Ratio, proportion and rates of change

Explanation

• Unit conversion means multiplying or dividing by a fixed conversion factor. Convert quantities to compatible units before substituting into a formula, and convert the numerator and denominator of a compound unit separately.
• Length factors must be raised to the correct power: because $1\,\text{m}=100\,\text{cm}$, then $1\,\text{m}^2=100^2\,\text{cm}^2$ and $1\,\text{m}^3=100^3\,\text{cm}^3$.
• In algebraic contexts the same factor applies to the whole expression.
• A sensible estimate can expose a conversion in the wrong direction.
• Examiners expect the conversion step to be visible, especially when the final unit is compound.

Explanation

• A scale written $1:n$ compares a diagram length with the corresponding real length in the same units. Multiply a diagram length by $n$ to obtain the real length, or divide the real length by $n$ to obtain the diagram length.
• Convert units only after establishing which direction the scale factor acts.
• For areas, square the linear scale factor; for volumes, cube it.
• On a map, measure between the stated points accurately and use any given scale bar or numerical scale.
• Examiners award method for applying the scale in the correct direction and then converting units correctly.

Explanation

• To express quantity $A$ as a fraction of quantity $B$, form $\dfrac{A}{B}$ and simplify.
• The order of the wording fixes the order of the fraction: the quantity before “as a fraction of” is the numerator.
• Quantities must first be written in the same units, so that the units cancel.
• The result may be less than $1$, equal to $1$, or greater than $1$; an improper fraction is valid when the first quantity is larger.
• Examiners expect a simplified exact fraction unless the question asks for another form.

Explanation

• Ratio notation compares quantities in a fixed order, for example $a:b$. Before simplifying, express every quantity in the same units.
• An equivalent ratio is produced by multiplying or dividing every part by the same non-zero value.
• For an integer ratio, divide all parts by their highest common factor; when decimals occur, first scale them to integers if helpful.
• Ratios can have more than two parts, and a shared part can be matched using a common multiple.
• Examiners require the stated order to be preserved and usually expect the simplest integer form.

Explanation

• In a part:part ratio $a:b$, the whole contains $a+b$ equal shares.
• Divide the total by the sum of the ratio parts to find one share, then multiply by each part.
• A part:whole statement needs different reading: if one part is $\dfrac{3}{8}$ of the whole, the other part is $\dfrac{5}{8}$, giving ratio $3:5$.
• In mixing or concentration problems, track the amount of the relevant ingredient rather than simply averaging percentages.
• Examiners expect both final parts to be labelled and to add back to the original total.

Explanation

• A multiplicative relationship compares quantities by division rather than subtraction. If $A=kB$, then $\dfrac{A}{B}=k$, so $A:B=k:1$ and $A$ is the fraction $k$ of $B$.
• Convert decimal multipliers to fractions when an integer ratio is required; for example, $1.75=\dfrac{7}{4}$ gives $A:B=7:4$.
• For a chain of relationships, substitute one equation into another or match the shared ratio part.
• Check that a multiplier greater than $1$ corresponds to the larger quantity.
• Examiners expect the direction of the comparison to agree with the wording, because reversing it produces the reciprocal.

Explanation

• Proportion means that two corresponding ratios are equal. Keep the quantities in the same order on both sides, then solve by scaling, finding a unit value, or cross-multiplying.
• If $\dfrac{a}{b}=\dfrac{c}{d}$ with non-zero denominators, then $ad=bc$.
• A unitary method is often clearest in context: find the amount for one unit and scale to the required number.
• Decimal answers involving money must be interpreted in pounds and pence.
• Examiners award method for a correct proportional relationship even if a later arithmetic slip occurs, so show the equation or unit value.

Explanation

• For parts in ratio $a:b$, the whole has $a+b$ equal parts, so their fractions of the whole are $\dfrac{a}{a+b}$ and $\dfrac{b}{a+b}$. A constant ratio between variables can also define a linear function.
• If $y:x=m:1$, then $\dfrac{y}{x}=m$ and $y=mx$.
• Its graph is a straight line through the origin, with gradient $m$.
• A line with a non-zero intercept does not represent a constant ratio.
• Examiners may ask you to move between ratio, fraction, equation and graph, so always state the link explicitly.

Explanation

• A percentage is a number of parts per hundred, so $p\%=\dfrac{p}{100}$. Percentage change is multiplicative: an increase of $r\%$ uses multiplier $1+\dfrac{r}{100}$ and a decrease uses $1-\dfrac{r}{100}$.
• Percentages above $100\%$ correspond to multipliers greater than $1$.
• To reverse a change, divide by the multiplier; do not apply the opposite percentage to the changed value.
• Simple interest is calculated each year from the original principal, so the yearly interest is constant.
• Examiners expect the original amount to be the denominator when calculating percentage change.

Explanation

• For direct proportion, $y\propto x$ means $y=kx$.
• The ratio $\dfrac{y}{x}$ is constant and the graph is a straight line through the origin.
• For inverse proportion, $y\propto\dfrac{1}{x}$ means $y=\dfrac{k}{x}$, so the product $xy$ is constant and the graph is a decreasing curve for positive values.
• Find $k$ from a known pair, write the equation, then substitute the required value.
• Examiners expect the equation with the constant of proportionality, not only a numerical scaling argument, when the command is “find a formula”.

Explanation

• A compound unit combines quantities, usually through division. Use $\text{speed}=\dfrac{\text{distance}}{\text{time}}$, $\text{density}=\dfrac{\text{mass}}{\text{volume}}$ and $\text{pressure}=\dfrac{\text{force}}{\text{area}}$.
• Rates of pay and unit prices are totals divided by the relevant time or number of items.
• Make the units compatible before substituting, and rearrange the formula if the unknown is in the numerator or denominator.
• The units provide a useful check on the operation and show which quantity should be divided by which.
• Examiners expect both the numerical value and the correct compound unit, such as $\text{kg}/\text{m}^3$.

Explanation

• Similar shapes have equal corresponding angles and proportional corresponding lengths. If the linear scale factor from one shape to another is $k$, the length ratio is $k$, the area ratio is $k^2$, and the volume ratio is $k^3$.
• Match corresponding measurements and keep the comparison order consistent.
• To recover a length factor from an area ratio take a square root; from a volume ratio take a cube root.
• Trigonometric ratios remain constant in similar right-angled triangles.
• Examiners expect you to identify whether the measurements are lengths, areas or volumes before applying the scale factor.

Explanation

• If $X$ is inversely proportional to $Y$, then $X\propto\dfrac{1}{Y}$ and $X=\dfrac{k}{Y}$ for a constant $k$; equivalently, $XY=k$. Foundation questions can require interpreting equations that describe direct and inverse proportion.
• Higher tier: construct equations involving a power, such as $y=kx^2$ or $y=\dfrac{k}{x^n}$.
• Substitute a known pair to find $k$, write the complete equation, then use it to find an unknown.
• Check any stated restrictions, such as a positive length, before choosing a root.
• Examiners usually award separate method marks for the correct proportional form and for finding the constant.

Explanation

• The gradient of a straight line is $\dfrac{\text{change in }y}{\text{change in }x}$, so it represents a rate of change with units taken from the axes.
• Choose two well-separated points on the line, not merely nearby grid intersections, and calculate rise over run.
• In context, state what the rate means: on a distance-time graph it is speed; on a cost-time graph it is cost per unit time.
• A direct-proportion graph is straight through the origin, while an inverse-proportion graph has constant product $xy$.
• Examiners expect both the gradient and its contextual interpretation.

Explanation

• For a curve, the gradient changes. The gradient of the chord joining two curve points gives the average rate of change over that interval.
• The gradient of a tangent at one point estimates the instantaneous rate of change there; GCSE questions use a drawn tangent, not calculus. Choose two well-separated, readable points on the chord or tangent and calculate $\dfrac{\Delta y}{\Delta x}$.
• The chosen tangent points need not lie on the original curve.
• Include compound units and compare rates using their values.
• Examiners allow a sensible range when answers depend on drawing and reading a tangent.

Explanation

• Repeated percentage growth by $r\%$ uses multiplier $1+\dfrac{r}{100}$ each period; repeated decay uses $1-\dfrac{r}{100}$. After $n$ equal periods, $\text{final}=\text{initial}\times(\text{multiplier})^n$.
• Compound interest is repeated growth because each period acts on the latest balance.
• Higher tier: an iterative process defines each new term from the previous term, so calculate successive values in order and identify the first one satisfying the condition.
• Keep full calculator precision until the requested final rounding.
• Examiners expect the multiplier, exponent or iteration trail, followed by an interpretation in context.