Tommy: "How old are you, mamma?"

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?

Mamma's Age – Amusements In Mathematics, Henry Ernest Dudeney.

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Hint

The answer isn't a whole number of years, and algebra might be required.

Answer

29 years 2 months.

Reasoning #1

This answer is taken directly from the original book.

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.

Reasoning #2

Here's my answer, with a little algebra.

If we call Tommy T, Mamma M and Papa P we can see that:

"our three ages add up to exactly seventy years" gives us:

(1) T + M + P = 70

"Just six times as old as you" gives us:

(2) P = 6 x T

In an unknown number of years (Y) "Shall I ever be half as old as you" gives us:

(3) P + Y = 2 x (T + Y)

and "our three ages will add up to exactly twice as much as today" gives us:

(T + Y) + (M + Y) + (P + Y) = 140

which can be written as

(4) T + M + P + 3Y = 140

We can see from (4) and (1) that

3Y = 70

so

(5) Y = 70 ÷ 3

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

(6) T = 70 ÷ 12

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.