Ratio, proportion and rates of change — Edexcel GCSE Mathematics

Ratio, proportion and rates of change covers comparing quantities multiplicatively, scaling recipes and maps, sharing amounts in a given ratio, direct and inverse proportion, compound measures such as speed and density, and percentage change including compound interest. It is a major strand of Pearson Edexcel GCSE Mathematics, assessed across all three papers and heavily applied in real-world problem-solving questions.

Ratio, proportion and rates of change is a fundamental GCSE Mathematics topic that develops your ability to compare quantities multiplicatively and solve problems involving scaling, sharing, and comparing different amounts. It covers ratio notation, dividing quantities in a given ratio, direct and inverse proportion, compound measures such as speed, density and pressure, and percentage change including simple and compound interest. These ideas appear in everyday contexts like recipes, maps, currency exchange, and financial calculations.

This topic is assessed across all three Pearson Edexcel GCSE Mathematics papers (non-calculator, calculator, and both tiers) and often forms the basis of multi-step problem-solving questions. Mastery of ratio and proportion is essential for success in algebra, geometry, and statistics, as well as for everyday life beyond the classroom. A strong understanding here will boost your confidence in tackling unfamiliar problems and improve your overall grade.

• →Ratio compares quantities of the same kind using division; it can be simplified like fractions and used to share a total into parts.
• →Direct proportion means as one quantity increases, the other increases at the same rate (e.g., cost of petrol vs litres); inverse proportion means as one increases, the other decreases (e.g., number of workers vs time taken).
• →A rate of change compares two different quantities with different units, such as speed (km/h), density (g/cm³), or pressure (N/m²).
• →Percentage change is calculated using (change / original) x 100, and compound interest uses repeated percentage multipliers over time.
• →Proportion problems can be solved using the unitary method (find the value of one unit first) or by setting up equations.

• a correct conversion factor, such as 1 m³ = 1 000 000 cm³ or 1 kg = 1000 g, wherever it appears in the working
• the method of multiplying or dividing by that factor in the right direction, even if the arithmetic afterwards is wrong
• carrying a two-stage conversion through both stages, for example seconds into hours and metres into kilometres
• the final value with its unit stated, because a rate without its unit is not an answer to a compound-unit question
• for a compound unit such as a price or pressure, converting both the numerator and denominator units consistently before combining them
• stating the scale as a multiplier, for example that 1 cm represents 25 000 cm
• the method of multiplying or dividing a measured length by that scale, even if the final unit is wrong
• a length measured within the tolerance allowed on the drawing, a small amount either way
• the real distance given in the unit the question asks for
• converting both quantities into the same unit before the fraction is formed
• the fraction written the right way round, with the quantity named after "of" as the denominator
• dividing both numbers by a common factor, even where the cancelling is not taken all the way
• the fraction in its simplest form, which is greater than 1 whenever the first quantity is the larger of the two
• converting the parts into a common unit before any cancelling
• dividing every part by the same number, even where the number chosen is not the highest common factor
• the fully simplified ratio, written with a colon rather than as a fraction
• a correct 1 : n form with the decimal kept, where the question asks for that form
• For a part:part ratio, calculating the total number of shares by adding the parts.
• For a part:whole ratio, treating the whole as the total shares and finding the other part by subtracting the given part from the whole.
• Finding the value of one share by dividing the total quantity by the number of shares.
• Correctly calculating both final amounts after finding the value of one share.
• Expressing two quantities as a ratio and simplifying it to its lowest terms.
• Applying ratio to a real context, such as scaling a recipe or a concentration, and interpreting the result.
• the multiplier, found by dividing one quantity by the other
• the relationship written as a ratio with the parts in the order the question names them
• the equivalent fraction, or for an equation linking the two quantities
• the answer in the form the question asks for, a ratio in its simplest form or a single fraction
• setting up two ratios that are equal, or an equation between two fractions, from the figures given
• the scale factor between the ratios, even where it is afterwards applied to the wrong part
• a unitary step such as the cost of one item, where that is the route taken
• the missing value, to the accuracy the question demands
• turning a given ratio into a fraction or a multiplier, distinguishing part-to-part (2/5) from part-to-whole (2/7)
• writing the equation of the line in the form y = kx, with k taken from the ratio as the multiplier between the two quantities
• reading a gradient from a graph as a change in y divided by the matching change in x
• putting an answer back into the context, such as an amount in euros rather than a bare number
• a correct multiplier, such as 1.15 for a 15% increase or 0.85 for a 15% decrease
• dividing rather than multiplying in an original value question, even if the figure that follows is wrong
• a percentage found by dividing one quantity by the other and multiplying by 100
• interest for one year before the total over several years in a simple interest question
• the final amount or percentage, followed through from a correct method even where an earlier figure was wrong
• a correct method that finds the multiplier or the constant, such as dividing a given output by its matching input
• a proportion equation such as y = kx or y = k/x with the value of k written into it
• substituting the value the question asks about into that equation, even if the final answer is wrong
• the final answer with its unit and any rounding the question demands
• choosing the correct relationship, for example writing density = mass ÷ volume before any numbers go in
• a correct conversion so that the two measures agree, such as changing minutes into hours
• the calculation carried out correctly, even if the unit is then left off or written wrongly
• the final answer stated with its compound unit
• a correct length scale factor, found by dividing a length on one shape by the length it corresponds to on the other
• squaring or cubing that scale factor as the question needs, even if the arithmetic afterwards goes wrong
• applying the factor in the right direction, multiplying to enlarge and dividing to reduce
• the final length, area or volume with its unit
• identifying the type of proportion from the equation or the table, supported by a reason such as a constant product
• a correct substitution into the given equation, even if the arithmetic that follows is wrong
• the value found, given to the accuracy the question asks for
• a correct statement of the effect of a change, such as saying that in an inverse relationship trebling one quantity divides the other by three
• a correct proportion statement or equation containing k, matching the power or root named, such as y = k/x for simple inverse proportion
• using the equivalence X inversely proportional to Y ⇔ X proportional to 1/Y to write X = k/Y
• substituting the given pair of values to form an equation in k
• the value of k, even if the answer that follows is wrong
• using the completed formula on the part asked, including rearranging it when the other variable is the unknown
• a correct gradient method, such as reading a change in the vertical value with the matching change in the horizontal value
• the value of the gradient, even if it is left without a unit
• interpreting that number in context, naming the quantity and the unit, for example a speed in kilometres per hour
• a correct description of the graph, such as identifying direct proportion from a straight line through the origin
• a tangent drawn accurately at the correct point, touching the curve rather than cutting across it
• a correct method for the gradient, using two points on the drawn tangent (or chord) to find the change in y and change in x
• a final gradient value inside the accepted tolerance, even if it is not interpreted in context
• the correct interpretation of the gradient in the given context, including the correct compound unit (e.g., metres per second, m/s)
• a correct multiplier for the percentage change, e.g., 1.03 for a 3% increase or 0.8 for a 20% decrease
• raising the multiplier to the correct power, matching the number of time periods in the question
• a correct final value after the full period, before any final subtraction if only the change is required
• the final answer correctly rounded as requested, typically to the nearest penny for monetary contexts
• a correct compound interest multiplier, such as 1.05 for 5% growth or 0.85 for 15% decay, used with the correct power
• the correct final amount after n periods, with the calculation shown rather than just an answer
• a correct first iteration, with the starting value substituted into the rule
• two further terms that follow correctly from your own previous answer, even where the first term was wrong
• the value the terms settle on, given to the number of decimal places asked for
• a correct multiplier raised to a trial power in a growth or decay problem, and one for naming the period at which the target is first passed

• 💡Write the conversion as an equation between the two units before you use it, so the direction is on the page rather than in your head.
• 💡Check the size of your answer: the same distance in centimetres must be a bigger number than in metres.
• 💡Convert both rates into the same units before you compare them, not after you have compared them.
• 💡Use a sharp pencil and measure to the nearest millimetre, because scale drawing marks are awarded inside a narrow tolerance
• 💡Write the unit beside every line of a map calculation; most lost marks here are centimetres reported as metres
• 💡A scale written as a ratio with no units has both sides in the same unit, so convert only once at the end
• 💡Underline the words that come after "of"; that quantity is the denominator every time
• 💡Convert the units before you write the fraction, so the one you cancel is already the right one
• 💡Leave the answer as a fraction unless the question asks for a decimal or a percentage
• 💡Look for one number that divides every part at once; if you cannot see one, halve or divide by 3 repeatedly until nothing divides again
• 💡Check a simplified ratio by multiplying it back up to the numbers you started with
• 💡Keep the parts in the order the question names them, because 3 : 5 and 5 : 3 are different answers
• 💡When sharing a quantity, check your answers by adding them together. They should sum to the original total.
• 💡When forming a ratio, always check if it can be simplified. Look for common factors of both numbers.
• 💡For a part:whole ratio, remember the whole is the total number of parts, so the other part is found by subtraction.
• 💡Test your multiplier on the pair of numbers you were given before you use it anywhere else
• 💡If the answer is a fraction, decide whether it should be smaller or larger than 1 before you write it down
• 💡Write the ratio in the order the words appear in the question, then simplify it
• 💡Write the two ratios one above the other with matching parts lined up, then look across for the multiplier
• 💡Keep a unit value as an exact fraction or a full decimal until the last line of working
• 💡Check the finished pair by simplifying both ratios; they must reduce to the same thing
• 💡Read a gradient using two points that sit exactly on grid intersections, not two points you estimated
• 💡Check the direction of a conversion: multiplying by the gradient goes one way and dividing goes back
• 💡If a ratio is given as part to whole, change it to part to part before you use it as a multiplier
• 💡Remember that the multiplier is the second part divided by the first part, so the ratio 2 : 5 gives 5/2 = 2.5, not 2/5
• 💡Turn every percentage change into one multiplier; it makes a two-stage question a single calculation
• 💡For an original value question, ask what the amount you were given is a percentage of, then divide by that multiplier
• 💡Write money answers to two decimal places and put the pound sign in once, at the front
• 💡Estimate before you calculate: 15% of £80 is about £12, so an answer of £120 has a decimal point in the wrong place
• 💡Check the direction before you calculate: if one quantity rising sends the other down so that their product stays the same, the relationship is inverse rather than direct.
• 💡Write the equation with k in it before any numbers go in, so the two-step method is visible to the examiner.
• 💡Keep the units consistent all the way through, and give money answers to two decimal places.
• 💡Write the relationship out in words first, then put the numbers in; the division you need becomes obvious and the method is on the page.
• 💡Convert everything into the units named on the answer line before you calculate, not afterwards.
• 💡When a rate is per hour and the time is in minutes, turn the minutes into a fraction of an hour, so 24 minutes becomes 0.4 of an hour.
• 💡Set the three factors out before you calculate: length k, area k², volume k³, then decide which line the question sits on.
• 💡Mark the pair of corresponding lengths on the diagram before dividing, so the scale factor is the right way up.
• 💡If an enlarged shape comes out smaller than the original, you divided where you should have multiplied; check that before moving on.
• 💡Test the equation with an easy value such as one, then double it and see what happens to the other quantity; that identifies the relationship quickly.
• 💡For an inverse relationship, multiply each pair in the table together; if every product agrees, you have found the constant.
• 💡Answer using the names of the quantities in the question, not only the letters.
• 💡Read the wording closely: the square of x means x², while the square root of x means √x, and the two give different formulas.
• 💡Keep k exact, as a fraction if it will not come out neatly, and round only the final answer.
• 💡Check the formula by putting the original pair of values back in; it should return the value you started with.
• 💡Pick two points where the line crosses grid corners exactly, so the values you read carry no estimating error.
• 💡Give the gradient as a rate with its unit, for example 60 kilometres per hour, rather than the number on its own.
• 💡A negative gradient means the quantity is falling; say so in words when the question asks what the graph shows.
• 💡Use a ruler and turn the page until the tangent looks balanced on either side of the point before you draw it.
• 💡Take the two points on your tangent as far apart as the grid allows; a bigger triangle gives a more reliable gradient.
• 💡State what the gradient means in the context given, with units, as this interpretation usually carries its own mark.
• 💡Write down the multiplier before using the calculator, then check it makes sense: above 1 for growth, below 1 for decay.
• 💡Use the power key (e.g., xʸ) on your calculator instead of multiplying repeatedly; it is faster and more accurate.
• 💡Keep the full calculator value on screen until the very end of the calculation, then round just once as the final step.
• 💡For compound interest, write the multiplier and the power before reaching for the calculator, so the method is visible even if the arithmetic slips.
• 💡Enter the starting value, press equals, then key the iteration rule using the answer button; each further press gives the next term.
• 💡Keep every digit on the calculator between steps and round only when you write a term down.
• 💡For growth with an unknown number of periods, work out the total for a few whole numbers of periods and pick the first that passes the target.
• 💡Always show your working clearly, especially in ratio sharing and proportion problems. Method marks are awarded even if the final answer is wrong due to a small arithmetic slip.
• 💡For compound measures, write down the formula you are using (e.g., speed = distance / time) and substitute carefully. This helps you gain method marks and reduces errors.
• 💡In percentage change and compound interest questions, use multipliers (e.g., 1.05 for 5% increase) rather than calculating step-by-step, as this is more efficient and less error-prone.

• converting m² to cm² by multiplying by 100 instead of by 100², so an area comes out 100 times too small; correct by squaring the length factor
• dividing where you should multiply, which usually shows as an answer far too small for the context; correct by checking the expected size first
• writing 1 hour 45 minutes as 1.45 hours when working in hours, rather than 1.75 hours; correct by converting the 45 minutes as 45 ÷ 60
• reporting a density as grams per cubic metre after working in cubic centimetres, so the unit does not match the figure; correct by keeping numerator and denominator units consistent
• reversing a scale of 1 : 50 000, so a map length is divided by the scale instead of multiplied by it
• turning 250 000 cm into kilometres by dividing by 1000 only, giving 250 km rather than 2.5 km
• measuring from the end of the ruler instead of the zero mark, so every length is out by the same amount
• taking a scale factor from a pair of sides that do not correspond, such as the base of one shape against the height of the other
• writing the fraction upside down, so 20 g as a fraction of 50 g is given as 50/20
• comparing 30 cm with 2 m without converting, giving 30/2 rather than 30/200
• assuming an answer above 1 must be wrong and inverting it to force a proper fraction
• cancelling part way and stopping at a fraction that still has a common factor
• dividing only one part, so 12 : 18 is written as 2 : 18
• stopping at 6 : 9 because the numbers look smaller, when 3 still divides both parts
• simplifying 30 minutes : 2 hours as 30 : 2 instead of 30 : 120
• reversing the order of the parts, so an answer of 3 : 5 is given for a question that named the quantities the other way round
• Dividing the total quantity by one of the parts of the ratio instead of the sum of the parts.
• Confusing a part:part ratio with a part:whole ratio (e.g., treating 3:5 as 3 parts to 5 parts, not 3 out of 5 total).
• When forming a ratio from two quantities, failing to simplify it to its lowest terms or writing the parts in the wrong order.
• In a part:whole ratio, adding the two numbers to find the total shares instead of using the whole as the total.
• inverting the multiplier, so saying the second quantity is 2/5 of the first when it is 5/2 of it
• reading "three times as many" as 3 : 1 when the question names the quantities in the other order
• using a difference instead of a multiplier, so 8 and 20 are described as twelve apart when the question asks for a ratio
• comparing a part with the whole when the question compares one part with the other part
• adding the same number to both parts instead of multiplying, so 3 : 4 is scaled to 6 : 7
• pairing the parts the wrong way round when the two ratios are written in different orders
• rounding a unit cost to the nearest penny early, so the scaled-up total is out by several pence
• scaling a quantity that does not scale, such as a fixed delivery charge added to a per-item price
• turning the ratio 2 : 5 into the fraction 2/7 when the question compares one part with the other part rather than with the whole
• using the fraction 2/5 as the multiplier when the ratio 2 : 5 gives the multiplier 5/2, so the equation is y = 2.5x
• reading the gradient as a change in x over a change in y, so the multiplier comes out inverted
• drawing a line that misses the origin for a relationship where zero of one quantity means zero of the other
• reading a value off the graph at a point that is not on a grid intersection and rounding it silently
• treating an original value question as a straightforward decrease, so £42 after a 20% rise is reduced by 20% to £33.60 instead of £35
• expecting a 10% rise followed by a 10% fall to return the starting amount, when the two multipliers together give 0.99
• writing 0.4 as 4% instead of 40% when changing a decimal into a percentage
• using simple interest where the question describes interest paid on the balance at the end of each year
• finding the second percentage of the original amount when the question means a percentage of the new amount
• treating every pair of quantities as directly proportional, so multiplying when the total amount of work is fixed and a division was needed
• finding the constant and then offering that constant as the final answer instead of using it
• dividing the wrong way round when finding the constant, writing k = x/y for a direct relationship when it should be k = y/x
• reading an inverse proportion curve as though it had one fixed gradient
• dividing the wrong way round, for example working out volume ÷ mass and calling the result a density
• mixing units inside one calculation, such as a distance in metres with a time in hours, without converting first
• reading a decimal time wrongly, treating 1.5 hours as one hour and fifty minutes
• averaging two speeds by adding them and halving, instead of using total distance ÷ total time
• using the length scale factor on an area or a volume, so multiplying by three rather than by nine when lengths are in the ratio 1 : 3
• taking a square root of a volume ratio, or a cube root of an area ratio, when converting back to lengths
• pairing sides that do not correspond, usually by not matching the shapes up by their equal angles first
• applying the scale factor the wrong way round, so the larger solid comes out smaller than the one you started from
• reading y = 24/x as direct proportion because both quantities are changing, then multiplying where a division was needed
• assuming any equation with the variable underneath is inverse proportion, including ones such as y = 2/x + 5 where no product stays constant
• swapping the roles of the two quantities, so working out x when the question asked for y
• calling a graph direct proportion because it passes through the origin, even when it is a curve
• writing y = kx for inverse proportion instead of y = k/x, so the variables move in the same direction rather than opposite directions
• squaring the whole product instead of the variable, writing y = (kx)² where the question says y is proportional to the square of x
• finding k and stopping there, without going back for the value the question actually wanted
• writing y = k/x where the relationship is inverse to the square of x, so dividing by x rather than by x²
• rounding k partway through, which pushes the final answer outside the accepted range
• counting squares instead of reading the axis scales, which fails whenever one square is not one unit
• dividing the horizontal change by the vertical change, which gives the reciprocal of the gradient
• calling a straight line direct proportion when it does not pass through the origin
• leaving the gradient as a bare number when the question asked what it represents
• drawing a chord between two points on the curve when asked for the rate at a single instant (which requires a tangent)
• reading the two points for the gradient calculation from the curve itself, rather than from the tangent line that was drawn
• using a very small triangle on the tangent, which makes the calculation very sensitive to small reading inaccuracies
• giving the y-coordinate of the point on the curve instead of the gradient of the tangent at that point
• calculating one period of interest and multiplying it by the number of periods (this is simple interest, not compound)
• using a multiplier of 0.2 for a decrease of 20% when it should be 1 - 0.2 = 0.8
• multiplying the initial value by the number of years instead of raising the multiplier to the power of the number of years
• giving the total final amount when the question asked only for the interest earned, or vice versa
• adding simple interest each year instead of multiplying by the same factor each period, so the total drifts below the compound answer
• rounding each term to two decimal places and feeding the rounded value back in, so later terms drift away from the true value
• starting again from the first value each time instead of using the term just found
• naming the last period before a target is passed when the question asked for the first period at which the amount is above it
• Students often think that a ratio 2:3 means 2/3 and 3/2 as fractions of the total, but it actually means 2/5 and 3/5 of the total. Correct by adding the parts to find the denominator.
• In inverse proportion, students sometimes multiply instead of dividing, or assume the relationship is linear. For example, if 4 workers take 6 days, 8 workers take 3 days (not 12 days).
• When converting between units in rates, students forget to convert both quantities consistently, e.g., using minutes with kilometres per hour. Always convert to matching units before calculating.

1. 1Day 1-2: Revise ratio basics: simplifying ratios, dividing amounts in a given ratio, and solving word problems. Practice with textbook exercises and online quizzes.
2. 2Day 3-4: Move on to direct and inverse proportion. Learn to recognise each type from context and solve using the unitary method. Complete past paper questions on these topics.
3. 3Day 5-6: Study compound measures (speed, density, pressure) and rates of change. Memorise the formulas and practice unit conversions thoroughly.
4. 4Day 7-8: Cover percentage change, including simple and compound interest, depreciation, and reverse percentages. Use multipliers and practice exam-style questions.
5. 5Day 9-10: Mixed practice: attempt a full past paper section on ratio, proportion and rates of change under timed conditions. Review mistakes and revisit weak areas.

• 📋Ratio sharing problems: Often ask to divide a total into a given ratio, sometimes with a twist such as one share being given. Advice: always find the value of one part first and check your shares add up.
• 📋Direct and inverse proportion: Presented as word problems or in a table. Advice: identify the type of proportion, then use the unitary method or set up an equation with a constant of proportionality.
• 📋Compound measures: Calculate speed, density, or pressure, sometimes requiring unit conversion. Advice: write the formula, convert units if needed, and include the correct unit in your answer.
• 📋Percentage change and compound interest: Calculate percentage increase/decrease, simple interest, or compound interest over multiple years. Advice: use multipliers for efficiency and show your working.