Section 1: Number and numeration

Convert $2314_{5}$ to a number in base ten.

Express $87$ (a base ten number) in base three.

Find $342_{5} + 234_{5}$, giving the answer in base five.

Find $1101_{2} \times 11_{2}$, giving the answer in base two.

In a certain base $n$, $24_{n} + 35_{n}$ $= 103_{n}$. Find $n$.

In base $n$, the square of $13_{n}$ is $171_{n}$. Find $n$.

Find the value of $5 \times 6$${}- 9 \; (\text{mod } 8)$.

A village market near Kano is held every $6$ days. One market day falls on a Thursday; call it the 1st market day. On which day of the week is the 10th market day?

Find the value of $x$ in $\{0, 1, 2, 3, 4, 5, 6\}$ for which $3x$ $\equiv 5 \; (\text{mod } 7)$.

A security guard in Tema works a repeating rota: $3$ nights on duty, then $2$ nights off. Night 1 of the rota is a Wednesday night, and it is his first night on duty. On which day of the week does his 50th night on duty fall?

Simplify $\left(2\frac{1}{3} - 1\frac{3}{4}\right) \div 1\frac{1}{6}$.

Write $0.0040362$ correct to three significant figures.

Evaluate $\frac{1.8 \times 10^{4}}{6 \times 10^{-3}}$, giving the answer in standard form.

Express $0.2\dot{7}$ (that is, $0.2777\ldots$, where only the $7$ recurs) as a fraction in its lowest terms.

One grain of sand has a mass of $6.4 \times 10^{-4}$ g. How many such grains have a total mass of $8$ kg? Give the answer in standard form.

Evaluate $0.0384 \times 0.25$, giving the answer in standard form.

Simplify $\left(8x^{6}y^{-3}\right)^{\frac{2}{3}}$.

Evaluate $\left(\frac{125}{64}\right)^{-\frac{2}{3}}$.

Solve $4^{x + 1}$ $= 8^{2x - 1}$.

Given that $25^{x} \times 5^{y}$ $= 625$ and $8^{x} \div 2^{y}$ $= 2$, find the value of $y$.

Simplify $\frac{3^{n + 2} - 3^{n}}{3^{n + 1} + 3^{n}}$.

Evaluate $\log_{8} 48$${}- \log_{8} 3$.

Simplify $\frac{\log 125}{\log 25}$ (logarithms to base ten).

Solve $\log_{3}(x + 5)$${}+ \log_{3}(x - 1)$ $= 3$.

Given that $\log 2$ $= 0.3010$ and $\log 3$ $= 0.4771$, find $\log 7.2$.

Find $x$ if $\log_{x} 64$ $= \frac{3}{2}$.

Simplify $\sqrt{75}$${}- \sqrt{12}$${}+ \sqrt{27}$.

Rationalise the denominator and simplify $\frac{6}{3 - \sqrt{3}}$.

Simplify $(2\sqrt{5} - \sqrt{3})(2\sqrt{5} + \sqrt{3})$.

Express $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}$ in the form $a + b\sqrt{6}$, where $a$ and $b$ are rational.

Find $413_{5} - 241_{5}$, giving the answer in base five.

Solve $3^{2x}$${}- 4 \times 3^{x}$${}+ 3$ $= 0$.

Given that $x$ $= 2 + \sqrt{3}$, find the value of $x^{2} + \frac{1}{x^{2}}$.

Given that $\log 2$ $= p$ and $\log 3$ $= q$ (logarithms to base ten), express $\log 45$ in terms of $p$ and $q$.

How many sacks does the harvest fill?

How much does the buyer pay for the whole harvest?

Estimate the number of beans in the harvest. Give your answer in standard form, correct to two significant figures.