Problem-solving practice 2 (AQA GCSE Maths): Revision Notes
Problem-solving practice 2
Problem-solving in ratio and proportion requires a systematic approach and clear understanding of the underlying mathematical relationships. This section focuses on three key areas where students commonly encounter challenges: ratio calculations, compound interest problems, and proportion reasoning.
Working with ratios to find totals
When dealing with ratio problems, the key strategy involves identifying the total number of parts in the ratio and using this to scale up to the actual quantities.
Method for ratio problems:
- Add up all parts in the ratio to find the total parts
- Use the given information to find what each part represents
- Multiply by the total parts to find the complete answer
For example, if you know one part of a ratio and need to find the total, work out what each individual part represents, then multiply by all parts combined. This approach ensures you account for all components in the ratio relationship.
Exam tip: Always check your answer by working backwards - divide your total by the ratio to verify it gives you the original information.
Compound interest calculations
Compound interest problems require step-by-step calculations where you apply the interest rate year by year, accounting for any additional investments along the way.
Key strategy for compound interest:
- Calculate interest for each year separately
- Add any extra investments at the specified times
- Apply the interest rate to the new total for subsequent years
This systematic approach prevents errors and ensures you don't miss any components of the calculation. The interest compounds because each year's interest becomes part of the principal for the following year's calculation.
Essential working notes to track:
- Interest earned in each year
- Total starting amount for each year
- Any additional investments made
- Final total at the end of the calculation period
Proportion and percentage problems
When working with proportional relationships and percentages, the most effective technique is to find what 1% represents, then scale this to find the required amount.
Worked Example: Percentage Problem Strategy
Step 1: Identify what percentage you know and its corresponding value Step 2: Calculate what 1% represents by dividing Step 3: Multiply by the target percentage to find your answer
For instance, if you know that a 3% increase equals a specific amount, divide this amount by 3 to find what 1% represents. Then multiply by 103 to find the total (original 100% plus the 3% increase).
This method works for any percentage problem and provides a clear pathway from the given information to the required answer.
Key Points to Remember:
- For ratio problems: Add all parts together, then use the total to scale up from your known information
- For compound interest: Work year by year, adding extra investments when specified and applying interest to the new totals
- For proportion problems: Find what 1% represents first, then multiply to get your target percentage
- Always show your working clearly and check your answers make sense in the context of the problem
- Break complex problems into smaller steps and note what you've calculated at each stage