If the game is played for the numbers 1 to 90, find out: a. How many times would the children say ‘idli’ (including the times they say ‘idli-vada’)? b. How many times would the children say ‘vada’ (including the times they say ‘idli-vada’)? c. How many times would the children say ‘idli-vada’?
Answer
Just count how many multiples of 3, of 5 and of 15 there are up to 90.
a. ‘idli’ → multiples of 3 up to 90 = 90 ÷ 3 = 30 times b. ‘vada’ → multiples of 5 up to 90 = 90 ÷ 5 = 18 times c. ‘idli-vada’ → multiples of 15 up to 90 = 90 ÷ 15 = 6 times
The 6 ‘idli-vada’ numbers are 15, 30, 45, 60, 75 and 90.
Why the divisions work: the multiples of 3 up to 90 are 3 × 1, 3 × 2, …, 3 × 30 — there are as many of them as there are counting numbers up to 30. So the count is simply 90 ÷ 3.
Careful: the question says including the ‘idli-vada’ turns. If you wanted the number of turns where only ‘idli’ is said, it would be 30 − 6 = 24, and only ‘vada’ would be 18 − 6 = 12.
Q3.
What if the game was played till 900? How would your answers change?
Answer
900 is exactly 10 times 90, so every count becomes 10 times bigger.
What is said
Numbers
Up to 90
Up to 900
‘idli’
multiples of 3
30
900 ÷ 3 = 300
‘vada’
multiples of 5
18
900 ÷ 5 = 180
‘idli-vada’
multiples of 15
6
900 ÷ 15 = 60
The 10th ‘idli-vada’ is still at 150, but now the game goes on till the 60th one, at 900.
Check it yourself: only-‘idli’ turns = 300 − 60 = 240, and only-‘vada’ turns = 180 − 60 = 120.
Q4.
Is this figure somehow related to the ‘idli-vada’ game? Hint: Imagine playing the game till 30. Draw the figure if the game is played till 60.
Answer
Yes — Fig. 5.1 is a picture of the game played till 30.
The left circle holds the multiples of 3 — the numbers for which the players say ‘idli’ (3, 9, 12, 18, 21, 24, 27).
The right circle holds the multiples of 5 — the numbers for which they say ‘vada’ (5, 10, 20, 25).
The overlapping middle part holds 15 and 30 — the common multiples, where the players must say ‘idli-vada’.
Playing till 60 gives the same picture with more numbers in it:
The ‘idli-vada’ game played up to 60. Left only → ‘idli’, right only → ‘vada’, overlap → ‘idli-vada’.
Check the counting: multiples of 3 up to 60 = 20, multiples of 5 = 12, common multiples = 4. So 20 − 4 = 16 numbers are ‘idli’ only and 12 − 4 = 8 are ‘vada’ only. ✔