Topic 3: Ratio, proportion and rates of change

Ratio and proportion sit at the heart of GCSE Maths and connect to almost every other topic, from algebra to geometry to real-life problem solving. This part of the specification rewards careful, organised working: if you can set out a ratio clearly, scale it sensibly, and keep track of your units, the marks follow. In this article you will learn how to simplify and use ratios, how to share an amount in a given ratio, and how to link ratios to fractions and percentages. You will then meet direct and inverse proportion as equations, tackle best-buy questions, and master percentage change including compound interest and depreciation. Finally you will work with compound measures like speed, density and pressure, convert between units, and interpret a gradient as a rate of change. Work through every example with a pen in hand.

Simplifying and using ratios

A ratio compares two or more quantities of the same kind. We write it with colons, for example $2:3$, meaning for every 2 of the first thing there are 3 of the second. To simplify a ratio, divide every part by the same number (the highest common factor). For example $12:18$ both divide by 6 to give $2:3$. Ratios must be written in the same units before simplifying: $50\\text{ cm}:2\\text{ m}$ becomes $50:200$ which simplifies to $1:4$. A ratio in the form $1:n$ is useful for comparisons and scales; to reach it, divide both parts by the left-hand value. For instance $4:10$ becomes $1:2.5$. You can also work the other way, scaling a ratio up: $3:5$ multiplied by 4 gives $12:20$. The key idea is that whatever you do to one part you must do to every part, so the comparison stays true. Always check your simplified ratio cannot be reduced further by looking for a common factor in all parts.

Dividing an amount in a given ratio

To share a quantity in a ratio, first add the parts to find the total number of shares, then find the value of one share, then multiply out. Suppose you share 60 pounds in the ratio $2:3$. The total is $2+3=5$ shares. One share is $60\\div 5=12$ pounds. So the amounts are $2\\times 12=24$ pounds and $3\\times 12=36$ pounds. Always check your answers add back to the original total: $24+36=60$. Some questions give you one part and ask for the whole. If the ratio of boys to girls is $4:5$ and there are 20 boys, then one share equals $20\\div 4=5$, so girls number $5\\times 5=25$ and the total is 45. Another common type gives the difference between two parts: if the ratio is $3:7$ and the difference is 24, the difference in shares is $7-3=4$, so one share is $24\\div 4=6$, giving values of 18 and 42. Lay your working out in clear lines so an examiner can follow it.

Ratios, fractions and percentages

A ratio can be rewritten as fractions of the whole. In the ratio $2:3$ the total is 5 parts, so the first quantity is $\\frac{2}{5}$ of the whole and the second is $\\frac{3}{5}$. To turn these into percentages, work out the fraction as a decimal and multiply by 100: $\\frac{2}{5}=0.4=40\\%$ and $\\frac{3}{5}=0.6=60\\%$. Be careful to distinguish a part-to-part ratio from a part-to-whole fraction. If the ratio of red to blue counters is $3:2$, the fraction that are red is $\\frac{3}{5}$, not $\\frac{3}{2}$. You may also be asked to combine ratios. If $a:b=2:3$ and $b:c=4:5$, make the shared term $b$ equal by scaling: multiply the first by 4 and the second by 3 to get $a:b=8:12$ and $b:c=12:15$, so $a:b:c=8:12:15$. This skill of finding a common value for the linking quantity is a frequent exam requirement.

Scale, maps and scale drawings

A scale tells you how a drawing or map relates to real life, and it is just a ratio written as $1:n$. A map scale of $1:25000$ means 1 cm on the map represents 25000 cm in reality. To convert that to a sensible unit, divide by 100 to reach metres, so $25000\\text{ cm}=250\\text{ m}$, then by 1000 for kilometres if needed. To find a real distance, multiply the map distance by the scale factor; to find a map distance, divide the real distance by the scale factor. For example, on a $1:50000$ map a real distance of $3\\text{ km}$ is $300000\\text{ cm}$, which becomes $300000\\div 50000=6\\text{ cm}$ on the map. Scale drawings of rooms or plans work the same way: read the scale, convert units carefully, and state units in your answer. Always keep both quantities in the same unit before dividing, and double-check whether the question wants the map measurement or the real-world measurement.

Direct and inverse proportion

Two quantities are in direct proportion when they increase or decrease in the same ratio, so doubling one doubles the other. This is written $y\\propto x$, which gives the equation $y=kx$ where $k$ is the constant of proportionality. To solve a problem, use a known pair of values to find $k$, then use $k$ for other values. If $y=12$ when $x=3$, then $k=\\frac{12}{3}=4$, so $y=4x$ and when $x=10$, $y=40$. Quantities are in inverse proportion when one increases as the other decreases in the same ratio, so doubling one halves the other. This is written $y\\propto\\frac{1}{x}$, giving $y=\\frac{k}{x}$. If 4 workers take 6 hours, then $k=4\\times 6=24$, so 3 workers take $\\frac{24}{3}=8$ hours. Higher-tier questions may use proportion to a power, such as $y=kx^2$ or $y=\\frac{k}{x^2}$; the method is identical, just find $k$ first. A direct proportion graph is a straight line through the origin, while an inverse proportion graph is a curve that never touches the axes.

Best buys and proportional reasoning

Best-buy questions ask you to compare value for money, and the reliable method is to find a common basis for comparison. Usually that means working out the price per unit, for example pounds per gram or pence per millilitre. Suppose a 500 g box costs 2.40 pounds and a 750 g box costs 3.45 pounds. The first is $240\\div 500=0.48$ pence per gram and the second is $345\\div 750=0.46$ pence per gram, so the larger box is better value because each gram is cheaper. Alternatively you can find how much you get per pound: divide the quantity by the price. Whichever direction you choose, be consistent across both options and state clearly which is the best buy and why. Many recipe and currency questions also use proportional reasoning: scale every ingredient by the same factor, or multiply by the exchange rate, keeping the units consistent throughout.

Percentage change, growth, decay and compound interest

Percentage change measures how much a quantity has increased or decreased relative to its starting value, using $\\text{percentage change}=\\frac{\\text{change}}{\\text{original}}\\times 100$. To increase an amount by a percentage, use a multiplier: increasing by 15 percent means multiplying by $1.15$, and decreasing by 15 percent means multiplying by $0.85$. Repeated percentage change uses powers of the multiplier. Compound interest pays interest on the growing total each year, so 200 pounds at 3 percent for 4 years becomes $200\\times 1.03^4=225.10$ pounds to the nearest penny. Depreciation is the same idea with a decrease: a car worth 8000 pounds losing 12 percent per year is worth $8000\\times 0.88^3=5451.78$ pounds after 3 years. The general formula is $\\text{final}=\\text{initial}\\times(\\text{multiplier})^n$ where $n$ is the number of periods. To find an original value before a change (reverse percentage), divide by the multiplier rather than subtracting the percentage: if a price after a 20 percent rise is 60 pounds, the original was $60\\div 1.2=50$ pounds.

Compound measures, units and gradient as a rate

A compound measure combines two different quantities, and the most common is speed, given by $\\text{speed}=\\frac{\\text{distance}}{\\text{time}}$. Rearranging gives $\\text{distance}=\\text{speed}\\times\\text{time}$ and $\\text{time}=\\frac{\\text{distance}}{\\text{speed}}$. Density links mass and volume through $\\text{density}=\\frac{\\text{mass}}{\\text{volume}}$, and pressure links force and area through $\\text{pressure}=\\frac{\\text{force}}{\\text{area}}$. Rate of flow measures volume per unit time, such as litres per second. Each compound measure has compound units that tell you the formula: metres per second is distance over time, and grams per cubic centimetre is mass over volume. Converting compound units needs care: to change km per hour to m per second, multiply by 1000 then divide by 3600, so $72\\text{ km/h}=72\\times\\frac{1000}{3600}=20\\text{ m/s}$. Converting areas and volumes uses squared and cubed factors: $1\\text{ m}^2=10000\\text{ cm}^2$ because $100^2=10000$. Finally, the gradient of a straight-line graph represents a rate of change; on a distance-time graph the gradient is speed, and on a graph of cost against quantity it is the price per unit. Steeper lines mean a greater rate.