Maths in Context: Ratios and Rates Notes

Jet Engines: Passenger jets operate at $10\,000\,\text{revs/minute}$ , intake air at $1200 - 1500\,\text{kg/second}$ , and have a compression ratio (CR) of $40:1$ .
Aspect Ratio (AR): The ratio of wing length to average wing width. Gliders use $15:1$ , commercial planes use $7:1$ to $9:1$ , and the Boeing 787 uses $11:1$ .
Glide Ratio (GR): The ratio of horizontal distance travelled to vertical height dropped without power. Modern gliders achieve $60:1$ ; the Boeing 787 has a GR of $20:1$ .
Fuel Consumption:
- Boeing 747: $4\,\text{L/s}$ ( $1200\,\text{L/100\,km}$ ) for $416\,\text{passengers}$ .
- Boeing 787: $1.5\,\text{L/s}$ ( $600\,\text{L/100\,km}$ ) for $270\,\text{passengers}$ .

Definition: Ratios compare related quantities of the same type and unit. The order is critical (e.g., teacher-to-student ratio of $1:15$ vs. $15:1$ ).
Simplest Form: A ratio is simplified when it consists of whole numbers and the highest common factor (HCF) is $1$ .
- Decimals/Fractions: Simplify by multiplying until whole numbers are achieved (e.g., multiply by the Lowest Common Denominator).
- Units: Quantities must be in identical units before simplification (e.g., $4\,\text{mm}$ to $2\,\text{cm}$ becomes $4:20$ , which simplifies to $1:5$ ).

Equivalent Ratios: Formed by multiplying or dividing each term by the same value.
Dividing a Quantity: Used to share a total based on a ratio.
- Unitary Method:
  1. Find total parts (sum of the ratio terms).
  2. Find the value of one part ( $\text{Total quantity} \div \text{Total parts}$ ).
  3. Multiply the single part value by the required terms.
- Fraction Method: Multiply the total amount by $\frac{\text{term}}{\text{total parts}}$ .

Scale Ratio: $\text{Drawing length : Actual length}$ .
Scale Factor: The value used to convert between drawing and real-life dimensions. It is the second term when the first term is normalized to $1$ .
Conversion Directions:
- Scaled distance to actual distance: $\text{Scaled length} \times \text{Scale factor}$ .
- Actual distance to scaled distance: $\text{Actual length} \div \text{Scale factor}$ .
Length Metrics: $1\,\text{km} = 1000\,\text{m}$ ; $1\,\text{m} = 100\,\text{cm}$ ; $1\,\text{cm} = 10\,\text{mm}$ .

Definition: Rates compare different types of quantities with different units. Units must be explicitly shown (e.g., $\$1.45/\text{L}$ ).
Simplified Rate: Expressed per one unit of the second quantity (denominator of $1$ ).
Average Rate: Calculated as $\frac{\text{Total change in quantity 1}}{\text{Total change in quantity 2}}$ .
Combination Rates: Used for solving tasks where multiple actors work together (e.g., multiple hoses filling a pool). Use the Least Common Multiple (LCM) of time to determine total output.

Formula: $Average\,speed (s) = \frac{Distance\,travelled (d)}{Time\,taken (t)}$ .
Derived Formulas:
- $d = s \times t$
- $t = \frac{d}{s}$
Standard Units: $\text{metres per second (m/s)}$ and $\text{kilometres per hour (km/h)}$ .

Unitary Technique: Solve complex problems by first calculating the value of a single unit (e.g., price per $1\,\text{kg}$ or distance per $1\,\text{L}$ ).
Speed Conversion Shortcuts:
- $\text{To convert km/h to m/s: Divide by } 3.6$
- $\text{To convert m/s to km/h: Multiply by } 3.6$

Classroom Ratios: How are ratios such as girls to boys, watch wearers to non-wearers, or eye colors compared and simplified?
Rate Estimation: What are standard rates for daily activities like television commercials per hour, words typed per minute, or laughter frequency for teenagers?
Variable Unit Rates: In cases like Teleconnect vs. Connectplus phone chargers, how does the tiered rate ( $50\text{c/min}$ vs $75\text{c/min}$ ) compare to a constant rate ( $60\text{c/min}$ ) for different call lengths?
Fun Run Investigation: Which rates are most useful for representing fitness: seconds per $100\,\text{m}$ , strides per minute, or heart beats per minute? Consider the performance of Mrs M, Mr P, and Mr A (e.g., Mrs M ran $10.44\,\text{km}$ in $41\,\text{minutes}$ , $50\,\text{seconds}$ ).