This task was made to give loads of practice of using decimal multipliers, while introducing reverse percentage problems, and developing a conceptual understanding of the various additive and multiplicative relationships underpinning percentages.

The follow-up task brings in ideas of prime factors, and combining decimal multipliers.

Some of the important ideas highlighted:

1) Scaling the original amount scales all three calculated amounts.

2) Scaling the percentage only scales P% of A, not the increased or decreased amount.

3) P% of A is equal to A% of P.

4) P% of A is equal to (2P)% of A/2. (etc.)

5) A decreased by P% is equal to (100-P)% of A. And vice versa.

6) If you decrease a positive amount by a percentage greater than 100%, you end up with a negative amount.

7) Decimal multipliers are really handy.

8) Decimal multipliers can be combined

9) If you put a task in a table, pupils end up doing more questions than they realise!

10) You can have fractions in percentages. But it is unwise to have percentages in fractions.

11) P% of (A+B) is equal to (P% of A) + (P% of B).

12) (P+Q)% of A is equal to P% of A + Q% of A.

13) Understanding prime factors is helpful when writing questions with nice numbers.

14) Excel is handy for creating sets of questions like this.

15) Sometimes the set of questions improvised under the visualiser is 98% as good as the ‘refined’ set that takes far longer…

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