The number of pears on my tree went up 5% in 2023, and went up another 25% in 2024. 1) If the number of pears at the end of 2022 was 80, how many were there at the end of 2024? Easy: up 5% = multiply by 1.05 for 2023 to get 84. Then increase that by 25% (multiply by 1.25) for 2024 so the total is 84*1.25 = 105

NOTE you can't add percentages because if you do you get 30% which gives you 80*1.3 = 104. Why is it different? Because in 2024 you added 25% of 84, not 80, which gave you one extra pear.

2) if the number of pears at the end of 2024 was 80, how many were there at the start of 2023?

Suppose it was P, then at the end of 2023 it was 1.05*P and at the end of 2024 it was 1.25*(1.05*P) So 1.25*1.05*P = 80 and P = 80/(1.25*1.05) = 61 (rounded). Another way to look at it is to say that at the end of 2023 it was 80/1.25 (=64), and at the start of 2023 it was that 64/1.05 = 61.

If you work backwards, you can see that you have to divide!

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