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## Quiz of the Month (July 2001)

### Hector C. Parr

***

#### SOLUTION TO LAST MONTH'S QUIZ

```         1).  1/2

2).  1/4

3).  1/8

Notes
Suppose x, y, z stand for the decimal parts of c, g, s
respectively. We may assume that each of x, y, z, is
distributed uniformly between 0 and 1.

1). x must lie between 0.25 and 0.75, so probability = 1/2.

2). The condition is satisfied if:
(i) neither c nor g is rounded up, but (c+g) is rounded
up, i.e. x < 0.5, y < 0.5, x+y > 0.5
or (ii) both c and g are rounded up, but (c+g) is not
doubly rounded up,
i.e. x > 0.5, y > 0.5, x+y < 1.5

The coloured areas on the diagram show the values of x and y
which satisfy these conditions. Total = 1/8 + 1/8 = 1/4.

3). The condition is satisfied only if c, g, s are all
rounded up, i.e. x > 0.5, y > 0.5, z > 0.5.
But x, y, x are not independent. In fact x+y+z must = 2.
So taking x and y as the independent variables, we
require:  x > 0.5, y > 0.5, 2-x-y > 0.5.
This corresponds to the red area in the diagram,
whose area is 1/8.
```

#### THIS MONTH'S QUIZ

```  1. The hot tap alone could fill a bath in 6 minutes, and the
cold tap alone could fill it in 3 minutes. How long will
it take to fill the bath with both taps running?

2. Water from the hot tap is at 75 degrees C, and from the
cold tap is at 15 degrees C. If the bath is filled with
both taps, as in question 1, what will be the temperature
of the bath water?

3. A dishonest innkeeper has a litre bottle full of whisky.
Each time he serves a measure of whisky he dispenses 5 cl,
but then secretly fills up the bottle with water. How many
times can he do this before the whisky becomes half strength?
```

***

(c) Hector C. Parr (2001)

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