1. Number

For figures and words, group digits in threes from the right and preserve zero place holders: six billion is $6\,000\,000\,000$, while $10\,007$ is ten thousand and seven.

A prime factorisation contains only prime factors. For example, $72=2^3\times3^2$; stopping at $8\times9$ is not complete because both factors are composite.

The rectangle represents the universal set, written $\mathscr{E}$. A complement is meaningful only after this universe is fixed: changing $\mathscr{E}$ can change $A'$ even when $A$ stays the same.

Place the intersection first. Then place members of $A$ that are not in $B$ in the $A$-only region, members of $B$ that are not in $A$ in the $B$-only region, and all remaining members of $\mathscr{E}$ outside both circles.

The intersection is subtracted once because it was counted once in $n(A)$ and again in $n(B)$. Also, $n(A')=n(\mathscr{E})-n(A)$ when every element is counted within the same universal set.

Read the square table in either direction: $13^2=169$ and $\sqrt{169}=13$. The radical $\sqrt{169}$ means the principal, non-negative square root.

Read the cube table in either direction: $4^3=64$ and $\sqrt[3]{64}=4$. Cubes and cube roots also preserve sign, so $(-3)^3=-27$ and $\sqrt[3]{-27}=-3$.

For another power or root, identify the index before calculating. For example, $5^4=5\times5\times5\times5=625$, and $\sqrt[4]{625}=5$ because $5^4=625$.

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

A fraction of a quantity must compare like units and use the whole as the denominator. For example, $32$ cm as a fraction of $2$ m is $\frac{32}{200}=\frac{4}{25}$ after converting metres to centimetres.

To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator: $2\frac{3}{5}=\frac{2\times5+3}{5}=\frac{13}{5}$. Reverse this by division: $13\div5=2$ remainder $3$, so $\frac{13}{5}=2\frac{3}{5}$.

Multiplying or dividing both parts of a fraction by the same non-zero number preserves its value. Thus $\frac{7}{12}=\frac{49}{84}$ because both numerator and denominator were multiplied by 7.

Choose a common form that preserves enough accuracy: convert percentages to decimals, divide numerator by denominator for fractions, and evaluate roots or standard-form values. Keep extra decimal places until the order is secure; do not round two close values to the same comparison value.

Therefore $58\%<\frac{7}{12}<0.6<\frac{8}{13}<\frac{2}{3}$. A chain uses a comparison between every neighbouring pair; each inequality must point consistently from smaller to larger.

To find a fraction between two fractions, a common denominator can expose a gap. Since $\frac{3}{25}=\frac{6}{50}$ and $\frac{4}{25}=\frac{8}{50}$, the value $\frac{7}{50}$ lies between them.

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

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

Translate practical wording before calculating. ‘$5^\circ$C lower than $-7^\circ$C’ means $-7-5=-12$, while ‘$8^\circ$C higher’ then means $-12+8=-4$. When a whole item is required, such as coaches or bags, interpret the remainder and round up rather than reporting a partial item.

The sequence $2^3=8$, $2^2=4$, $2^1=2$, $2^0=1$, $2^{-1}=\frac12$ divides by 2 at every step. This explains both the zero-index and negative-index definitions rather than treating them as disconnected rules.

The expressions $0^0$ and $0^{-n}$ are not defined here because the negative-index rule would require division by zero. This objective uses integer indices only; fractional indices are outside C1.7 Core scope.

When the bases already match, equate indices after simplifying. For $5^n\div5^4=5^6$, the quotient law gives $5^{n-4}=5^6$, so $n-4=6$ and $n=10$.

$85.1\times10^4$ has the correct value but is not in standard form because $85.1$ is not less than 10. Renormalising gives $8.51\times10^5$.

For positive standard-form numbers, compare powers of 10 first. Thus $2.04\times10^9>9.78\times10^8$ because $10^9$ is ten times the scale of $10^8$. If powers match, compare the coefficients.

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 records the reverse of the movement used to create the coefficient.

Check both value and format: the coefficient must satisfy $1\le A<10$, and converting back must reproduce every zero and significant digit of the ordinary number.

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

Keep full calculator precision during the calculation, normalise first, and round only the final coefficient when a degree of accuracy is requested. For example, $4.6\times10^2\times6.7\times10^5=3.082\times10^8$, which is $3.1\times10^8$ to 2 significant figures.

Keep all digits before the rounding position unchanged. If rounding up creates a 10, carry left through place values: $9876$ to the nearest thousand is $10000$.

Zeros can communicate accuracy. The value $57.3997$ to 4 significant figures is $57.40$: the final zero must remain because it is the fourth significant figure. Similarly, $0.0050$ has 2 significant figures.

Use this order: round each input separately; rewrite the complete expression with the rounded values and the same brackets, powers and operations; calculate that simpler expression; then compare its scale with the original values.

For $\frac{423.8-78.4}{23.5}$, rounding to 1 significant figure gives $\frac{400-80}{20}=16$. The subtraction remains in the numerator; removing its grouping would estimate a different calculation.

An estimate is not always above or always below the exact answer. Its direction depends on how each rounded input affects the operation: rounding both positive factors down makes their estimated product smaller, but rounding a denominator down can make a quotient larger.

Context can control rounding direction rather than nearest rounding. If $5.25$ bags are needed and only whole bags can be bought, the answer is 6 bags; rounding to 5 would leave too little material.

For a calculator value $4.285714\ldots$ requested to 4 significant figures, retain the full value until the end and report $4.286$. Early rounding of intermediate values can change the final digit.

Here $x$ is the stated rounded value, $r$ is one unit at the stated accuracy, and $v$ is the original value. The lower bound is included; the upper bound is excluded because an exact upper-bound value rounds to the next stated value.

Convert the stated accuracy into the variable's unit before halving. If $p$ is in kilograms and 12.4 kg is correct to the nearest 100 g, then $r=0.1$ kg, so $12.35\le p<12.45$.

For significant figures, use the place value of the final significant digit as the rounding step. For example, 350 correct to 2 significant figures has step 10, giving $345\le v<355$.

For a direct proportion, every quantity changes by the same scale factor. A recipe using 550 g for 8 cakes needs $550\times(360\div8)=24\,750$ g, or 24.75 kg, for 360 cakes. Scale from a known pair; do not add the same amount.

For map scales, first use matching units. A scale of 1 cm to 8 km is $1:800\,000$ because 8 km is 800 000 cm. At a scale of $1:250\,000$, 3.5 cm represents $3.5\times250\,000=875\,000$ cm, or 8.75 km.

To determine best value, compare like with like: calculate the cost for one common unit, or scale every option to the same quantity. The lowest cost per common unit is the best value, provided the units and quantities are equivalent.

Convert the reference unit before applying the rate. For example, $84$ km/h is $84\,000\div60=1400$ m/min. Multiplying by 1000 changes kilometres to metres; dividing by 60 changes 'per hour' to 'per minute'.

1. Copy the formula given in the question. 2. Convert values into compatible units. 3. Substitute with units. 4. Rearrange only if the unknown is not already isolated. 5. State the compound unit in the same order as the formula.

For a population of 735 000 across 1 477 300 km², population density is $735\,000\div1\,477\,300\approx0.498$ people/km². A value below 1 is possible: it means fewer than one person per square kilometre on average, not a fraction of a person at each location.

1. Add every distance that belongs to the journey. 2. Add every travel time and included stop time. 3. Convert all time to the unit required by the speed. 4. Divide total distance by total time. 5. Attach the compound unit, such as km/h or m/s.

To convert m/s to km/h, multiply by $60\times60\div1000=3.6$. To convert km/h to m/s, divide by 3.6. A decimal hour is not decimal minutes: 2.4 hours is 2 hours 24 minutes.

1. Identify the part being described. 2. Identify the complete reference quantity. 3. Convert both to the same units. 4. Divide part by whole and multiply by 100. 5. Round only as requested.

The part can exceed the reference whole. For example, $150\div120\times100=125$, so 150 is 125% of 120.

For profit or loss, the original cost price is the denominator. Buying for \2.50$and selling for$$4.20$gives profit$$1.70$. Since$1.70\div2.50\times100=68$, the percentage profit is 68%.

Here $P$ is the principal, $r$ is the annual percentage rate, $n$ is the number of years, $I$ is total interest and $A$ is the final amount. These formulas are not supplied in the Core examination.

1. Estimate the size and sign of the answer. 2. Use brackets to preserve the intended numerator, denominator and order of operations. 3. Keep the full displayed value, or use the calculator's answer memory, in later steps. 4. Round only the final answer to the requested accuracy. 5. Compare with the estimate.

If an intermediate result is $7.428571\ldots$, do not replace it with 7.43 before the next operation. Reuse the full value; early rounding can change the final digit.

Enter 2 hours 30 minutes as 2.5 hours when the calculation uses decimal hours, because $30/60=0.5$. On a calculator with a degrees–minutes–seconds key, the equivalent entry is $2^\circ30'0''$.

Read the calculator's expression line before pressing equals. Check that every opening bracket has a closing bracket and that a negative sign belongs to the intended number or exponent.

To convert a decimal-hour display, keep the whole number as hours and multiply only the decimal part by 60. If seconds are needed, multiply the remaining fraction of a minute by 60 again.

For money, write the correct currency symbol and two decimal places: a display of 12 means $12.00, while 12.5 means$12.50. Apply the rounding rule before adding the trailing zero.

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

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.

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.

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.

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.

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.

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.