Number

• Coverage promise. Every one of the eighteen official subtopics has its own teaching section below, with its own definition panel, method, fully checked worked example and quick check. Nothing in Topic 1 is covered by a heading alone.
• The pattern behind eight of the ten. Each one is a moment where a quantity was silently converted into something it is not: an interval into a value, an exact number into a decimal, a base into a different base, a rate into a mean. Before you write a final answer, name what kind of object it is.
• The pattern. Fourteen of these twenty are the same underlying slip: an operation was applied to the wrong base or the wrong object — a percentage of the new value instead of the old, a cancel across a sum instead of a product, an index across a sum, a bound treated as a value. Before every line of working, name what you are operating on.
• One habit above all. Whatever else you drop from this plan, keep this: work exactly for as long as you can, and round only once, at the end. It costs nothing, it applies to every chapter of the course, and it is the single largest source of avoidable lost marks in Topic 1.

• I can identify and use natural numbers, integers, prime numbers, square numbers, cube numbers, common factors, common multiples, rational and irrational numbers, real numbers and reciprocals. 1.1
• I can write a positive integer as a product of prime powers and use that form to find an HCF and an LCM. 1.1
• I can read and write large numbers by grouping digits in threes, and say them correctly. 1.1
• I can use the notation \(n(A)\), \(\in\), \(\notin\), \(A'\), \(\varnothing\), \(\xi\), \(\subseteq\), \(\nsubseteq\), \(\cup\) and \(\cap\) correctly. 1.2
• I can describe a set by listing its elements and by a rule in set-builder notation. 1.2
• I can draw, complete and read a two-set and a three-set Venn diagram, and use \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\). 1.2
• I can calculate squares, cubes, square roots, cube roots and other roots, and I can recall \(1^2\) to \(15^2\), \(1^3\) to \(5^3\), \(10^3\) and the corresponding roots. 1.3
• I can convert between proper fractions, improper fractions, mixed numbers, decimals and percentages in every direction. 1.4
• I can add, subtract, multiply and divide fractions and mixed numbers, and cancel only common factors. 1.4
• I can read and write the dot notation for a recurring decimal, convert one to an exact fraction, including one with a non-recurring prefix, and convert a fraction to a recurring decimal. 1.4
• I can order quantities given in different forms and use \(=\), \(\ne\), \(<\), \(>\), \(\le\) and \(\ge\) correctly, including with negative numbers. 1.5
• I can use the four operations on integers, fractions and decimals with brackets and the correct order of operations, including signed arithmetic. 1.6
• I can use positive, zero, negative and fractional indices and apply \(a^ma^n=a^{m+n}\), \(a^m\div a^n=a^{m-n}\) and \((a^m)^n=a^{mn}\). 1.7
• I can evaluate expressions such as \(27^{2/3}\) and \(16^{-3/4}\) without a calculator. 1.7
• I can convert to and from standard form \(A\times10^n\) with \(1\le A<10\), and calculate with numbers in standard form. 1.8
• I can round to a given number of decimal places or significant figures and choose a sensible final accuracy for a context. 1.9
• I can make an estimate by rounding inputs to one significant figure, and use it to test whether a calculated answer is reasonable. 1.9
• I can state the upper and lower bounds of a rounded measurement, using \(\le\) for the lower endpoint and \(<\) for the upper. 1.10
• I can find the bounds of a sum, difference, product or quotient built from rounded measurements. 1.10
• I can simplify a ratio, divide a quantity in a given ratio, and use a map or model scale including area and volume scale factors. 1.11
• I can use direct and inverse proportion and compare best value. 1.11
• I can calculate a rate, a unit rate and an average speed, and use given formulas such as density and population density. 1.12
• I can find a percentage of a quantity, express one quantity as a percentage of another, and calculate percentage increase and decrease with a multiplier. 1.13
• I can handle profit and loss, discount, deposit and instalments, earnings, and percentages greater than \(100\%\). 1.13
• I can calculate simple interest, compound interest and repeated percentage change, and solve reverse-percentage problems by dividing by the multiplier. 1.13
• I can use a calculator efficiently, enter a whole expression with brackets, keep full precision, and interpret the display in context. 1.14
• I can calculate with time in the 12-hour and 24-hour systems, read an analogue clock face, work out elapsed time across midnight, read a timetable including waiting time and a change of date, and handle time zones and calendar boundaries. 1.15
• I can calculate with money, work out change and unit cost, and convert currency in the correct direction. 1.16
• I can model exponential growth, decay and depreciation with a repeated multiplier \(Q_n=Q_0(1\pm r)^n\). 1.17
• I can simplify a surd using its largest square factor, combine like surds, use conjugates and rationalise one-term and two-term denominators. 1.18

Why it matters: the classification is not decoration. It decides what you are allowed to do. You may only write \(\frac pq\) for a rational; you may only build a prime factorisation from a positive integer; and you may only leave \(\sqrt2\) as \(\sqrt2\) because it is irrational and no decimal is exact. Getting the name right first prevents the method being wrong later.

Standard Form

A way of writing any number as A times ten to the power n, where A is at least 1 and less than 10 and n is an integer. It separates the significant digits of a quantity from its scale, which makes very large and very small numbers easy to compare, to multiply and to divide without counting zeros.

Estimation

Replacing the numbers in a calculation with nearby values that are easy to handle mentally, usually rounded to one significant figure, so that the size of the true answer is known before the calculation is performed. An estimate is a test of reasonableness, not a substitute for the accurate answer, and it is what makes a keying error or a misplaced decimal point visible.

Rates

A rate compares two quantities of different kinds, such as distance and time or mass and volume, and is written as the amount of the first per one unit of the second. Average speed is the rate that matters most in this topic and is always total distance divided by total time, never the mean of separate speeds.

Sets

A set is a well-defined collection of distinct objects called its elements, described either by listing the elements inside braces or by a rule in set-builder notation. Set language provides the operations union, intersection and complement, all taken relative to a stated universal set, and a Venn diagram is the standard picture used to count the elements of each region.

Types of Number

The classification of the real numbers into natural numbers, integers, rational numbers and irrational numbers, together with the derived descriptions prime, square, cube, factor, multiple and reciprocal. Each classification decides which methods are legal: only a rational number can be written as an integer fraction, and only a positive integer has a prime factorisation.

Limits of Accuracy

The interval of values that a rounded measurement could actually have taken. A quantity stated as a to the nearest unit u lies between a minus half u and a plus half u, with the lower endpoint included because it rounds to a, and the upper endpoint excluded because it rounds to the next stated value.

Ratio and Proportion

A ratio compares two or more quantities of the same kind measured in the same unit, written with colons and simplified by dividing every part by a common factor. Proportion is the relationship between those quantities when one changes: in direct proportion their quotient stays constant, and in inverse proportion their product stays constant.

Money

Calculation with currency, in which answers are normally written to two decimal places in the currency's main unit. Converting between currencies uses a quoted exchange rate, and the direction of the conversion decides whether to multiply or divide: multiply when moving from the currency named as one unit, divide when moving towards it.

Powers and Roots

A power is the result of multiplying a number by itself a stated number of times, written with an index; a root is the inverse operation, so the nth root of a number is the value which, raised to the power n, returns it. The principal square root of a non-negative number is itself non-negative, which is why the square root of x squared equals the modulus of x rather than x.

Indices I

An index, or power, records how many times a base is used as a factor. Three laws govern them: multiplying powers of the same base adds the indices, dividing subtracts them, and raising a power to a power multiplies them. Extending those laws consistently forces a zero index to equal one, a negative index to mean a reciprocal, and a fractional index to mean a root.

Ordering

Arranging quantities by size, from smallest to largest or the reverse, after converting them all into one comparable form or locating them on a number line. On a number line the smaller of two numbers is always the one further to the left, which is what makes negative eight smaller than negative three.

Using a Calculator

The disciplined use of a scientific calculator: entering a whole expression with brackets rather than in separate pieces, retaining unrounded values throughout, checking that the angle mode matches the question, and translating the display back into the units and form the context requires. It is a judgement skill rather than a keying skill, and Paper 1 does not assess it because that paper is taken without a calculator.

The Four Operations

Addition, subtraction, multiplication and division, applied to integers, fractions and decimals under a fixed order of precedence: brackets first, then indices and roots, then multiplication and division worked from left to right, then addition and subtraction worked from left to right. The order is a convention that makes every written expression have exactly one value.

Exponential Growth and Decay

A quantity grows or decays exponentially when it is multiplied by the same factor in every equal period, so after n periods it equals the starting value times that factor to the power n. Growth uses a multiplier greater than one and decay a multiplier between zero and one, and the exponent counts completed periods rather than elapsed time.

Percentages

A percentage is a number of hundredths, so a percentage of a quantity is that quantity multiplied by the percentage over one hundred. Any percentage increase or decrease can be carried out in a single step by multiplying by one plus or minus the percentage as a decimal, and reversing such a change means dividing by that same multiplier rather than applying the opposite percentage.

Fractions, Decimals and Percentages

Three interchangeable ways of writing the same rational quantity: a fraction as a ratio of two integers, a decimal as a place-value expansion, and a percentage as a number of hundredths. Every rational number has a terminating or recurring decimal expansion, and every terminating or recurring decimal can be converted back to an exact fraction.

Surds

A surd is a root that is irrational and is therefore kept in exact root form rather than written as a decimal. Surds are simplified by extracting the largest perfect-square factor, combined only when they contain the same root, and removed from a denominator by multiplying numerator and denominator by the root itself or by the denominator's conjugate.

Time

Time is measured in a mixed-base system: sixty seconds to a minute, sixty minutes to an hour, twenty-four hours to a day. Because the bases are not ten, elapsed time is found by counting in stages to a convenient boundary rather than by ordinary decimal subtraction, and a decimal part of an hour must be multiplied by sixty to become minutes.

• 1. “1 is a prime number.” Why wrongA prime has exactly two positive factors. The only positive factor of 1 is 1 itself, so it has one. Correct1 is neither prime nor composite. Prime factorisations therefore never contain a 1, and 2 is the smallest prime. Say this“A prime number has exactly two positive factors, so 1 is not prime.” CheckList the primes below 12. (2, 3, 5, 7, 11.)
• 2. “The HCF takes the smaller power of every prime you can see.” Why wrongFor \(72=2^3\times3^2\) and \(90=2\times3^2\times5\), that rule would include the 5 and give 90 — which does not divide 72 at all. CorrectOnly primes appearing in both factorisations go into the HCF. The LCM, by contrast, takes every prime present. Say this“The common primes are 2 and 3 at their lower powers, so the HCF is \(2\times3^2=18\).” CheckHCF of \(2^4\times5\) and \(2^2\times3\times5^2\). (\(2^2\times5=20\).)
• 3. “\(n(A)=22\) means 22 students study only Art.” Why wrong\(n(A)\) counts the whole circle, including the overlap. Putting 22 in the Art-only region double-counts everyone in the intersection. CorrectEnter the intersection first, then subtract it from each set total to obtain the “only” regions. Say this“Art only is \(n(A)-n(A\cap B)\).” Check\(n(A)=31\), \(n(A\cap B)=15\). How many are in \(A\) only? (16.)
• 4. “\(\sqrt{64}=4\), because \(4^3=64\).” Why wrong\(64\) is both \(8^2\) and \(4^3\). The index on the root sign decides which fact is needed, and a plain \(\sqrt{\phantom{x}}\) means a square root. Correct\(\sqrt{64}=8\) and \(\sqrt[3]{64}=4\). Say this“A bare root sign is a square root.” CheckEvaluate \(\sqrt{729}\) and \(\sqrt[3]{729}\). (27 and 9.)
• 5. “\(\frac12+\frac13=\frac25\).” Why wrongAdding numerators and denominators is not addition of fractions. \(\frac25=0.4\), but the true sum is \(0.8\overline3\) — the “answer” is smaller than one of the things being added. CorrectRewrite over a common denominator: \(\frac36+\frac26=\frac56\). Say this“Fractions are added over a common denominator; only the numerators combine.” Check\(\frac34+\frac25\). (\(\frac{15}{20}+\frac{8}{20}=\frac{23}{20}\).)
• 6. “In \(\dfrac{3+x}{3}\) the threes cancel.” Why wrongThe 3 on top is a term, not a factor of the whole numerator. Test with \(x=6\): the true value is \(3\), not \(6\). CorrectCancel only a factor of the entire numerator against a factor of the entire denominator. \(\frac{3(1+x)}{3}\) does cancel, to \(1+x\). Say this“Cancel factors, never terms joined by addition.” CheckSimplify \(\dfrac{5x+10}{5}\). (\(x+2\), after factorising the numerator as \(5(x+2)\).)
• 7. “\(-7\) is bigger than \(-3\) because 7 is bigger than 3.” Why wrongSize and position are different things. On the number line \(-7\) is further left, and further left always means smaller. Correct\(-7<-3\). Among negatives, the larger the digits the smaller the number. Say this“The smaller number is the one further to the left on the number line.” CheckOrder \(-2,-9,0,-5\) smallest first. (\(-9,-5,-2,0\).)
• 8. “\(24\div6\times2=2\), because multiplication comes before division.” Why wrongMultiplication and division share one level of precedence. They are performed in the order they appear, left to right. Correct\(24\div6=4\), then \(4\times2=8\). The same applies to addition and subtraction: \(10-4+3=9\). Say this“Multiplication and division are equal in rank and taken left to right.” Check\(36\div9\times3\). (12, not 1.)
• 9. “\(-3^2=9\).” Why wrongWithout a bracket, the index binds only to the 3, so the expression means \(-(3^2)\). Correct\(-3^2=-9\), while \((-3)^2=9\). Write the bracket whenever the negative is part of the base. Say this“The bracket decides whether the minus sign is part of the base.” CheckEvaluate \(-2^4\) and \((-2)^4\). (\(-16\) and \(16\).)
• 10. “\((a+b)^2=a^2+b^2\).” Why wrongIndex laws describe products, quotients and powers — not sums. Test it: \((3+4)^2=49\) but \(9+16=25\). CorrectExpand the bracket properly. The same warning applies to \(\sqrt{a+b}\ne\sqrt a+\sqrt b\). Say this“The index laws apply to products and powers, never across a sum.” CheckIs \(\sqrt{25+144}=5+12\)? (No: \(\sqrt{169}=13\), not 17.)
• 11. “A negative index means a negative answer.” Why wrongThe negative sign is instruction, not value: it says “take the reciprocal”. \(2^{-3}\) is \(\frac18\), which is positive. Correct\(a^{-n}=\frac{1}{a^n}\). For a positive base the answer is always positive. Say this“A negative index means a reciprocal.” CheckEvaluate \(5^{-2}\). (\(\frac{1}{25}\).)
• 12. “\(18\times10^{3}\) is in standard form.” Why wrongStandard form requires \(1\le A<10\), and \(18\) fails that. The number is right but the form is not, so the final accuracy mark is lost. Correct\(18\times10^3=1.8\times10^1\times10^3=1.8\times10^4\). Say this“The coefficient must be at least 1 and less than 10, so I renormalise.” CheckWrite \(0.42\times10^{6}\) in standard form. (\(4.2\times10^{5}\).)
• 13. “\(0.004786\) to 2 significant figures is \(0.00\).” Why wrongThat is the answer to 2 decimal places. Significant figures start at the first non-zero digit, which here is the 4. Correct\(0.0048\). The leading zeros stay because they hold the place value. Say this“Significant figures start counting at the first non-zero digit.” Check\(0.0906\) to 2 s.f. (\(0.091\).)
• 14. “A rounded measurement is an exact value.” Why wrong“\(8.4\) cm to the nearest \(0.1\) cm” describes any length from \(8.35\) up to (but not including) \(8.45\). Treating it as exactly \(8.4\) makes every bound in the question wrong. CorrectWrite the interval first: \(8.35\le L<8.45\). Note the asymmetry of \(\le\) and \(<\). Say this“The lower bound is included because it rounds to the stated value; the upper bound rounds to the next one.” CheckBounds of \(240\) to the nearest 10. (\(235\le x<245\).)
• 15. “The largest quotient uses both upper bounds.” Why wrongDividing by a larger number makes the result smaller. Upper over upper is a perfectly possible value, but it is not the maximum. CorrectLargest quotient \(=\dfrac{\text{upper}}{\text{lower}}\); smallest quotient \(=\dfrac{\text{lower}}{\text{upper}}\). The same asymmetry applies to differences. Say this“To make it as large as possible, divide as much as possible by as little as possible.” Check\(d=150\) m (nearest 10 m), \(t=20\) s (nearest 1 s). Upper bound of speed? (\(155\div19.5=7.95\) m/s to 3 s.f.)
• 16. “\(45\ \text{min}:2\ \text{h}\) simplifies to \(45:2\).” Why wrongA ratio compares quantities in the same unit. Leaving mixed units compares minutes with hours and scales one side by 60. CorrectConvert first: \(45:120=3:8\). Say this“Convert to a common unit before simplifying a ratio.” CheckSimplify \(1.5\ \mathrm{kg}:400\ \mathrm{g}\). (\(15:4\).)
• 17. “Average speed is the mean of the speeds.” Why wrongThe two legs usually take different times, so they carry different weights. \(60\) km at \(40\) km/h then \(60\) km at \(60\) km/h averages \(48\) km/h, not \(50\). CorrectAverage speed \(=\dfrac{\text{total distance}}{\text{total time}}\). Find each leg's time first. Say this“Average speed is total distance over total time.” Check\(30\) km at \(60\) km/h then \(30\) km at \(90\) km/h. (72 km/h.)
• 18. “To reverse a \(30\%\) discount, add \(30\%\) back on.” Why wrongThe \(30\%\) was a percentage of the original price, and the \(30\%\) you would add back is a percentage of the sale price. Different bases, so the operations are not inverses. CorrectDivide by the multiplier: \(84\div0.70=\$120\). Adding \(30\%\) would give \(\$109.20\). Say this“A reverse percentage divides by the multiplier.” CheckA price is \(\$156\) after a \(20\%\) rise. Original? (\(156\div1.20=\$130\).)
• 19. “Three years of \(15\%\) depreciation is \(45\%\).” Why wrongEach year's \(15\%\) is taken from a smaller amount than the year before, so the reductions cannot be added. CorrectMultiply: \(0.85^3=0.614125\), a total fall of about \(38.6\%\), not \(45\%\). Say this“Repeated percentage changes multiply their multipliers.” CheckA \(10\%\) rise then a \(10\%\) fall. (\(1.1\times0.9=0.99\): a \(1\%\) net decrease.)
• 20. “\(\sqrt{288}=16.97\) is a complete answer.” Why wrongWhen a question asks for an exact value, a decimal is an approximation of it, not equal to it. Correct\(\sqrt{288}=\sqrt{144\times2}=12\sqrt2\). Keep fractions, surds and multiples of \(\pi\) exact whenever exactness is asked for or the data is exact. Say this“The largest square factor of 288 is 144, so \(\sqrt{288}=12\sqrt2\).” CheckSimplify \(\sqrt{98}\). (\(7\sqrt2\).)

• One rule that applies to every section below. Do the mathematics exactly for as long as you can, and round only on the final line. An intermediate value rounded to 3 significant figures and then reused will often push the final answer outside the accepted range, and the mark is lost even though the method was right.
• Read the AO split carefully. More than half of the assessment may involve analysis, interpretation and communication rather than routine execution. Being fast at the arithmetic of Topic 1 is necessary but not sufficient — the marks sit in choosing the right representation and explaining the choice.
• Learn 64 twice. \(64\) is both \(8^2\) and \(4^3\), so \(\sqrt{64}=8\) while \(\sqrt[3]{64}=4\). Reading the root sign carelessly here is a very cheap mark to lose.
• Work in improper fractions. Multiplying or dividing mixed numbers directly does not work: \(1\frac12\times2\frac12\) is \(\frac32\times\frac52=\frac{15}{4}\), not \(2\frac14\). Convert to improper form first, operate, then convert back only if the question asks for a mixed number.
• Read the dots before you start. \(0.\dot{1}2\dot{3}\) and \(0.1\dot{2}\dot{3}\) differ only in where the first dot sits, and they are different numbers — \(\frac{41}{333}\) and \(\frac{61}{495}\). Write out three or four decimal places in full before choosing the power of 10 to multiply by. This chapter uses the overbar as well, purely because it is easier to read on screen; the two notations mean exactly the same thing.
• The distinction between \(<\) and \(\le\) carries marks in 1.10. A bounds answer written \(8.35<L<8.45\) loses the lower endpoint, which is attainable. The correct statement is \(8.35\le L<8.45\).
• Difference and quotient are the two that catch people out, because the “largest” answer mixes an upper with a lower. Say it as a sentence rather than memorising a formula: to make a difference as large as possible, start with as much as possible and take away as little as possible.
• Minutes are not decimals of an hour. \(3\) h \(45\) min is \(3.75\) h, not \(3.45\) h. Minutes are sixtieths, not hundredths, so every mixed time has to be converted before it is substituted — and the same conversion reappears in 1.14 when reading a calculator display.
• Cambridge writes 24-hour times without a colon — \(03\,15\) and \(15\,15\). Follow the notation used in the question. Whichever you use, always write four digits: “\(3\,15\)” is ambiguous.
• Service D changes date. It leaves Greenhill at \(22\,55\) and reaches Marston at \(00\,04\) — the next day. Times in a timetable column always run forwards, so a time that appears to go backwards is telling you that midnight has been crossed.
• Before you look. Give yourself 25 minutes and write full working, not just answers. Several of these carry method marks that survive an arithmetic slip — but only if the method is on the page.
• Carry unrounded values between parts. Several of these questions use an earlier answer in a later part. If you round part (a) and reuse the rounded value, part (b) will drift outside the accepted range.