ESAT (Engineering and Science Admissions Test)

Standard units and prefixes

Every quantity has an SI unit: mass in kilograms (kg), length in metres (m), time in seconds (s). Other units are built from these, and a unit with a negative index says "per": a speed of $5\ \mathrm{m\,s^{-1}}$ is 5 metres for every second. The prefixes change a unit by a power of ten.

A litre is $1000\ \mathrm{cm^3}$, so $1\ \mathrm{m^3} = 1000$ litres, and $1\ \mathrm{dm^3} = 1$ litre. A tonne is 1000 kg.

Changing units of area and volume

A length conversion must be applied once for each dimension. Since $1\ \mathrm{m} = 100\ \mathrm{cm}$, an area converts with $1\ \mathrm{m^2} = 100^2\ \mathrm{cm^2} = 10\,000\ \mathrm{cm^2}$, and a volume with $1\ \mathrm{m^3} = 10^6\ \mathrm{cm^3}$. Converting $3.5\ \mathrm{cm^2}$ to $\mathrm{mm^2}$ gives $3.5 \times 10^2 = 350\ \mathrm{mm^2}$, because each centimetre is 10 millimetres and there are two lengths in an area.

Compound units

A compound unit divides or multiplies two basic ones, and it converts one part at a time.

• Speed: $72\ \mathrm{km\,h^{-1}}$ becomes metres per second by $72 \times 1000 \div 3600 = 20\ \mathrm{m\,s^{-1}}$.
• Density: $8.0\ \mathrm{g\,cm^{-3}}$ becomes $\mathrm{kg\,m^{-3}}$ by $8.0 \times 10^{-3}\ \mathrm{kg} \div 10^{-6}\ \mathrm{m^3} = 8000\ \mathrm{kg\,m^{-3}}$.
• Pressure: force divided by area, so $200\ \mathrm{N}$ on $0.5\ \mathrm{m^2}$ is $400\ \mathrm{N\,m^{-2}}$, which is 400 pascals.
• Rates of pay and unit pricing divide money by time or by quantity. A 750 g jar at £1.50 costs $1.50 \div 750$ = £0.002 per gram, and so £2 per kilogram, which lets jars of different sizes be compared.

A useful check is to write the units alongside the numbers and cancel them as if they were fractions: $\dfrac{72\ \mathrm{km}}{1\ \mathrm{h}} \times \dfrac{1000\ \mathrm{m}}{1\ \mathrm{km}} \times \dfrac{1\ \mathrm{h}}{3600\ \mathrm{s}}$ leaves metres over seconds.

Ordering numbers

The symbols are $=$ (equal), $\neq$ (not equal), $<$ (less than), $>$ (greater than), $\le$ and $\ge$ (less than or equal to, greater than or equal to). On a number line, numbers increase to the right, so $-7 < -2$, and a larger negative number is smaller.

To order fractions, write them over a common denominator. For $\dfrac{3}{4}$, $\dfrac{5}{8}$ and $\dfrac{7}{10}$, use 40: they become $\dfrac{30}{40}$, $\dfrac{25}{40}$ and $\dfrac{28}{40}$, so the order from smallest is $\dfrac{5}{8} < \dfrac{7}{10} < \dfrac{3}{4}$. To order decimals, compare digits from the left, place by place, after padding with zeros so they have the same length: $0.305$, $0.35$ and $0.3$ become $0.305$, $0.350$ and $0.300$.

The four operations

Fractions

Addition and subtraction need a common denominator. Multiplication multiplies numerators and denominators, and division multiplies by the reciprocal of the divisor.

A mixed number such as $2\dfrac{1}{3}$ is best turned into an improper fraction, $\dfrac{7}{3}$, before multiplying or dividing. With negative numbers the signs follow the usual rules: two negatives multiplied give a positive, so $-\dfrac{2}{3} \times -\dfrac{9}{4} = \dfrac{3}{2}$.

Negative numbers and place value

Subtracting a negative is adding: $5 - (-3) = 8$. Each digit's value is its face value times a power of ten, so in $4.706$ the 7 is worth $7 \times 10^{-1}$ and the 6 is worth $6 \times 10^{-3}$. Multiplying by 10 moves every digit one place to the left, and dividing moves it to the right.

Order of operations

Brackets are worked first, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right. A fraction bar and a root sign both act as brackets. For example:

Inverse operations and cancellation

Addition and subtraction undo each other, as do multiplication and division, and squaring and taking a square root. Inverses solve equations by working backwards. Cancellation removes a factor common to a numerator and a denominator before multiplying, which keeps numbers small: $\dfrac{18 \times 35}{14 \times 27} = \dfrac{2 \times 9 \times 5 \times 7}{2 \times 7 \times 3 \times 9} = \dfrac{5}{3}$. Division by zero has no meaning.

Primes, factors and multiples

A prime number has exactly two factors, 1 and itself, so 1 is not prime and 2 is the only even prime. A factor (divisor) of $n$ divides it exactly, and a multiple of $n$ is in its times table. Every integer greater than 1 is a product of primes in exactly one way apart from order, which is the unique factorisation theorem. Written with powers, $360 = 2^3 \times 3^2 \times 5$, and a product notation such as $\prod$ is simply shorthand for a long product.

To find a prime factorisation, divide repeatedly by the smallest prime that goes in: $360 \to 180 \to 90 \to 45 \to 15 \to 5 \to 1$, using 2, 2, 2, 3, 3, 5. A number is prime if it has no prime factor up to its square root, so to test 97 it is enough to check 2, 3, 5 and 7, none of which divides it.

The highest common factor (HCF) takes the lowest power of each shared prime, and the lowest common multiple (LCM) takes the highest power of every prime that appears in either number. For $72 = 2^3 \times 3^2$ and $84 = 2^2 \times 3 \times 7$:

For two numbers, $\mathrm{HCF} \times \mathrm{LCM}$ equals their product, which is a check: $12 \times 504 = 6048 = 72 \times 84$. The number of factors of a number with factorisation $p^a q^b r^c$ is $(a+1)(b+1)(c+1)$, so 360 has $4 \times 3 \times 2 = 24$ factors. A number is a perfect square exactly when every index in its factorisation is even.

An LCM solves repeating-event problems: two lights flashing every 12 s and every 18 s flash together every 36 s.

Systematic listing and counting

If one task can be done in $m$ ways and, for each, a second task can be done in $n$ ways, the two together can be done in $m \times n$ ways. A code made from a choice of 3 letters then 2 digits then 4 symbols has $3 \times 2 \times 4 = 24$ possibilities. If repeats are not allowed the number of choices drops at each stage: arranging 4 different books on a shelf gives $4 \times 3 \times 2 \times 1 = 24$ orders, and choosing a president, secretary and treasurer from 8 people gives $8 \times 7 \times 6 = 336$.

When a list is needed, go through it in a fixed order so nothing is missed or repeated. The three-digit numbers that can be made from the digits 1, 2 and 3 with no repeats are 123, 132, 213, 231, 312 and 321, listed by the first digit, then the second. When the order does not matter, each group is counted several times in the product above, and is divided out: choosing 2 people from 5 gives $\dfrac{5 \times 4}{2} = 10$ pairs.

Squares, cubes and roots

The squares from $1^2$ to $15^2$ are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, and the cubes from $1^3$ to $5^3$ and $10^3$ are 1, 8, 27, 64, 125 and 1000. A positive number has two square roots, one positive and one negative, since $(-7)^2 = 49$ as well as $7^2$. The symbol $\sqrt{49}$ means the positive root only, 7, so $x^2 = 49$ gives $x = \pm 7$. A cube root $\sqrt[3]{-64} = -4$ is unique and may be negative. Negative numbers have no real square roots.

Indices

For a base $a$ and indices $m$ and $n$:

Zero and negative indices follow from these: $a^0 = 1$ for $a \ne 0$, and $a^{-n} = \dfrac{1}{a^n}$. A fractional index is a root, $a^{1/n} = \sqrt[n]{a}$, and more generally $a^{m/n} = \left(\sqrt[n]{a}\right)^m$. Take the root first, since it keeps the numbers small:

The laws apply only to a common base. To simplify $\dfrac{4^3 \times 8^2}{2^5}$, put everything in powers of 2: $\dfrac{2^6 \times 2^6}{2^5} = 2^7 = 128$. Note that $(ab)^n = a^n b^n$, but $(a+b)^n$ is not $a^n + b^n$.

Recurring decimals

A fraction in lowest terms terminates as a decimal only when its denominator has no prime factors other than 2 and 5, so $\dfrac{7}{40} = 0.175$ terminates and $\dfrac{1}{7} = 0.\overline{142857}$ recurs. A dot or bar marks the repeating block.

To turn a terminating decimal into a fraction, write the digits over a power of ten and cancel: $0.36 = \dfrac{36}{100} = \dfrac{9}{25}$.

A recurring decimal is turned into a fraction by multiplying so that the repeating part lines up, then subtracting. For $x = 0.\overline{27}$, multiply by 100:

If some digits come before the repeat, multiply twice so that the repeats line up. For $y = 0.4\overline{3}$, $10y = 4.333\ldots$ and $100y = 43.333\ldots$, so $90y = 39$ and $y = \dfrac{13}{30}$.

Fractions, decimals and percentages

A percentage is a fraction with denominator 100, so $35\% = \dfrac{35}{100} = \dfrac{7}{20} = 0.35$. To convert a fraction, divide top by bottom and multiply by 100: $\dfrac{3}{8} = 0.375 = 37.5\%$. Equivalent fractions come from multiplying or dividing the numerator and denominator by the same number. In calculations the three forms can be mixed freely, and the convenient one is used: $25\%$ of 360 is a quarter of it, 90, and $\dfrac{3}{5}$ of $0.5$ is $0.6 \times 0.5 = 0.3$.