How many stickers did Tsz-ching have at first?

Question

Tsz-ching, Ka-lok and Ho-yin have 188 stickers in all.

First Tsz-ching gives a third of hers to Ka-lok.

Then Ka-lok sticks 6 in his handbook.

Next Ka-lok gives a third of what he has then to Ho-yin.

Now the three have stickers in the ratio 5 :

3 :

5.

Accepted Answer

Answer: 105 stickers. With 6 used, the three end with 188 − 6 = 182. In 5 : 3 : 5, that is 13 parts of 14: Tsz-ching 70, Ka-lok 42, Ho-yin 70. Undo step 3: after giving a third Ka-lok kept 2/3, 42, so he had 42 ÷ 2 × 3 = 63 and gave 21 to Ho-yin. Ho-yin started with 70 − 21 = 49. Undo step 2: before using 6, Ka-lok had 63 + 6 = 69. Undo step 1: Tsz-ching never changed after it; she kept 2/3, 70, so she had 70 ÷ 2 × 3 = 105 and gave 35 to Ka-lok (who started with 69 − 35 = 34). Check: 105 + 34 + 49 = 188 ✓; forwards: Tsz-ching 105 → 70, Ka-lok 34 → 69 → 63 → 42, Ho-yin 49 → 70; 70 : 42 : 70 = 5 : 3 : 5 ✓ Common mistake: sharing 188 in the ratio, forgetting the 6 used are gone; or undoing "gave a third" as 42 × (1 + 1/3). What's left is 2/3: divide.

Working backwards: olympiad question 8

The question

Worked solution