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Which square, which circle, and which triangle has the closest area to the doughnut shape on the left? The drawings are to scale, so you might be able to judge it, as well as working the actual areas out.

The larger circle has diameter = 4, therefore the radius is 2, and the area is π x 2^{2} = 4π.

The smaller circle has diameter = 2, therefore the radius is 1, and the area is π x 1^{2} = π.

Therefore the shaded area is 4π - π = 3π ≈ 9.42.

Circle

The area of a circle is π x Radius^{2}.

The circle with diameter 3.1 has a radius of 1.55 and an area of π x 1.55^{2} = 7.55.

The circle with diameter 3.5 has a radius of 1.75 and an area of π x 1.75^{2} = 9.62 (** closest match).

The circle with diameter 3.9 has a radius of 1.95 and an area of π x 1.95^{2} = 11.95.

Square

The area of a square is Side x Side.

The square with side 3.1 has an area of 3.1 x 3.1 = 9.61 (** closest match).

The square with side 3.5 has an area of 3.5 x 3.5 = 12.25.

The square with side 3.9 has an area of 3.9 x 3.9 = 15.21.

Triangle

The area of a triangle is ½ x Base x Height. Using Pythagoras' theorem it can be shown that the area of an equilateral triangle is Sqrt(3) x Base^{2} ÷ 4.

The triangle with side 3.9 has an area of Sqrt(3) x 3.9 x 3.9 ÷ 4 = 6.59.

The triangle with side 4.3 has an area of Sqrt(3) x 4.3 x 4.3 ÷ 4 = 8.01.

The triangle with side 4.7 has an area of Sqrt(3) x 4.7 x 4.7 ÷ 4 = 9.57 (** closest match).

Puzzle 402

Using all of the letters A to Z, each once only, complete these common words:

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.

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 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

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.

QED.

Puzzle 405

Using the BrainTracker grid below, how many words can you find? Each word must contain the central P and no letter can be used twice, however, the letters do not have to be connected. Proper nouns are not allowed, however, plurals are. Can you find the nine letter word?