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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 today."
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.
Answer:
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 today" 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
so
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.
Puzzle 10
What number is...
...three quarters of eight ninths of one half of 2001?
Many years ago, a cruise liner sank in the middle of the Atlantic Ocean. The survivors luckily landed on a remote desert island.
There was enough food for the 135 people to last four weeks.
Nine days later a rescue ship appeared, unluckily this ship also sank, leaving an additional 36 people stranded on the island to now share the original rationed food.
The food obviously had to be rerationed, everyone was now on threequarters of the original ration, so how many days in total would the food last, from the day of the original sinking?
Originally there was enough food for 135 people for 28 days, which totals 3780 rations.
After 9 days, 1215 rations had been eaten.
Therefore there were now 2565 rations left for 171 people, which would last for another 20 days at threequarter rations per person = (2565 ÷ (3 ÷ 4)) / 171.
Which is 29 days in total from the original sinking.
Puzzle 12
I recently travelled from my home town to a distant music concert, on a pedal tricycle of all things! My wonderful, three wheeled tricycle.
I knew that the epic 2,345 mile trip would wreak havoc on the tyres, but luckily I took along 4 spares!
Instead of waiting for any single tyre to fail, I decided that I would rotate the tyres evenly, making sure that by the end of the trip all seven tyres had travelled exactly the same distance.
A total of 2,345 miles were travelled and at any one time, three tyres were on the tricycle.
Therefore 3 x 2,345 = 7,035 tyre miles were travelled, which was shared equally by the 7 tyres.
And 7,035 ÷7 = 1,005.
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