X.

Exercise 60.

Ex. A courier who travels $6$ miles an hour is followed, after $2$ hours, by a second courier who travels $7\frac{1}{2}$ miles an hour. In how many hours will the second courier overtake the first?

lint Let x = the number of hours the first travels. lint Then x - 2 = the number of hours the second travels, 6x = the number of miles the first travels, lintand (x - 2) 712 = the number of miles the second travels. [1] They both travel the same distance. 6x = (x - 2) 712, lintor 12x = 15x - 30. x = 10.

Therefore the second courier will overtake the first in $10-2$, or $8$ hours.

1. A sets out from Boston and walks towards Portland at the rate of $3$ miles an hour. Three hours afterward B sets out from the same place and walks in the same direction at the rate of $4$ miles an hour. How far from Boston will B overtake A?

2. A courier who goes at the rate of $6\frac{1}{2}$ miles an hour is followed, after $4$ hours, by another who goes at the rate of $7\frac{1}{2}$ miles an hour. In how many hours will the second overtake the first?

3. A person walks to the top of a mountain at the rate of two miles an hour, and down the same way at the rate of $4$ miles an hour. If he is out $6$ hours, how far is it to the top of the mountain?

4. In going a certain distance, a train travelling at the rate of $40$ miles an hour takes $2$ hours less than a train travelling $30$ miles an hour. Find the distance.

Exercise 61.

Ex. A hare takes $4$ leaps to a greyhound's $3$; but $2$ of the greyhound's leaps are equivalent to $3$ of the hare's. The hare has a start of $50$ leaps. How many leaps must the greyhound take to catch the hare?

lint Let 3x = the number of leaps taken by the greyhound. lint Then 4x = the number of leaps of the hare in the same time.

lint Also, let a = the number of feet in one leap of the hare. [1] lint Then 3a2 = the number of feet in one leap of the hound. [1] lint Therefore, 3x x 3a2 9ax2 = the whole distance.

As the hare has a start of $50$ leaps, and takes $4x$ leaps more before she is caught, and as each leap is $a$ feet,

(50 + 4x)a = the whole distance. 9ax2 = (50 + 4x)a. lint Multiply by $2$, 9ax = (100 + 8x)a, lint Divide by $a$, 9x = 100 + 8x, x = 100, 3x = 300.

Therefore the greyhound must take $300$ leaps.

1. A hound makes $3$ leaps while a rabbit makes $5$; but $1$ of the hound's leaps is equivalent to $2$ of the rabbit's. The rabbit has a start of $120$ leaps. How many leaps will the rabbit take before she is caught?

2. A rabbit takes $6$ leaps to a dog's $5$, and $7$ of the dog's leaps are equivalent to $9$ of the rabbit's. The rabbit has a start of $60$ of her own leaps. How many leaps must the dog take to catch the rabbit?