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Mamma: "Let me think, Tommy. Well, our three ages add up to exactly seventy years."
Tommy: "That's a lot, isn't it? And how old are you, Papa?"
Papa: "Just six times as old as you, my son."
Tommy: "Shall I ever be half as old as you, Papa?"
Papa: "Yes, Tommy; and when that happens our three ages will add up to exactly twice as much as to-day."
Tommy: "And supposing I was born before you, Papa; and supposing Mamma had forgot all about it, and hadn't been at home when I came; and supposing..."
Mamma: "Supposing, Tommy, we talk about bed. Come along, darling. You'll have a headache."
Now, if Tommy had been some years older he might have calculated the exact ages of his parents from the information they had given him. Can you find out the exact age of Mamma?
[Ref: ZPYU] Mamma's Age. Amusements In Mathematics by Henry Ernest Dudeney (1917).
Hint: The answer involves years and months.
The age of Mamma must have been 29 years 2 months.
That of Papa, 35 years; and that of the child, Tommy, 5 years 10 months. Added together, these make seventy years. The father is six times the age of the son, and, after 23 years 4 months have elapsed, their united ages will amount to 140 years, and Tommy will be just half the age of his father.
The answer above is taken from the original book, here is another version of the answer:
If we call Tommy T, Mamma M and Papa P we can see that:
"our three ages add up to exactly seventy years" leads to
T + M + P = 70 (1)
"Just six times as old as you" leads to
P = 6 x T (2)
In an unknown number of years (Y) "Shall I ever be half as old as you" leads to:
P + Y = 2 x (T + Y) (3)
and "our three ages will add up to exactly twice as much as to-day" leads to:
(T + Y) + (M + Y) + (P + Y) = 140
which can be written as
T + M + P + 3Y = 140 (4)
We can see from (4) and (1) that
3Y = 70
Y = 70 ÷ 3 (5)
Using (2) and (5) in (3) we have
P + Y = 2 x (T + Y)
6 x T + 70÷3 = 2 x (T + 70÷3)
4 x T = 70÷3
T = 70÷12 (6)
We can now use (6) in (2) to see that:
P = 6 x T
P = 6 x 70÷12
P = 70÷2
And using the values for T and P in (1) we have:
T + M + P = 70
70÷12 + M + 70÷2 = 70
Multiply throughout by 12 to give:
70 + 12 x M + 420 = 840
12 x M = 840 - 420 - 70
12 x M = 350
M = 350÷12
So: Tommy = 70÷12 = 5.83333 = 5 years 10 months.
Papa = 70÷2 = 35 = 35 years.
Mamma = 350÷12 = 29.1666 = 29 years 2 months.
Imagine a bottle recycling skip, empty other than one lonely bottle.
Every hour, on the hour, people come and put bottles into the skip.
The first hour, at Noon, one person came and put a bottle in.
One hour later, two people placed a bottle each into the skip.
An hour later four people placed a bottle each into the skip.
This doubling of people continued until 11pm, when the skip was finally full.
When was the skip exactly half full?
Hint: Remember that the bottle count doubles each time.
The skip started with 1 lonely bottle.
At Noon: 1 person came along and added a bottle, making the total 2 bottles.
At 1pm: 2 people came along and added a bottle each, making the total 2 + 2 = 4 bottles.
At 2pm: 4 people came along and added a bottle each, making the total 4 + 4 = 8 bottles.
At 3pm: 8 people came along and added a bottle each, making the total 8 + 8 = 16 bottles.
Therefore the number of bottles in the skip is doubling every hour, as it was full at 11pm, it must have been half full at 10pm.