Stage 9 - Q1 - 2 marks
The garden has 58 seedlings and donates 4. How many remain before the sale?
Hint
This is the stock to which the later fraction will apply.
Stage 9 - Q2 - 2 marks
From 70 seedlings, donate 4, then sell \(\frac{1}{3}\) of the remainder. How many are left?
Hint
The retained fraction is one minus the sold fraction.
Stage 9 - Q3 - 4 marks
Start with 84 seedlings. Each day donate 4; then sell half on day 1 and one third on day 2. Find the stock after each day.
Hint
Do not apply day 2’s fraction to the original stock.
Stage 9 - Q4 - 4 marks
Each day donate 4 seedlings; then sell half on day 1 and one third on day 2. There are 24 left. Recover the original stock.
Hint
The order of inverse operations matters.
Stage 9 - Q5 - 5 marks
Start with 190 seedlings. On each of three days donate 18, then sell these fractions in order: \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{3}\). Find the stock after each day.
Hint
A small stock table prevents restarting from the original number.
Stage 9 - Q6 - 5 marks
Start with 48 units of seed mix. Each day remove 2 units, then sell a fraction. Compare selling half then one third with one third then half. Fractional units are allowed.
Hint
Multiplying retained fractions commutes, but the fixed removals interrupt the products.
Stage 9 - Q7 - 3 marks
A pupil models “donate 4, then sell one third” by \(R=\frac23n-4\). Correct the model.
Hint
Test both formulae with a small starting number.
Stage 9 - Q8 - 4 marks
For a donation of 4 followed by selling one quarter, describe all starting whole-number stocks for which the sale is an integer.
Hint
Express the stock after the donation as four equal groups.
Stage 9 - Q9 - 5 marks
With a daily donation of 4, compare retained fractions a then b with b then a. Derive the difference “second order minus first”.
Hint
Keep the order labels fixed while expanding.
Stage 9 - Q10 - 6 marks
Challenge: Start with 190 seedlings. On each of three days donate 18, then sell one of the fractions one half, one third and one quarter, each used once. The final stock is 22. All sales and stocks are whole numbers. Recover the unique order and justify it.
Hint
There are only six permutations; a table of successive remainders keeps the search systematic.