EduNinja

E1.1 Types of number

Number types describe properties, and one number may belong to several types. Prime factorisation then exposes the building blocks used to find common factors and multiples.

Type	Meaning	Examples
natural number	positive whole number	1, 2, 3, …
integer	whole number, including zero and negatives	−4, 0, 19
prime	integer greater than 1 with exactly two positive factors	2, 3, 5, 7
square / cube	$n^2$ / $n^3$ for an integer $n$	49 / 64
rational	can be written $a/b$ for integers $a,b$ , $b\ne0$	$-3$ , $2/5$ , 0.125, $0.\overline{3}$
irrational	cannot be written as an integer fraction	$\sqrt3$ , $\pi$
reciprocal of $x$	number whose product with $x$ is 1	reciprocal of $2/3$ is $3/2$

Place value controls number words. Six billion is 6 000 000 000, while 10 007 is ten thousand and seven: zero placeholders preserve the missing hundreds and tens.

To express an integer as a product of prime factors, divide repeatedly by primes until every factor is prime, then collect repeats with indices. For example, $180=2\times2\times3\times3\times5=2^2\times3^2\times5$ .

Job from prime factors	Selection rule
HCF	take only primes common to every number, using the lowest power
LCM	take every prime that appears, using the highest power

$126=2\times3^2\times7$ and $180=2^2\times3^2\times5$ . Their HCF is $2\times3^2=18$ . Their LCM is $2^2\times3^2\times5\times7=1260$ .

The number 1 is not prime, and 0 has no reciprocal. A square root is not automatically irrational: $\sqrt9=3$ is rational, whereas $\sqrt3$ is irrational. HCF selects shared factors; LCM builds the smallest number containing both factorisations.

E1.2 Sets

A set is a collection of distinct elements. In a Venn diagram, the rectangle is the universal set $\mathscr E$ and each circle contains the elements of one set. Overlaps show elements that satisfy more than one set description; regions outside a circle show elements that do not belong to that set.

Notation	Meaning
$n(A)$	number of elements in $A$
$x\in A$ / $x\notin A$	$x$ is / is not an element of $A$
$A'$	complement: elements of $\mathscr E$ that are not in $A$
$\varnothing$	empty set: a set with no elements
$A\subseteq B$ / $A\nsubseteq B$	every element of $A$ is / is not also in $B$
$A\cup B$	union: in $A$ or $B$ , including the overlap
$A\cap B$	intersection: in both $A$ and $B$

Sets may be listed, such as $D=\{a,b,c,\ldots\}$ , or defined by a rule. Examples are $A=\{x:x\text{ is a natural number}\}$ , $C=\{x:a\le x\le b\}$ , and $B=\{(x,y):y=mx+c\}$ . The colon means ‘such that’; an ordered pair $(x,y)$ is one element of $B$ .

To translate notation into a Venn region:

Work inside brackets first, then apply any complement mark.
For an intersection, keep only regions that satisfy every condition.
For a union, include regions that satisfy at least one condition.
For a complement, switch to the part of the universal set outside the named region.
With three sets, identify the most specific overlap first; for example, $A\cap B'\cap C$ is inside $A$ and $C$ but outside $B$ .

Let $\mathscr E=\{1,2,3,4,5,6,7,8,9,10\}$ , $A=\{2,4,6,8,10\}$ and $B=\{3,6,9\}$ . Then $A\cap B=\{6\}$ , $A\cup B=\{2,3,4,6,8,9,10\}$ , and $A'=\{1,3,5,7,9\}$ . Also $A\cap B'=\{2,4,8,10\}$ , so $n(A\cap B')=4$ .

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$

The intersection is subtracted because it was counted once in $n(A)$ and once in $n(B)$ . When filling a numerical Venn diagram, place the deepest intersection first, then pair-only regions, then single-set regions, and finally the region outside all sets. This prevents an overlap from being counted twice.

$x\in A$ says that $x$ is one element; $X\subseteq A$ says that every element of set $X$ lies in $A$ . The empty set $\varnothing$ has no elements, whereas $\{0\}$ contains one element. A complement is always relative to the stated universal set, and ‘or’ in a union includes the intersection unless the question explicitly excludes it.

E1.3 Powers and roots

A power tells you how many times to use a number as a factor. A matching root reverses that operation: $\sqrt{a}$ asks for the non-negative number whose square is $a$ , while $\sqrt[3]{a}$ asks for the number whose cube is $a$ .

$(a^2)^{1/2}=a\quad(a\ge0),\qquad (a^3)^{1/3}=a$

$n$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
$n^2$	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225

Read every square fact in both directions: $13^2=169$ , so $\sqrt{169}=13$ . The symbol $\sqrt{169}$ means the principal square root, so its value is $13$ , not $\pm13$ .

$n$	1	2	3	4	5	10
$n^3$	1	8	27	64	125	1000

Read cube facts in both directions: $4^3=64$ , so $\sqrt[3]{64}=4$ . Cubing and cube-rooting preserve sign: $(-3)^3=-27$ and $\sqrt[3]{-27}=-3$ .

For any other power or root, match the index. For example, $5^4=5\times5\times5\times5=625$ , and $\sqrt[4]{625}=5$ because $5^4=625$ . A root with no small index shown is a square root.

Calculation	Working and check	Result
$\sqrt[4]{39\frac1{16}}$	$39\frac1{16}=39.0625$ ; check $2.5^4$	$2.5$
$\dfrac{4.87-2.7}{-0.2+\sqrt[3]{0.729}}$	$\sqrt[3]{0.729}=0.9$ , then evaluate numerator and denominator	$2.17\div0.7=3.1$
$6^3$	$6\times6\times6$	$216$

Do not multiply the base by the exponent: $6^3$ is not $18$ . Do not halve a number to find its square root. When entering a compound expression on a calculator, keep each root and its radicand grouped, and check a root by raising the result to the matching power. Index laws, negative indices and fractional-index rules are taught separately in E1.7.

E1.4 Fractions, decimals and percentages

Fractions, decimals and percentages are different ways to describe a quantity relative to one whole. The appropriate form depends on what needs to be clear: equal parts, decimal place value, or comparison with 100.

Form	Meaning	Example
proper fraction	numerator is smaller than denominator, so the value is less than 1	$\frac35$
improper fraction	numerator is at least the denominator, so the value is at least 1	$\frac74$
mixed number	a whole-number part plus a proper fraction	$1\frac34=1+\frac34$
decimal	digits after the point represent tenths, hundredths, and so on	$0.25$ is 25 hundredths
percentage	a number of parts per 100	$25\%=25$ out of 100

In $\frac ab$ , the denominator $b$ tells how many equal parts make one whole and the numerator $a$ tells how many of those parts are taken. The denominator cannot be zero.

The whole and the units matter. For example, 24.60 dollars as a fraction of 2870 dollars is $\frac{24.60}{2870}$ because both amounts use the same unit. A percentage can be greater than $100\%$ when a quantity exceeds the chosen whole.

A mixed number is a sum, so $2\frac13$ means $2+\frac13$ , not $2\times\frac13$ . A decimal is not automatically a percentage: $0.4$ means four tenths, while $0.4\%$ means four tenths of one percent.

Equivalent fractions, decimals and percentages name the same value. A conversion changes the notation, not the quantity; finish fractional answers in simplest form.

Conversion	Method	Example
fraction $\to$ decimal	numerator $\div$ denominator	$\frac38=0.375$
decimal $\to$ fraction	use place value, then simplify	$0.24=\frac{24}{100}=\frac6{25}$
decimal $\to$ percentage	multiply by 100 and add $\%$	$0.375=37.5\%$
percentage $\to$ decimal	divide by 100	$62\%=0.62$
improper $\leftrightarrow$ mixed	divide, or combine wholes and parts	$\frac74=1\frac34$

Dots identify the recurring block. For example, $0.1\dot7=0.1777\ldots$ , $0.1\dot2\dot3=0.1232323\ldots$ , and $0.\dot1 2\dot3=0.123123\ldots$ . To convert a recurring decimal to a fraction, multiply by powers of 10 until two versions have the same recurring tail, then subtract so the tail cancels.

Let $x=0.\overline{37}=0.373737\ldots$ . Then $100x=37.373737\ldots$ . Subtracting gives $99x=37$ , so $x=\frac{37}{99}$ .

For a non-recurring prefix, let $x=0.4\overline{19}=0.4191919\ldots$ . Then $1000x=419.1919\ldots$ and $10x=4.1919\ldots$ . Subtract: $990x=415$ , so $x=\frac{415}{990}=\frac{83}{198}$ .

To convert a fraction to a decimal, divide. If the digits repeat, mark the shortest complete recurring block: $\frac2{11}=0.181818\ldots=0.\overline{18}$ . Check any conversion by evaluating the final fraction as a decimal.

Multiply both versions of $x$ by powers of 10 that align the same recurring tail; subtracting mismatched tails gives a false fraction. Dots belong over the first and last digits of the repeating block, not over a non-recurring prefix.

E1.5 Ordering

To order quantities reliably, express them on one common scale and compare their positions from smallest to largest. On a number line, values increase to the right; this remains true for negatives, fractions, decimals and percentages.

Symbol	Meaning	Example
$=$	equal to	$0.5=50\%$
$\ne$	not equal to	$0.5\ne5\%$
$>$	greater than	$0.5>5\%$
$<$	less than	$-7<-5$
$\ge$	greater than or equal to	$x\ge3$ includes 3
$\le$	less than or equal to	$x\le3$ includes 3

Choose a common form that preserves enough accuracy: divide numerator by denominator for fractions and divide percentages by 100. Keep extra decimal places until the order is secure; rounding close values too early can make unequal quantities appear equal.

Original value	Comparable decimal
$58\%$	$0.58$
$\frac7{12}$	$0.5833\ldots$
$0.6$	$0.6$
$\frac8{13}$	$0.6153\ldots$
$\frac23$	$0.6666\ldots$

Therefore $58\%<\frac7{12}<0.6<\frac8{13}<\frac23$ . In an inequality chain, every neighbouring comparison must point consistently from the smaller value towards the larger value.

A common denominator can expose a value between two fractions. Since $\frac3{25}=\frac6{50}$ and $\frac4{25}=\frac8{50}$ , $\frac7{50}$ lies strictly between them.

The symbols $\ge$ and $\le$ include equality; $>$ and $<$ do not. For negative values, the number with the greater absolute size can be smaller: $-8<-3$ because $-8$ lies farther left on the number line.

E1.6 The four operations

The four operations describe different relationships: addition combines, subtraction finds change or difference, multiplication scales, and division shares or finds how many groups fit. Choose the relationship first, then preserve its structure through the agreed calculation order.

Priority	Calculate	Boundary
1	brackets	work from innermost brackets outward
2	powers and roots	evaluate each complete powered or rooted value
3	multiplication and division	work left to right when tied
4	addition and subtraction	work left to right when tied

For $9+5\times7-4\div2$ , multiplication and division come first: $9+35-2=42$ . Brackets change the structure: $(9+5)\times7-4\div2=14\times7-2=96$ .

Signed operation	Reliable rule	Example
add or subtract	rewrite subtraction as adding the opposite	$-7-5+8=-12+8=-4$
multiply or divide	same signs give positive; different signs give negative	$-18\div(-4)=4.5$
difference	when context asks how far apart, use the positive distance	$5-(-7)=12$

Fraction operation	Method
add or subtract	use a common denominator, combine numerators, simplify
multiply	convert mixed numbers to improper fractions, multiply, simplify
divide	multiply by the reciprocal of the divisor, simplify

$1\frac56+\frac25=\frac{11}{6}+\frac25=\frac{55}{30}+\frac{12}{30}=\frac{67}{30}=2\frac7{30}$

For decimal addition and subtraction, align place values. For multiplication or division, track place-value scale and estimate first: $1.6\times0.02$ is near $2\times0.02=0.04$ , so the exact result $0.032$ has a sensible size.

Translate practical wording before calculating. ‘ $5^\circ$ C lower than $-7^\circ$ C’ means $-7-5=-12$ ; ‘ $8^\circ$ C higher’ then means $-12+8=-4$ . If the context requires whole coaches or bags, interpret the remainder and round up rather than reporting part of an item.

Do not work mechanically from left to right across different operation priorities, and do not add denominators when adding fractions. A negative sign belongs to its number; subtracting a negative changes the operation to addition.

E1.7 Indices I

An index tells how a base is powered. Positive whole-number indices repeat multiplication; zero, negative and fractional indices extend the same pattern so powers can also represent 1, reciprocals and roots.

Index	Meaning	Conditions	Example
$a^n$	multiply $n$ factors of $a$	$n$ positive integer	$3^4=81$
$a^0$	1	$a\ne0$	$7^0=1$
$a^{-n}$	reciprocal $1/a^n$	$a\ne0$	$2^{-4}=1/16=0.0625$
$a^{1/n}$	principal $n$ th root	real even roots need $a\ge0$	$64^{1/3}=4$
$a^{m/n}$	take the $n$ th root and raise to power $m$	same root condition	$64^{2/3}=4^2=16$

$a^{-m/n}=\frac{1}{a^{m/n}}=\frac{1}{(\sqrt[n]{a})^m}$

For $27^{-2/3}$ , take the cube root first: $27^{1/3}=3$ . Square to get $27^{2/3}=9$ , then use the negative index as a reciprocal: $27^{-2/3}=1/9$ .

The radical or a fractional power gives its principal value, not a plus-or-minus pair: $16^{1/2}=4$ . The two solutions of the equation $x^2=16$ are $x=\pm4$ , but that is a separate equation-solving statement. A negative index does not make the value negative; it makes a reciprocal.

Index laws preserve repeated multiplication. They apply when powers share the same base; rewrite numbers to a common base before combining their indices.

Structure	Index law	Condition
multiply same base	$a^m\times a^n=a^{m+n}$	same base
divide same base	$a^m\div a^n=a^{m-n}$	same non-zero base
power of a power	$(a^m)^n=a^{mn}$	multiply the indices
power of a product	$(ab)^n=a^n b^n$	power applies to every factor

Expression	Index step	Result
$2^{-3}\times2^4$	$2^{-3+4}$	$2$
$(2^3)^2$	$2^{3\times2}$	$64$
$2^3\div2^4$	$2^{3-4}=2^{-1}$	$1/2$

To write $243\times27^{2n}$ as one power of 3, first rewrite both bases: $243=3^5$ and $27^{2n}=(3^3)^{2n}=3^{6n}$ . Then multiply equal bases: $3^5\times3^{6n}=3^{6n+5}$ .

Do not add indices when adding powers: $2^3+2^4=8+16=24$ , not $2^7$ . In $(a^m)^n$ , multiply the indices; do not raise $m$ to the power $n$ . Multi-variable algebraic simplification is developed later in E2.4.

E1.8 Standard form

Standard form writes a non-zero number as a coefficient multiplied by an integer power of 10. The coefficient carries the significant digits and the power records the scale.

$A\times10^n,\qquad 1\le A<10,\qquad n\in\mathbb Z$

Part	Requirement	Meaning
$A$	at least 1 and less than 10	exactly one non-zero digit before the decimal point
$10^n$	$n$ is an integer	positive $n$ gives a large scale; negative $n$ gives a scale below 1

The expression $85.1\times10^4$ has a clear value but is not in standard form because $85.1\ge10$ . Move the decimal one place left and increase the exponent by 1: $85.1\times10^4=8.51\times10^5$ .

A power of 10 alone does not make a representation standard form: $0.3\times10^{-2}$ is invalid because its coefficient is below 1. Standard form preserves the exact value; it is not automatically a rounded approximation.

To convert into standard form, move the decimal point until the coefficient lies from 1 inclusive to 10 exclusive, then choose the power of 10 that restores the original place value.

Ordinary number	Coefficient	Movement used to make it	Standard form
$510100000$	$5.101$	8 places left	$5.101\times10^8$
$0.0605$	$6.05$	2 places right	$6.05\times10^{-2}$
$0.0000000347$	$3.47$	8 places right	$3.47\times10^{-8}$

A positive exponent restores a large number by moving the decimal point right. A negative exponent restores a small number by moving it left. The exponent reverses the movement used to make the coefficient.

Standard form	Apply the scale	Ordinary number
$4.73\times10^6$	move 6 places right	$4730000$
$2.06\times10^{-2}$	move 2 places left	$0.0206$
$3.47\times10^{-8}$	move 8 places left	$0.0000000347$

Do not choose the exponent by counting visible zeros alone; count place-value moves from the original decimal point. Check both value and format by converting back and confirming $1\le A<10$ .

In standard-form calculations, operate on coefficients and powers separately, then normalise the result so the coefficient returns to $1\le A<10$ .

Operation	Method	Example before final normalisation
multiply	multiply coefficients; add exponents	$(4.1\times10^{-3})(8.9\times10^7)=36.49\times10^4$
divide	divide coefficients; subtract exponents	$(6.39\times10^4)\div(2.45\times10^6)=2.608\ldots\times10^{-2}$
add or subtract	first rewrite terms with the same power	$3\times10^5+4\times10^4=3\times10^5+0.4\times10^5$

$36.49\times10^4=3.649\times10^5$

$3\times10^5+0.4\times10^5=3.4\times10^5$

For a power, apply it to both parts: $(3\times10^{-3})^3=3^3\times10^{-9}=27\times10^{-9}=2.7\times10^{-8}$ .

Units must be consistent before calculating. If one atom has mass $7.95\times10^{-23}$ g, then 1 kg is 1000 g and the atom count is $1000\div(7.95\times10^{-23})=1.257\ldots\times10^{25}$ .

Keep full calculator precision, normalise first, and round only the final coefficient when accuracy is requested. Never add coefficients until powers match, and never add exponents when adding numbers.

E1.9 Estimation

Rounding replaces a value with a nearby value at a stated place. Locate the final digit to keep, inspect the next digit, then keep it unchanged for 0–4 or increase it by 1 for 5–9.

Accuracy request	Where counting starts	Example
2 decimal places	first digit after the decimal point	$4.376\to4.38$
3 significant figures	first non-zero digit	$0.004376\to0.00438$
nearest thousand	thousands place	$5764\to6000$

After rounding a whole number, replace removed place-value digits with zeros. For a decimal, remove digits beyond the stated accuracy. Leading zeros only locate the decimal point and are not significant figures.

Trailing zeros may communicate the requested precision: $2.50$ has 3 significant figures. For a large value where zeros are ambiguous, standard form can make the accuracy explicit, for example $2.5\times10^4$ to 2 significant figures.

Round once from the original value. Repeated rounding can change the result: $2.449$ is $2.4$ to 2 significant figures, even though rounding first to $2.45$ and then again would incorrectly give $2.5$ .

An estimate is a deliberately approximate result used to judge size and reasonableness. When an instruction specifies 1 significant figure, round every input to 1 significant figure before calculating.

Step	Action	Check
1	round each input as instructed	every rounded number has the requested accuracy
2	copy the operations and brackets unchanged	the structure still matches the original calculation
3	calculate with the simpler values	write $\approx$ , not $=$ , between the original and estimate
4	compare scale and sign	the estimate is plausible for the original values

$\frac{8.2\times3.8}{29.6-9.7}\approx\frac{8\times4}{30-10}=\frac{32}{20}=1.6$

If no accuracy is prescribed, choose nearby numbers that simplify the calculation without changing its scale. Avoid choices that make a denominator zero or erase an important small quantity.

Estimating the calculation is different from calculating exactly and merely rounding the final answer. In an estimate, the inputs are simplified first; in an exact calculation, retain full precision until the end.

A final answer should match both the instruction and what the context can meaningfully represent. Keep full calculator precision during the working, then round once at the end.

Context or instruction	Usually sensible final form
people, trees or complete objects	a whole number, interpreted according to the situation
money in ordinary transactions	two decimal places
a measured quantity	no more precision than the data justify
nearest 1000, stated decimal places or significant figures	exactly the requested accuracy

Suppose 150 trees each produce 52.4 kg of fruit and one fruit has mass 180 g. Convert kilograms to grams before dividing: $150\times52.4\times1000\div180=43666.6\ldots$ . To the nearest thousand, the estimate is $44000$ fruits.

A context can also control direction. A count of complete boxes needed may require rounding up, whereas the number of complete groups that can be made may require rounding down. Follow an explicit accuracy instruction when one is given.

Do not round intermediate values unless the question asks for an estimate. Premature rounding accumulates error, and excessive decimal places can claim a precision that the measurements do not support.

E1.10 Limits of accuracy

A rounded measurement stands for a range of possible original values. Identify one unit at the stated accuracy, halve it, then subtract and add that half-unit to locate the two boundaries.

$a-\frac{r}{2}\le x<a+\frac{r}{2}$

Here $a$ is the reported value, $r$ is the rounding step and $x$ is the original value. For positive data rounded to the nearest value, the lower boundary is included but the upper boundary is excluded: the upper boundary would round to the next reported value.

Reported accuracy	Step $r$	Half-step	Possible original values
470 to the nearest 10	10	5	$465\le x<475$
6.2 to 1 decimal place	0.1	0.05	$6.15\le x<6.25$
830 to 2 significant figures	10	5	$825\le x<835$

Keep the step in the same unit as the data before halving. A length of 3.6 m correct to the nearest 20 cm has step $0.2$ m and half-step $0.1$ m, so $3.5\le L<3.7$ m.

Bounds are exact boundary values, so do not round them again. Do not use the full rounding step on each side, and do not include the upper boundary with $\le$ .

To bound a result, first replace every rounded input by its interval. Then choose the endpoint combination that makes the required result as small or as large as possible; do not automatically choose all lower bounds or all upper bounds.

Positive-quantity calculation	Smallest result uses	Largest result uses
$A+B$ or $A\times B$	lower $A$ , lower $B$	upper $A$ , upper $B$
$A-B$	lower $A$ , upper $B$	upper $A$ , lower $B$
$A\div B$	lower $A$ , upper $B$	upper $A$ , lower $B$

A rectangle is reported as 8.4 cm by 5 cm, correct to the nearest 0.1 cm and nearest centimetre. Its area approaches its greatest value as both dimensions approach their upper boundaries, so the upper bound is $8.45\times5.5=46.475$ cm $^2$ ; the actual area is less than this value.

For speed $=\text{distance}\div\text{time}$ , the smallest speed uses the lower distance and upper time. If 84.6 km is correct to 0.1 km and 6 h is correct to the nearest hour, the lower bound is $84.55\div6.5=13.007\ldots$ km/h.

Keep boundary values exact throughout and round only if the question requests a final degree of accuracy. State whether the result is a lower or upper bound and retain its units.

The table assumes positive quantities and expressions that change monotonically. With negative values, squares or a denominator whose interval crosses zero, analyse how the expression changes instead of applying the table mechanically.

E1.11 Ratio and proportion

A ratio compares quantities as equal-sized parts. In $a:b:c$ , the three quantities can be written $ka$ , $kb$ and $kc$ for the same multiplier $k$ . Preserve the stated order and convert quantities to the same units before comparing them.

Information given	Use the ratio parts	Example
simplify	divide every term by the same highest common factor	$45:75:120=3:5:8$
total shared in $a:b:c$	divide the total by $a+b+c$ to find one part	share 540 in $2:3:4$ : one part $=540\div9=60$ , so the shares are 120, 180 and 240
one quantity known	divide it by its matching ratio number	blue:red $=5:7$ and blue $=45$ : one part $=9$ , so red $=63$
difference known	divide the difference by the difference in ratio parts	$A:B=4:9$ and $B-A=35$ : five parts $=35$ , so $A=28$ and $B=63$

$\text{one part}=\frac{\text{known quantity}}{\text{matching number of ratio parts}}$

To combine linked ratios, make the shared quantity use the same number of parts. If $P:Q=2:5$ and $Q:R=3:4$ , scale the first ratio by 3 and the second by 5: $P:Q:R=6:15:20$ .

Direct proportion keeps a constant multiplier. A map scale of $1:250000$ uses the same units, so 3.2 cm represents $3.2\times250000=800000$ cm, or 8 km. For recipes or best value, scale to the required quantity or compare each option at one common quantity.

The ratio $2:3$ does not mean the first share is $2/3$ of the total: there are $2+3=5$ parts, so the shares are $2/5$ and $3/5$ . Adding the same amount to both quantities does not preserve a ratio; proportional scaling multiplies both by the same factor.

E1.12 Rates

A rate compares quantities with different units. Read 'per' as division: 18 litres per minute means 18 litres for each minute. The written unit shows which quantity is produced and which reference quantity it is measured against.

$\text{rate}=\frac{\text{quantity}}{\text{reference quantity}}\qquad \text{quantity}=\text{rate}\times\text{reference quantity}$

Context	Rate unit	Use
hourly pay	dollars/hour	pay $=$ rate $\times$ hours
currency exchange	new currency/old currency	multiply in the stated direction; divide to reverse it
flow	litres/minute	volume $=$ rate $\times$ time
fuel use	litres/100 km	fuel per 100 km $=\dfrac{\text{total fuel}}{\text{total distance}}\times100$

A machine makes 11 components every 15 minutes. In 6 hours there are $6\times60\div15=24$ intervals, so it makes $24\times11=264$ components. Converting the time first keeps the rate's reference unit consistent.

Units also guide conversion. A flow of 3.6 m $^3$ /h is $3600$ litres per hour, then $3600\div60=60$ litres per minute. Apply a conversion to the quantity named by that part of the compound unit.

Do not multiply automatically: divide when finding a rate from two totals, and multiply when a rate and reference quantity are known. Fuel consumption in litres/100 km is not the same quantity as efficiency in km/litre.

Pressure, material density and population density each measure one quantity per unit of another. The required formula is supplied in the question; identify its numerator and denominator, then make their units compatible before substituting.

Measure	Typical supplied formula	Meaning of result
pressure	$P=F/A$	force per unit area, such as N/m $^2$
density	$\rho=m/V$	mass per unit volume, such as g/cm $^3$
population density	$d=N/A$	people per unit area, such as people/km $^2$

Copy the given formula, convert the data to the units required by the answer, substitute, and rearrange only if the unknown is not isolated. Preserve the numerator/denominator order in the compound unit.

A cuboid of sand measures $0.60$ m by $0.30$ m by $0.25$ m and has mass 54 kg. Its volume is $0.60\times0.30\times0.25=0.045$ m $^3$ , so $\rho=54\div0.045=1200$ kg/m $^3$ .

If density and volume are known, $m=\rho V$ ; if density and mass are known, $V=m/\rho$ . Area and volume conversions must be squared or cubed: $1$ m $^2=10000$ cm $^2$ and $1$ m $^3=1000000$ cm $^3$ .

A density below 1 person/km $^2$ is meaningful as an average over an area. Do not interpret it as a fractional person at one location, and do not mix kilograms with cubic centimetres unless the final unit deliberately combines them.

Average speed describes an entire journey: divide total distance travelled by total elapsed time. Include every journey section and any stop whose time is included in the stated journey.

$\text{average speed}=\frac{\text{total distance}}{\text{total time}}\qquad d=st\qquad t=\frac{d}{s}$

Add the relevant distances, add the relevant times, convert time to the unit required by the speed, then divide. For 72 km in 1 h 36 min, $36/60=0.6$ h, so the average speed is $72\div1.6=45$ km/h.

For 30 km at 60 km/h followed by 40 km at 40 km/h, the times are $30/60=0.5$ h and $40/40=1$ h. The whole-journey average is $(30+40)\div(0.5+1)=46.7$ km/h, not the mean of 60 and 40.

To convert m/s to km/h, multiply by $3600/1000=3.6$ ; reverse the conversion by dividing by 3.6. A decimal hour must be converted by multiplying its fractional part by 60: 2.4 h is 2 h 24 min.

Never average stage speeds unless the stage times are equal. Average speed is total distance divided by total time, and its unit must match the distance and time units used in that division.

E1.13 Percentages

A percentage is a number of hundredths. To calculate $p$ percent of a quantity $Q$ , convert $p\%$ to the multiplier $p/100$ and multiply.

$p\%\text{ of }Q=\frac{p}{100}\times Q$

To find 37% of 640 dollars, calculate $0.37\times640=236.8$ , so the percentage amount is 236.80 dollars. For 135% of 80, $1.35\times80=108$ ; percentages above 100% are valid.

Percentage	Efficient route	Example
10%	divide by 10	10% of 470 is 47
1%	divide by 100	1% of 470 is 4.7
5%	half of 10%	5% of 470 is 23.5
25%	divide by 4	25% of 360 is 90

The percentage amount is not automatically the final value. A tax or increase amount is added to the original; a discount or decrease amount is subtracted. Keep money to cents only when reporting the final monetary result.

To express one quantity as a percentage of another, divide the quantity being described by the reference whole. The wording after 'of' usually identifies that whole.

$\text{percentage}=\frac{\text{part}}{\text{reference whole}}\times100\%$

Identify the part and whole, convert them to matching units, divide in that order, multiply by 100 and round only as requested.

In a box of 96 items, 18 are damaged. The damaged percentage is $18\div96\times100=18.75\%$ . To compare 1.92 with 1.60, $1.92\div1.60\times100=120\%$ .

A percentage has no physical unit, but the compared quantities must use the same unit. For example, 450 g as a percentage of 2 kg is $450\div2000\times100=22.5\%$ .

Do not reverse the fraction. A result above 100% simply means the part exceeds the chosen reference whole; it is not an error.

Percentage change compares the change with the original value. The original value is the denominator whether the result is an increase, decrease, profit or loss.

$\text{percentage change}=\frac{|\text{new}-\text{original}|}{\text{original}}\times100\%$

Required result	Multiplier or comparison
increase by $p$ %	multiply by $1+p/100$
decrease by $p$ %	multiply by $1-p/100$
percentage profit	profit $\div$ cost price $\times100$
percentage loss	loss $\div$ cost price $\times100$

A price rises from 72 dollars to 90 dollars. The increase is 18 dollars, so $18\div72\times100=25\%$ . Reducing 560 dollars by 18% uses multiplier $0.82$ , giving $560\times0.82=459.2$ dollars.

For repeated changes, multiply the factors rather than adding the percentages. A 12% rise followed by a 5% fall gives factor $1.12\times0.95=1.064$ , an overall 6.4% increase.

Equal percentage increases and decreases do not cancel because the second change uses a different base. Keep the change amount, change percentage and final value distinct.

Simple interest is calculated from the original principal every period, so it adds a constant amount. Compound interest is calculated from the current balance, so each period multiplies the balance and previous interest also earns interest.

Type	Formula	Pattern
simple	$I=P(r/100)n$ , then $A=P+I$	equal interest added each period
compound	$A=P(1+r/100)^n$	balance multiplied each period

$P$ is principal, $r$ is the percentage rate per period, $n$ is the number of matching periods, $I$ is total interest and $A$ is final amount. These formulas are not supplied, and monthly or daily rates require months or days in the exponent.

For a principal of 2400 dollars at 3% simple interest for 5 years, $I=2400\times0.03\times5=360$ and $A=2760$ . At 3% compound interest, $A=2400(1.03)^5=2782.58\ldots$ .

For compound interest with known $P$ , $A$ and $n$ , the period multiplier is $(A/P)^{1/n}$ , so $r=100[(A/P)^{1/n}-1]$ . For a minimum number of complete periods, test integer powers until the threshold is first reached.

Do not use $P(r/100)n$ for compound interest. Distinguish total interest $A-P$ from the final balance $A$ , and keep full precision until the requested final rounding.

A reverse-percentage problem gives a final amount and asks for the original. Write the forward multiplier first, then divide the final amount by it.

$\text{original}=\frac{\text{final}}{\text{forward multiplier}}$

Final amount described as	Forward multiplier	Reverse calculation
after a $p$ % increase or profit	$1+p/100$	final $\div(1+p/100)$
after a $p$ % decrease or discount	$1-p/100$	final $\div(1-p/100)$
$p$ % of the original	$p/100$	final $\div(p/100)$

A sale price of 44.80 dollars follows a 20% discount, so it is 80% of the original: $44.80\div0.80=56$ . A selling price of 270 dollars after a 35% profit gives cost price $270\div1.35=200$ dollars.

Reverse repeated changes by dividing by their product. If a value rises 10% and then falls 20% to 440, the original is $440\div(1.10\times0.80)=500$ .

Do not subtract the stated percentage from the final amount: that percentage was taken from the unknown original. Tax included, discount, profit and capacity questions all use the same multiplier logic.

E1.14 Using a calculator

Efficient calculator work begins before pressing keys: predict the sign and approximate size, preserve the written expression with brackets, and keep unrounded values until the final answer.

Stage	Controlled action	Check
plan	identify powers, roots, numerator and denominator	estimate sign and size
enter	use brackets or a fraction template	read back the expression line
continue	reuse the full display or answer memory	do not copy a shortened decimal
finish	round once to the requested accuracy	compare with the estimate

For $\sqrt{45.7-2.4^2}\div3.1$ , keep the whole difference under the root. The display is $2.038648\ldots$ , which is reasonable because the numerator is a little above 6 and the denominator is a little above 3.

If an intermediate value is $7.428571\ldots$ , using 7.43 in the next step can alter the final digit. Use the calculator's full stored value, then apply the stated decimal-place or significant-figure rule only at the end.

A calculator evaluates the keys entered, not the expression intended. An unexpected sign or scale should trigger a bracket, exponent and mode check—not a forced change to the displayed answer.

A safe calculator entry must represent exactly the same mathematical object as the written value. Group complete numerators, denominators, powers and time values before evaluation.

Written value	Safe entry	What it protects
$\dfrac{18.4}{7.2-1.6}$	`18.4 ÷ (7.2 − 1.6)` or a fraction template	the whole denominator
$(3.1-0.8)^4$	`(3.1 − 0.8)` then power 4	the whole base of the power
$6.4\times10^{-5}$	standard-form exponent key with exponent −5	a negative exponent, not subtraction

For decimal-hour calculations, 2 hours 45 minutes is $2+45/60=2.75$ hours. On a calculator with a degrees–minutes–seconds key, the same time can be entered as $2^\circ45'0''$ .

Before evaluating, confirm every opening bracket is closed, the negative sign belongs to the intended exponent or value, and the calculator is in the mode required by the expression.

Do not enter 2.45 to mean 2 hours 45 minutes: 2.45 hours is 2 hours 27 minutes. Decimal hours and hours–minutes notation use different place values.

A calculator display is only a numerical value; the context decides its unit, formatting and whether it must be rounded or converted.

Display	Context	Appropriate interpretation
4.8	dollars	4.80 dollars
3.25	hours	3 hours 15 minutes
2.4	hours	2 hours 24 minutes
7.6666…	whole objects available	7 complete objects, if partial objects cannot be used
0.333333…	answer to 3 significant figures	0.333

For a decimal-hour display, keep the whole-number part as hours and multiply only the fractional part by 60. If a fractional minute remains, multiply that fraction by 60 again to obtain seconds.

For money, apply the required rounding and then show two decimal places: 12 becomes 12.00 dollars and 12.5 becomes 12.50 dollars. The trailing zero communicates cents even though it does not change the numerical value.

A decimal point does not separate hours from minutes, and the calculator cannot decide whether a contextual count should round up, round down or use an ordinary rounding rule. Interpret the situation before formatting the answer.

E1.15 Time

Time units do not use one place-value system. Convert through the exact relationship between neighbouring units, multiplying toward smaller units and dividing toward larger units.

Relationship	Toward the smaller unit	Toward the larger unit
1 minute = 60 seconds	minutes × 60	seconds ÷ 60
1 hour = 60 minutes	hours × 60	minutes ÷ 60
1 day = 24 hours	days × 24	hours ÷ 24
1 week = 7 days	weeks × 7	days ÷ 7
1 year = 12 months	years × 12	months ÷ 12

For day calculations in this syllabus, use 1 year = 365 days unless the question supplies other information.

July has 31 days, so it contains $31\times24\times60\times60=2\,678\,400$ seconds. For $2.15$ hours, multiply the whole decimal by 60: $2.15\times60=129$ minutes.

For a mixed-unit answer, convert to the smallest useful unit first. Eighteen trips of 23 minutes total 414 minutes; $414\div60=6$ remainder 54, so the time is 6 hours 54 minutes.

A decimal hour is not hours and minutes: 2.15 hours is 2 hours 9 minutes, not 2 hours 15 minutes. Months have different numbers of days, so do not invent a fixed month-to-day conversion unless the question provides one.

Clock times name positions in a day; durations measure the interval between them. Convert both clock formats consistently, then add, subtract or bridge midnight in hours and minutes.

12-hour time	24-hour time	Rule
3.25 a.m.	03 25	keep the hour and add a leading zero
3.25 p.m.	15 25	add 12 to the hour
12.00 noon	12 00	noon starts the p.m. half of the day
12.00 midnight	00 00	midnight starts a new day

To add a duration, add minutes first and exchange every 60 minutes for 1 hour. Starting at 19 50, adding 2 hours 42 minutes gives 21 92, which normalises to 22 32.

For an interval across midnight, split at 24 00. From 21 15 to 24 00 is 2 hours 45 minutes; from 00 00 to 04 33 is 4 hours 33 minutes. The total is 7 hours 18 minutes.

To find a start time, reverse the process and borrow 1 hour as 60 minutes when necessary. A film ending at 23 05 after 2 hours 50 minutes started at 20 15.

Do not subtract clock digits as ordinary base-10 numbers. There are 60 minutes in an hour, and 24 00 is the same boundary instant as 00 00 on the next day.

A timetable links places or events to clock times. Read the correct row and column first; then distinguish travel time, waiting time and local-time differences.

For each stage, pair its departure with its arrival. A boat departing Millwater at 11 45 and arriving Westbridge at 13 07 travels for 1 hour 22 minutes. A wait between an arrival and the next departure is not travel time, but it is included if the question asks for the whole journey.

Statement	Conversion at the same instant
destination is $k$ hours ahead	destination time = source time + $k$ hours
destination is $k$ hours behind	destination time = source time − $k$ hours

For a flight: 1. Start with the departure day and local time. 2. Add the flight duration in the departure time zone. 3. Apply the destination's ahead/behind offset. 4. Move the day or date whenever the running time crosses 24 00 or 00 00.

A plane leaves Seattle at 07 30 on Tuesday, flies for 10 hours 55 minutes, and Seoul is 16 hours ahead. In Seattle time it lands at 18 25 Tuesday; adding 16 hours gives 10 25 Wednesday in Seoul.

A time-zone offset changes the local clock label, not the flight duration. Do not add the offset to the elapsed flying time, and always state the new day or date after a midnight crossing.

E1.16 Money

Money calculations combine quantities, unit prices and payment rules. Keep every amount in one currency and one unit, model the whole bill, then format the final amount appropriately.

Money job	Calculation structure
Cost of several items	quantity × unit price
Total bill	add every item or charge once
Change	amount paid − total bill
Fixed fee plus usage	fixed fee + number of additional units × extra-unit price
Maximum whole items	divide budget by unit price, then take the whole-number part

With 20 dollars and pineapples costing 1.45 dollars each, $20\div1.45=13.79\ldots$ , so at most 13 can be bought. Their cost is $13\times1.45=18.85$ dollars, leaving $20-18.85=1.15$ dollars change.

Convert cents and dollars before combining them: 47 cents is 0.47 dollars, while 5 dollars is 500 cents. For mass or volume prices, also match the quantity unit; 125 ml is 0.125 litres before multiplying by a price per litre.

Read special pricing literally. “Buy 3 for the price of 2” means every complete group of 3 costs 2 unit prices. “First hour 15.50 dollars, each additional hour 7.25 dollars” means the first hour is not charged again at 7.25 dollars.

Round a final money answer to the smallest stated currency unit, normally two decimal places for dollars. Do not round a unit price or intermediate total early, and never round a maximum item count upward beyond the available budget.

An exchange rate is a unit rate. Read its direction before calculating: if 1 unit of currency A equals $r$ units of currency B, then $r$ converts one A into B.

$1\text{ A}=r\text{ B}$

Conversion	Operation	Unit check
A to B	multiply by $r$	A × B/A = B
B to A	divide by $r$	B ÷ B/A = A

If 1 rupee = 0.016 dollars, a 20-dollar ticket costs $20\div0.016=1250$ rupees. Division is required because the given rate tells how many dollars one rupee is worth, not how many rupees one dollar buys.

For two currencies with a common bridge, convert in two labelled stages. If 1 dollar = 0.615 euros and 1 krona = 0.087 euros, then 2000 dollars becomes $2000\times0.615=1230$ euros, then $1230\div0.087=14\,137.93\ldots$ krona, or 14 138 krona to the nearest krona.

To compare prices in different currencies, convert both prices into the same currency first, subtract to find the difference, and round only the final requested amount.

Do not choose multiply or divide from whether the number should become larger. Currency values vary; let the written rate and units determine the operation. Keep full precision until the final money rounding.

E1.17 Exponential growth and decay

Exponential change applies the same percentage multiplier to the current amount in every equal time period. Because the amount changes each time, the numerical increase or decrease is not constant.

$A_n=A_0m^n$

Change per period	Multiplier $m$	Model after $n$ periods
growth by $r\%$	$1+\dfrac{r}{100}$	$A_n=A_0\left(1+\dfrac{r}{100}\right)^n$
decay by $r\%$	$1-\dfrac{r}{100}$	$A_n=A_0\left(1-\dfrac{r}{100}\right)^n$

A population of 250 000 decreases by 1.7% each year. After 5 years, $A_5=250000(0.983)^5=229460.9\ldots$ , so the population is 229 500 to the nearest hundred. The exponent is 5 because the multiplier is applied five times.

To recover an earlier amount, divide by the complete multiplier power. If a car is worth 6269.40 dollars after 3 years of 10% annual decay, its earlier value was $6269.40\div0.9^3=8600$ dollars.

For a missing rate, first isolate the multiplier: if 550 grows to 736 in 5 years, $m=\sqrt[5]{736/550}=1.06$ , so the growth rate is 6%. For the first whole period above or below a target, evaluate consecutive integer powers and choose the first one that crosses the target.

Use the number of percentage-change periods, not merely the difference between printed year numbers without checking the endpoints. Keep full calculator precision until the requested rounding, and do not replace repeated percentage change with simple change $A_0(1+nr/100)$ . Knowledge of $e$ is not required here.

E1.18 Surds

A surd is an irrational root kept in exact form, such as $\sqrt3$ . Simplify it by extracting square factors, then combine only terms with the same remaining surd.

$\sqrt{ab}=\sqrt a\sqrt b,\qquad \sqrt a\sqrt a=a\quad(a,b\ge0)$

Choose the largest square factor. Since $300=100\times3$ and $48=16\times3$ , $\sqrt{300}+\sqrt{48}=10\sqrt3+4\sqrt3=14\sqrt3$ . The matching $\sqrt3$ terms combine like algebraic terms.

Operation	Safe move	Example
add or subtract	simplify first, then combine like surds	$5\sqrt2-3\sqrt2=2\sqrt2$
multiply	multiply coefficients and radicands	$2\sqrt3\times4\sqrt6=8\sqrt{18}=24\sqrt2$
expand brackets	use ordinary distributive multiplication	$\sqrt5\times\sqrt5=5$

For $(3-\sqrt5)(2+3\sqrt5)$ , expand to $6+9\sqrt5-2\sqrt5-15$ . Collect rational and surd terms separately to obtain $-9+7\sqrt5$ .

Do not add unlike surds or split a root across addition: $\sqrt2+\sqrt3$ cannot be simplified, and $\sqrt{a+b}$ is not generally $\sqrt a+\sqrt b$ . Extract only genuine square factors and leave the answer exact.

To rationalise a denominator, multiply the fraction by a form of 1 that removes every surd from the denominator while preserving the fraction's value.

Denominator	Multiply numerator and denominator by	Why it works
$\sqrt c$	$\sqrt c$	$\sqrt c\times\sqrt c=c$
$a+\sqrt c$	$a-\sqrt c$	conjugates give $a^2-c$
$a-\sqrt c$	$a+\sqrt c$	conjugates give $a^2-c$

For a single surd, $\dfrac{6}{\sqrt{10}}\times\dfrac{\sqrt{10}}{\sqrt{10}}=\dfrac{6\sqrt{10}}{10}=\dfrac{3\sqrt{10}}5$ . Simplify the numerical fraction after the denominator becomes rational.

$(a+b\sqrt c)(a-b\sqrt c)=a^2-b^2c$

For $\dfrac1{\sqrt7+2}$ , use the conjugate $\sqrt7-2$ : $\dfrac1{\sqrt7+2}\times\dfrac{\sqrt7-2}{\sqrt7-2}=\dfrac{\sqrt7-2}{7-4}=\dfrac{\sqrt7-2}{3}$ .

Multiplying only the denominator changes the value; multiply the numerator by the same non-zero expression. For a two-term denominator, using the same sign creates another surd term, so use the conjugate with the opposite sign and simplify fully.

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