The distance of a number from zero, without reference to its positive or negative sign.
• Additive inverse — A number that gives zero when added to the original number.
• Commutativity — The property that exchanging the order of two numbers leaves an operation's result unchanged.
• Associativity — The property that regrouping three numbers leaves an operation's result unchanged.
• Distributivity — The property connecting multiplication of a sum to the sum of the separate products.
• Highest common factor — The greatest positive factor that divides each of the given positive integers exactly.
• Lowest common multiple — The smallest positive number that is a multiple of each given positive integer.
• Rational number — A number expressible as an integer numerator divided by a non-zero integer denominator.
• Reciprocal — A number that gives one when multiplied by the original non-zero number.
• Terminating decimal — A decimal expansion that ends after a finite number of decimal places.
• Exponent — A natural-number power showing how many times a base occurs as a factor.
• Cardinality — The number of distinct elements belonging to a given finite set.
• Equivalent finite sets — Finite sets with the same number of distinct elements, whether or not their elements match.
• Universal set — The set containing all elements under consideration within a particular discussion or problem.

Common errors and misconceptions

• Misconception: Two negative factors produce a negative answer. Correct: Their product is positive; unlike signs give a negative product.
• Misconception: Integer division must produce an integer. Correct: Integers are not closed under division; 5 ÷ 8 gives 5/8.
• Misconception: A number divided by zero equals zero. Correct: Division by zero is not defined. Zero divided by a non-zero number is zero.
• Misconception: Denominators should be added when adding fractions. Correct: Use a common denominator, add the signed numerators and retain that denominator.
• Misconception: Dividing fractions requires reversing both fractions. Correct: Keep the dividend and multiply by the reciprocal of the divisor.
• Misconception: Multiplication increases every positive number. Correct: Multiplication by a positive fraction smaller than one reduces the other positive number.
• Misconception: The exponents should be multiplied in 3⁴ × 3³. Correct: Add them because the base is the same; multiply exponents for a power raised to a power.
• Misconception: Sets with equal cardinalities must be equal. Correct: They are equivalent; equality additionally requires exactly the same elements.

Exam-style questions with model answers

Q1. Calculate (−100) ÷ 25 and verify the result by multiplication. [2 marks]

1. The quotient is −4: the numbers have unlike signs, and dividing their magnitudes gives 100 ÷ 25 = 4.
2. Check the result by multiplying the divisor by the quotient: 25 × (−4) = −100, which reproduces the dividend.

Q2. Find the HCF of 12 and 16 by prime factorisation. Show the factorisation of each number and identify the common factors used. [3 marks]

1. Write 12 as a product of primes: 12 = 2 × 2 × 3. Its factorisation contains two occurrences of the prime factor 2.
2. Write 16 as a product of primes: 16 = 2 × 2 × 2 × 2. It contains four occurrences of 2.
3. Take the two occurrences of 2 shared by both factorisations. Their product is 4, so the highest common factor is 4.

Q3. Calculate 3/8 + (−5/7), showing a common denominator, the equivalent fractions and the final sum. [4 marks]

1. Choose 56 as a common denominator. It is divisible by both 8 and 7, so both fractions can be expressed in fifty-sixths.
2. Multiply the numerator and denominator of 3/8 by 7. This gives the equivalent fraction 21/56.
3. Multiply the numerator and denominator of −5/7 by 8. Keeping its negative sign gives −40/56.
4. Add the signed numerators and retain the denominator: (21 + (−40))/56 = −19/56. This is the required sum.

Q4. Calculate (2/3) ÷ (3/5). Identify the divisor's reciprocal, show the multiplication and state the answer. [3 marks]

1. The dividend is 2/3 and the divisor is 3/5. The reciprocal of the non-zero divisor is 5/3, obtained by interchanging its numerator and denominator.
2. Retain the dividend and replace division by multiplication by this reciprocal: (2/3) ÷ (3/5) = (2/3) × (5/3).
3. Multiply the numerators and denominators separately. The result is (2 × 5)/(3 × 3) = 10/9, already in lowest terms.

Q5. Mala answers all 50 questions in a test. Each correct answer earns 5 marks and each wrong answer earns −2 marks. She has 30 correct and 20 wrong answers. Calculate and explain her total score. [5 marks]

1. Represent the score for each correct answer by the positive integer 5, since it increases the total score.
2. The contribution from 30 correct answers is 30 × 5 = 150 marks. This is the positive part of her score.
3. Represent each wrong answer by −2. The contribution from 20 wrong answers is 20 × (−2) = −40 marks.
4. Combine both contributions using addition: total score = 150 + (−40). The negative contribution reduces the positive contribution by 40.
5. Therefore, Mala's total score is 110 marks. This includes all 50 answers, because the given counts of 30 correct and 20 wrong answers add to 50.

Q6. A car travels 12.5 kilometres per litre of petrol. At the same rate, how far can it travel using 7.5 litres? Show the decimal calculation. [4 marks]

1. The required distance is the distance per litre multiplied by the number of litres. Write the calculation as 12.5 × 7.5 kilometres.
2. Express the decimal factors as fractions: 12.5 = 125/10 and 7.5 = 75/10. Each denominator is ten.
3. Multiply the numerators and denominators: (125/10) × (75/10) = 9375/100. The combined denominator accounts for two decimal places.
4. Convert the product back to decimal notation: 9375/100 = 93.75. The car can therefore travel 93.75 kilometres at the stated rate.

Q7. Simplify 3⁴ × 3³ and evaluate it, explaining the exponent rule used. [3 marks]

1. Both powers have the same base, 3. The first contains four factors of 3 and the second contains three factors of 3.
2. Combine these factors by adding the exponents: 3⁴ × 3³ = 3⁷. There are seven factors of 3 in the product.
3. Evaluate using 3⁴ = 81 and 3³ = 27. Their product is 81 × 27 = 2187, so the expression equals 2187.

Q8. Let V = {a, e, i, o, u} be the vowel set and A = {1, 3, 5, 7, 9}. Find both cardinalities. Are the sets equivalent? Are they equal? Give reasons. [4 marks]

1. The set V contains five distinct vowels. Its cardinality, written n(V), is therefore 5.
2. The set A contains five distinct numbers, namely 1, 3, 5, 7 and 9. Its cardinality n(A) is also 5.
3. The sets are equivalent because their cardinalities are equal: each contains the same number of distinct elements.
4. The sets are not equal because their elements differ. V contains vowels, while A contains the five listed numbers.

Key takeaways

• Integer multiplication and division use positive results for like signs and negative results for unlike signs, with division by zero excluded.
• Commutativity changes order and associativity changes grouping; integer addition and multiplication satisfy both properties.
• Use the HCF for shared factors and the LCM for shared multiples, including common denominators.
• A fraction can operate on a quantity; division by a non-zero fraction means multiplication by its reciprocal.
• Rational numbers include integers and signed fractions, with a non-zero denominator in every valid fractional representation.
• Decimal calculations follow place value; scaling both dividend and divisor by the same non-zero factor preserves the quotient.
• Natural-number exponents describe repeated multiplication, and each exponent law has conditions about bases and operations.
• Equal sets have identical elements; equivalent finite sets have equal cardinalities, while a universal set depends on context.

Test yourself

What is the additive inverse of −18?

It is 18, because adding 18 to −18 gives zero.

What is (−4) × (−2), and why is its sign positive?

The product is 8. Multiplying two negative integers gives a positive result.

What is the LCM of 14 and 35?

It is 70, formed from the prime factors 2 × 5 × 7.

How many quarters fit into one half?

Two quarters fit into one half, so (1/2) ÷ (1/4) = 2.

Why can a rational number not have zero as its denominator?

The fraction represents division by its denominator, and division by zero is not defined.

What is 4.68 ÷ 0.13?

Multiplying both numbers by 100 gives 468 ÷ 13, so the quotient is 36.

In 5⁴ = 625, which number is the base and which is the exponent?

The base is 5 and the exponent is 4, indicating four factors of 5.

Why does the set of letters in SCHOOL have cardinality five?

Its distinct elements are S, C, H, O and L. Repeating O does not add another element.