Ratio, proportion and rates of change

A charity walk is $3.6\,\text{km}$ long. Change this distance to metres.

A pump moves water at $540$ litres per minute. Change this rate to cubic metres per second.

A solid block measures $2.4\,\text{m}$ by $35\,\text{cm}$ by $80\,\text{mm}$ and has mass $26.88\,\text{kg}$. Work out its density in $\text{kg}/\text{m}^3$.

A ranger's sketch uses scale $1:25\,000$. A footpath trace measures $6.4\,\text{cm}$. Find the real distance in kilometres.

A model theatre uses scale $1:40$. A real doorway is $2.04\,\text{m}$ high. Work out the model doorway height in centimetres.

A site plan has scale $1:2500$. A garden occupies $96\,\text{cm}^2$ on the plan. Calculate its real area in hectares. Use $1$ hectare $=10\,000\,\text{m}^2$.

Express $18$ as a fraction of $30$. Give the fraction in its simplest form.

Express $84$ minutes as a fraction of $35$ minutes. Simplify your answer.

Express $0.84\,\text{m}^2$ as a fraction of $2400\,\text{cm}^2$. Give the fraction in its simplest form.

Write the ratio $42:63$ in its simplest form.

Write $1.8\,\text{m}:75\,\text{cm}$ as a ratio in its simplest form.

Given $x:y=4:7$ and $y:z=6:5$, find $x:y:z$ in its simplest integer form.

Divide £84 in the ratio $3:4$.

A box holds $240$ counters. Red counters make up $\frac{3}{8}$ of the whole. Find the number of red and non-red counters, and state their ratio.

A technician mixes a $20\%$ solution with a $50\%$ solution to make $18$ litres of a $40\%$ solution. Find the volume of each starting solution.

Quantity $A$ is $18$ and quantity $B$ is $30$. Express $A$ as a fraction of $B$ and write $A:B$ in simplest form.

The mass of parcel $P$ is $1.75$ times the mass of parcel $Q$. Express $P:Q$ as an integer ratio and express $P$ as a fraction of $Q$.

Quantities $P$, $Q$ and $R$ satisfy $P=\frac{5}{3}Q$ and $Q=\frac{3}{4}R$. Express $P:R$ in simplest form and write $P$ as a fraction of $R$.

Solve the proportion $\frac{x}{15}=\frac{6}{10}$.

Eight identical notebooks cost £11.20. Use proportion to find the cost of $14$ notebooks.

Solve $\frac{2x+3}{18}=\frac{x+6}{12}$ and verify that the two ratios are equal.

The ratio of cats to dogs at a shelter is $3:5$. What fraction of the animals are cats?

Quantities $y$ and $x$ are always in the ratio $5:2$. Write $y$ as a linear function of $x$, then find $y$ when $x=14$.

A mixture contains concentrate and water in the ratio $2:7$. Let $c$ litres be the concentrate and $V$ litres be the total mixture. Express $V$ as a linear function of $c$, then find both component volumes when $V=54$.

Work out $35\%$ of $240$.

After a $12\%$ decrease, a machine is valued at £704. Work out its value before the decrease.

A saver deposits £2500 in an account paying $3.6\%$ simple interest each year. After $5$ years, a fee equal to $2\%$ of the final balance is charged. Calculate the amount left after the fee.

A direct variation links $x$ and $y$. The pair $x=6$, $y=18$ is known. Determine $y$ at $x=10$.

Variables $x$ and $y$ vary inversely. One recorded pair is $x=3$, $y=14$. Determine $y$ at $x=7$.

$x$ and $y$ are inversely proportional. Initially $x=6$ and $y=12$. The value of $x$ is increased by $25\%$. Find the new value of $y$ and the percentage decrease in $y$.

A shift lasting $7$ hours pays £52.50. Work out the hourly rate of pay.

A coach travels $156\,\text{km}$ in $2$ hours $24$ minutes. Calculate its average speed in $\text{km}/\text{h}$.

A force of $3.6\,\text{kN}$ acts uniformly on a rectangular pad measuring $24\,\text{cm}$ by $15\,\text{cm}$. Calculate the pressure in pascals, where $1\,\text{Pa}=1\,\text{N}/\text{m}^2$.

Two similar shapes have corresponding sides of $6\,\text{cm}$ and $15\,\text{cm}$. Write the smaller-to-larger length ratio in simplest form.

The corresponding length ratio of two similar tiles is $3:7$. The smaller tile has area $54\,\text{cm}^2$. Find the area of the larger tile.

Two similar solids have smaller-to-larger volume ratio $125:216$. The smaller solid has surface area $275\,\text{cm}^2$. Work out the larger surface area.

Variables $p$ and $q$ vary inversely, with recorded values $p=12$ and $q=5$. Write their connecting equation.

The relationship between $y$ and $x^2$ is direct proportion. Given $x=3$ when $y=45$, form the equation and evaluate $y$ at $x=4$.

For positive $b$, the variable $a$ varies inversely with $b^2$. Given $a=20$ at $b=3$, determine $b$ when $a=7.2$.

A straight distance-time graph passes through $(2,10)$ and $(7,35)$, where time is in seconds and distance in metres. Find and interpret its gradient.

A phone-call cost graph is modelled by $C=18+0.12m$, where $C$ is cost in pounds and $m$ is time in minutes. Interpret the gradient and calculate the cost of a $35$-minute call.

An inverse-proportion model contains the points $(2,18)$, $(3,12)$ and $(6,6)$. Check that all three coordinates are consistent with the model, write its equation, and find $y$ when $x=9$.

A curve passes through $(2,5)$ and $(8,23)$. Calculate the average rate of change of $y$ with respect to $x$ between these points.

A tangent to a curve at $x=7$ passes through the grid points $(4,11)$ and $(10,32)$. Estimate the instantaneous rate of change at $x=7$.

A curve shows water volume $V$ litres after $t$ minutes and passes through $(2,46)$ and $(8,118)$. The tangent at $t=5$ passes through $(4,70)$ and $(7,112)$. Find the average rate from $t=2$ to $t=8$, estimate the instantaneous rate at $t=5$, and compare them.

£600 is invested at $4\%$ compound interest per year. Work out the value after $2$ years.

A culture initially contains $960$ cells and decreases by $15\%$ each hour. Calculate the expected number after $3$ hours.

A cooling model uses $T_{n+1}=0.65T_n+12$ with $T_0=80$. Find the first value of $n$ for which $T_n<40$, and give that temperature to one decimal place.