Overview
This lecture covers key concepts and problem-solving methods in ratio, proportion, recipe scaling, value for money, density, speed, and related algebraic applications.
Ratio Calculations
- To split an amount in a ratio, add the ratio parts and divide the total by this sum to get one part's value.
- Multiply each part by the value to find each share, then check by summing shares.
- For ratios involving given shares, determine the scaling factor and adjust all parts accordingly.
- When extra differences are provided (e.g., one share is £21 more), relate this difference to the ratio parts and calculate the value of one part.
Combining Ratios & Unit Conversions
- To combine ratios with overlapping terms, scale ratios so the overlapping term matches, then merge.
- For currency conversion, use the exchange rate: divide or multiply depending on direction.
- Compare values by converting both amounts to the same unit (pounds or dollars).
Recipe Scaling & Value for Money
- Scale recipes up by adding recipes for smaller groups as needed.
- For value for money, find price per unit by dividing total price by quantity; compare per-unit prices.
Mass, Density, and Speed Problems
- Use tables to organize values for mass, density, and volume.
- Density formula: density = mass / volume.
- For mixtures, add masses and volumes, then use total density = total mass / total volume.
- Speed formula: speed = distance / time, distance = speed × time, time = distance / speed.
- For multi-part journeys, add distances and times to find total averages.
Proportion, Taps, and Estimation
- Work out single machine/tap rates, then divide by the number of machines for group work.
- State assumptions (e.g., all machines/taps work equally).
- For proportional sampling (capture-recapture), set up equations based on ratios and solve for the unknown.
Direct & Inverse Proportionality
- Direct proportion: a = k × b; inverse: a = k / b.
- Substitute given values to find k, then use it to solve for unknowns.
- For direct proportion to roots/powers, modify formula accordingly (e.g., a ∝ √b → a = k√b).
Probability and Fractions in Ratios
- Convert ratios to fractions and use multiplication for combined probabilities.
- Add results with common denominators; simplify if needed.
Algebra with Ratios
- Equivalent ratios can be written as fractions and set equal; cross-multiply and solve resulting equations.
- Factorize quadratics to find possible variable values.
Distance, Speed Graphs, and Acceleration
- Estimate distance by calculating the area under speed-time graphs (split into shapes and sum areas).
- Estimate acceleration by drawing a tangent at a point and calculating the gradient (rise/run).
Key Terms & Definitions
- Ratio — Comparison of quantities, showing the relative sizes of two or more values.
- Proportion — An equation stating two ratios are equal.
- Density — Mass per unit volume (density = mass / volume).
- Direct Proportion — Relationship where one quantity increases with another.
- Inverse Proportion — Relationship where one quantity increases as another decreases.
Action Items / Next Steps
- Practice ratio and proportion problems, including recipe scaling and combining ratios.
- Review direct and inverse proportion formulas and their applications.
- Complete any assigned homework or textbook problems on these topics.