Lecture Notes on Ratios, Proportions, Density, and Speed

Key Topics Covered

1. Splitting Amounts in Ratios

   • Example: Splitting £240 in a ratio of 5:7.
     • Total parts = 5 + 7 = 12 parts.
     • Each part = £240 / 12 = £20.
     • Amounts split as: 5 parts = 5 x £20 = £100 and 7 parts = 7 x £20 = £140.
     • Verification: £100 + £140 = £240.

2. Solving Ratio Problems with Given Amounts

   • Example: Ratio of 8:3 where the first part is £32.
     • £32 corresponds to 8 parts, therefore 1 part = £32 / 8 = £4.
     • For 3 parts: 3 x £4 = £12.

3. Solving Ratio Problems with Differences

   • Example: Ratio of 4:7, James receives £21 more than Emily.
     • Difference of 3 parts = £21, so 1 part = £21 / 3 = £7.
     • Calculating individual amounts: Emily = 4 x £7 = £28, James = 7 x £7 = £49.
     • Total shared amount: £28 + £49 = £77.

4. Combining Ratios

   • Example: A to B is 3:5, B to C is 2:1.
     • Align the middle term by making both equivalent, find common multipliers.
     • Resultant combined ratio: A:B:C = 6:10:5.

5. Currency Exchange and Cost Comparison

   • Evaluating cost difference between two countries using exchange rates.
   • Example: £480 vs. $600 with exchange rate £1 = $1.29.
   • Conversion: $600 / 1.29 = £465.12, cheaper than £480.

6. Increasing Recipe Quantities

   • Scale recipes accordingly by dividing existing quantities to find new targets.
   • Example: Alice has 140g butter but needs 150g to bake for 24 people.

7. Evaluating Best Value Purchases

   • Compare based on unit cost for equalized quantities.
   • Example: Compare 2kg for £3.40 vs. 5kg for £7.20.
   • Lower cost/kg indicates better value.

8. Solving Direct and Inverse Proportion Problems

   • Understand direct (A = kB) versus inverse proportion (A = k/B).
   • Solve for unknown constants and apply to find outcomes with new inputs.

9. Handling Proportions in Problem Solving

   • Capture-recapture method for estimating population using proportions.
   • Apply logical assumptions, such as no change in population size.

10. Density Calculations in Alloys

• Use mass, volume, and density relationships to find alloy densities.
• Combine masses and volumes to find new densities after mixing.

11. Speed and Time Calculations

• Apply average speed formulas and break down journeys into segments.
• Example: Calculating average speed from combined journey segments.

12. Estimate Distance and Acceleration from Graphs

• Use graph areas to estimate total distances.
• Use tangents to find acceleration at specific points on speed-time graphs.

Practical Applications

• Ratio Simplification: Simplifying complex ratios for ease of calculation.
• Fraction Comparison: Ensuring consistent units when comparing fractions or ratios.
• Estimation Techniques: Using visual aids or logical breakdowns for estimation in non-calculator scenarios.

These notes summarize key principles in solving problems involving ratios, proportions, currency conversions, density, and speed-time relationships, with practical examples and methods to apply these concepts.