Ratio, Proportion and Units

A ratio compares two or more quantities of the same type. It is written using a colon: $a:b$.

Simplifying a ratio — divide all parts by the highest common factor:

$24:36=2:3\text{(HCF = 12)}$

$1.5:2=3:4\text{(multiply by 2 to remove the decimal)}$

$\frac{3}{4}:\frac{1}{2}=\frac{3}{4}:\frac{2}{4}=3:2\text{(common denominator first)}$

Connecting ratio to fractions (R8): in a ratio $a:b$, the fraction of the total that the first part represents is $\frac{a}{a+b}$.

Ratio $3:5$ — total 8 parts; first quantity is $\frac{3}{8}$ of the total. ✓

Relating to linear functions (R8): if $y:x=3:1$, then $y=3x$ — a direct proportion.

Method:

1. Add the ratio parts to get the total number of parts.
2. Divide the quantity by the total to find 1 part.
3. Multiply each ratio value by the unit part.

Worked example — divide £240 in the ratio $3:5$:

Total parts: $3+5=8$. One part: $\pounds 240÷8=\pounds 30$.

Shares: $3\times \pounds 30=\pounds 90$ and $5\times \pounds 30=\pounds 150$. Check: $90+150=240$ ✓

Worked example — share 600 g of mortar in the ratio cement : sand $=1:4$:

One part: $600÷5=120$ g. Cement: $120$ g; sand: $480$ g. ✓

Part:whole ratio: if the ratio of red to the total is $2:7$, and the total is 35, red $=35\times \frac{2}{7}=10$.

Standard unit conversions (examples):

Length	Mass	Capacity
$1\text{ km}=1000\text{ m}$	$1\text{ kg}=1000\text{ g}$	$1\text{ litre}=1000\text{ ml}$
$1\text{ m}=100\text{ cm}$	$1\text{ t}=1000\text{ kg}$	$1{\text{ m}}^3=1000\text{ litres}$

Compound units combine two or more base units:

• Speed $=\frac{\text{distance}}{\text{time}}$ (km/h, m/s)
• Density $=\frac{\text{mass}}{\text{volume}}$ (g/cm³, kg/m³)
• Pressure $=\frac{\text{force}}{\text{area}}$ (N/m², Pa)

Converting between compound units:

Convert 72 km/h to m/s: $72\times \frac{1000}{3600}=72\times \frac{5}{18}=20$ m/s ✓

Algebraic context (R1 underlined): $v=\frac{d}{t}$ — rearranges to $d=vt$ and $t=\frac{d}{v}$.

A car travels at $v$ m/s for $t$ seconds. Distance $=vt$ metres; to convert to km, divide by 1000.

A scale factor $k$ enlarges or reduces all lengths by the same multiple.

Map scales: a scale of $1:25000$ means 1 cm on the map represents 25 000 cm $=250$ m in reality.

Worked example — two towns are 4.6 cm apart on a map with scale $1:50000$. Find the real distance in km.

Real distance $=4.6\times 50000=230000$ cm $=2300$ m $=2.3$ km ✓

Scale diagrams — a drawing of a room at scale $1:20$ means every real centimetre is drawn as $\frac{1}{20}$ cm.

Worked example — a wall is 3.6 m long. On a 1 : 50 scale drawing, its length is $\frac{360\text{ cm}}{50}=7.2$ cm.

R3 — express one quantity as a fraction of another:

12 out of 40: $\frac{12}{40}=\frac{3}{10}$. The fraction may be greater than 1 if the first quantity is larger.

$50\text{ as a fraction of }30=\frac{50}{30}=\frac{5}{3}$

R6 — multiplicative relationship: if $a=kb$ for some constant $k$, then $a$ and $b$ are in a multiplicative relationship. Express as ratio $a:b=k:1$ or as fraction $\frac{a}{b}=k$.

R7 — proportion as equality of ratios:

$a$ and $b$ are in the same proportion as $c$ and $d$ if $\frac{a}{b}=\frac{c}{d}$ (or equivalently $a:b=c:d$).

Worked example — if 3 tins of paint cover 24 m², how many tins are needed for 56 m²?

$\frac{3}{24}=\frac{n}{56}⟹n=\frac{3\times 56}{24}=7$ tins ✓

1. Simplifying a ratio — dividing only one part

$8:12$ simplifies to $2:3$. Both parts must be divided by the HCF. Writing $8:3$ (only dividing the second) is incorrect.

2. Dividing in a ratio — dividing the quantity by one part, not the total

For $3:5$, there are $3+5=8$ parts, so one part $=\frac{\text{quantity}}{8}$, not $\frac{\text{quantity}}{3}$ or $\frac{\text{quantity}}{5}$.

3. Map scales — forgetting to convert units

Scale $1:50000$: a map length of 3 cm means 150 000 cm in reality. Convert to m or km for a sensible answer.

4. Fractions of quantities — using the wrong denominator

In ratio $2:5$, the first part is $\frac{2}{7}$ of the total (denominator $=2+5=7$), not $\frac{2}{5}$.

Mistake	Correction
"Simplify $9:12$ → $3:12$"	Divide both by 3: $3:4$
"Divide £180 in ratio $2:3$: one part $=90$"	Total parts $=5$; one part $=36$; shares: £72 and £108
"Map scale 1:20000; 5 cm → 100 000 cm, so 100 000 m"	$100000$ cm $=1000$ m, not 100 000 m — divide by 100 to convert cm to metres