Challenging extension questions for high-achieving students. These questions require multi-step reasoning, algebraic thinking, and justification of answers. Show full working throughout.

1. 1. A value is increased by 20% and then decreased by 20%. Prove algebraically that the result is always 4% less than the original, regardless of the starting value.

2. 2. A value is increased by p% and then decreased by p%. Show algebraically that the overall change is always a decrease, and find an expression for the percentage decrease in terms of p.

3. 3. An investment of £P grows at r% compound interest per year. After 2 years the interest earned is £1,664. After 3 years the interest earned is £2,597.12. Find the values of P and r.

4. 4. A price is increased by 30% and then reduced by a percentage d so that it returns exactly to its original value. Find the value of d as an exact fraction and as a percentage to 2 decimal places.

5. 5. £10,000 is invested at 5% compound interest per year. £10,000 is also invested at 5% simple interest per year. Show that after n years the compound account always has more money than the simple account for n greater than 1, and find the difference after 10 years to the nearest penny.

6. 6. A car is worth £V. It depreciates at 15% in year 1, 12% in year 2, and 10% per year thereafter. Write an expression for the value of the car after n years (where n is greater than or equal to 2). Find the value after 5 years if V = £24,000. Give your answer to the nearest penny.

7. 7. Two shops sell the same item. Shop A sells it for £180 and is offering a 20% discount followed by a further 5% off the discounted price. Shop B sells it for £165 and is offering a straight 10% off. Which shop is cheaper, and by how much?

8. 8. A quantity grows at a fixed compound percentage rate each year. In year 1 it grows from 400 to 440. In year 3 it is 532.40. Verify that the growth rate is consistent and find the value at the end of year 5 to 2 decimal places.

9. 9. A value is increased by 50% and then increased by a further 50%. A student says this is the same as a 100% increase. Show with algebra whether the student is correct or not, and find the actual overall percentage increase.

10. 10. £P is invested at r% compound interest. After 2 years it is worth £11,236. After 3 years it is worth £11,697.44. Find P and r. Show all algebraic steps.

11. 11. The Rule of 72 states that the number of years for an investment to double at compound interest is approximately 72 divided by the interest rate. Use your knowledge of compound interest to verify this rule for rates of 6%, 8%, and 12%. Comment on the accuracy of the approximation in each case.

12. 12. A population P decreases by a fixed percentage d% each year. After 3 years the population is 0.857375 of its original value. Find d, and find after how many complete years the population will be less than half its original value. Use trial and improvement or logarithms.

13. 13. A sequence of three percentage changes is applied to a value: first an increase of a%, then a decrease of b%, then an increase of c%. The combined multiplier is exactly 1 (the value returns to its original). Given that a = 20 and b = 10, find the exact value of c. Justify your answer algebraically.

14. 14. Account A holds £5,000 at 6% compound interest per year. Account B holds £6,000 at 3% compound interest per year. After how many complete years does Account A first hold more money than Account B? Show your working year by year.

15. 15. A shop increases the price of an item by 25% and then applies a 25% discount to the increased price. A different shop takes the original price, applies a 25% discount first, and then increases the result by 25%. Show algebraically that both methods give the same final price, and find the percentage change from the original price.

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