105. There is a number whose three digits are the same. If 7 times the sum of the digits is subtracted from the number, the remainder is 180. What is the number?

106. A and B can do a piece of work in m days, B and C in n days, A and C in p days. In what time can all together do it? How long will it take each alone to do it?

107. Two passengers together have 400 pounds of baggage and are charged, for the excess above the weight allowed free, 40 and 60 cents respectively. If the baggage had belonged to one of them, he would have been charged $1.50. How much baggage is one passenger allowed without charge?

108. Divide 20 into two parts such that the sum of the two fractions formed by dividing each part by the other is 44.

109. It takes 1000 square tiles of a certain size to pave a hall, or 1440 square tiles whose dimensions are one inch less. the area of the hall floor.

Find

110. The sum of two numbers is 16, and the difference of their squares is 128. What are the numbers?

111. Find two numbers such that their sum, their product, and the difference of their squares are all equal.

112. Divide 25 into two parts such that the difference of their square roots is 1.

113. The difference of two numbers is 6, and their product is equal to twice the cube of the less number. What are the numbers?

114. It took a number of men as many days to pave a sidewalk as there were men. Had there been 3 men more, the work would have been done in 4 days. How many men were there?

115. The product of two numbers is 8, and the sum of their squares is 14 greater than the sum of the numbers. What are the numbers?

116. A rectangular lawn 50 feet long and 40 feet wide has a path of uniform width around it. If the area of the walk is 64 square yards, what is its width?

117. A merchant sold goods for 56 dollars and gained as many hundredths of the cost as there were dollars in the cost. Find the cost of the goods.

118. A person swimming in a stream that runs 11⁄2 miles per hour finds that it takes him 3 times as long to swim a certain distance up the stream as it does to swim the same distance down. What is his rate of swimming in still water?

119. A drover bought some oxen for $900. After 5 had died, he sold the rest at a profit of $ 20 each and thereby gained $350. How many oxen did he buy?

120. A detachment from an army was marching in regular column with 5 men more in depth than in front. On approaching the enemy, the front was increased by 845 men, and the whole was thus drawn up in 5 lines. Find the number of men in the detachment.

121. A round iron bar weighed 36 pounds. If it had been 1 foot longer and of uniform diameter, each foot of it would have weighed a pound less. Find the length of the iron bar and its weight per foot.

122. A farmer has two cubical granaries. The side of one is 3 yards longer than the side of the other, and the difference of their solid contents is 117 cubic yards. What is the length of the side of each?

123. Two workmen, A and B, were employed at different wages. At the end of a certain number of days A received $30, but B, who had been idle two days in the meantime, received only $19.20. If B had worked the whole time, and A had been idle two days, they would have received equal sums. Find the number of days, and the daily wages of each.

124. By traveling 5 miles an hour less than its usual rate a train was 50 minutes late in running 300 miles. Find the usual rate of speed and the time required to make the trip.

125. Find two numbers such that their sum, their product, and the sum of their squares are all equal.

126. A merchant bought two lots of tea, paying for both $34. One lot was 20 pounds more than the other, and the number of cents paid per pound was in each case equal to the number of pounds bought. How many pounds of each did he buy?

127. A and B hired a pasture into which A put 4 horses, and B as many as cost him 18 shillings per week. Afterward B put in 2 additional horses, and found that he must pay 20 shillings per week. How much was paid for the pasture per week?

128. By lowering the selling price of apples 1 cent a dozen, an apple woman finds that she can sell 60 more than she used to sell for 60 cents. At what price per dozen did she sell them at first?

129. A and B are two stations 300 miles apart. Two trains start at the same time, one from A, the other from B, and travel to the opposite station. If the first train reaches B 9 hours after the trains meet, and the second train reaches A 4 hours after they meet, when do they meet, and what is the rate of each train?

130. If a carriage wheel 143 feet in circumference takes one second longer to revolve, the rate of traveling will be 23 miles less per hour. How fast is the carriage traveling?

131. A railway train, after traveling 2 hours, is detained 1 hour by an accident. It then proceeds at of its former rate, and arrives 72 hours behind time. If the accident had occurred 50 miles farther on, the train would have arrived 6 hours behind time. What was the whole distance traveled by the train?

132. A person rents a certain number of acres of land for $200. He retains 5 acres for his own use and sublets the rest at $1 an acre more than he gave. If he receives $10 more than he pays for the whole, how many acres does he rent, and at what rate per acre?

133. A and B left Chicago and walked in the same direction at uniform rates. B started 2 hours after A and overtook him at the 30th milestone. Had each traveled half a mile more per hour, B would have overtaken A at the 42d milestone. At what rate did each travel?

RATIO AND PROPORTION

308. 1. What is the relation of 10 x to 5 x? of 3 x to 12 x? of 8 a to 2 a? of 14 m to 7 m? of 4 a to 8 a? of 2 b to 6 b?

2. In finding the relation, or ratio, of 10 a to 5 a, which is the dividend, the number that precedes, or the number that follows? Which is the divisor?

3. What is the ratio of a to b? Since b may not be exactly contained in α, how may the ratio be expressed?

4. Since the ratio of two numbers may be expressed in the form of a fraction, what operations may be performed upon the terms of a ratio without changing the ratio?

309. The relation of two numbers that is expressed by the quotient of the first divided by the second is called their Ratio.

310. The Sign of Ratio is a colon (:).

A ratio is also expressed in the form of a fraction.

The ratio of a to b is written ab or

α

b

The colon is sometimes regarded as derived from the sign of division by omitting the line.

311. The first term of a ratio is called the Antecedent.

It corresponds to a dividend, or numerator.

312. The second term of a ratio is called the Consequent.

It corresponds to a divisor, or denominator.

313. The antecedent and consequent form a Couplet.

α

In the ratio a: b, or a is the antecedent, b the consequent, and the

terms a and b form a couplet.

314. The ratio of the reciprocals of two numbers is called the Reciprocal, or Inverse Ratio of the numbers.

It may be expressed by interchanging the terms of the ratio.

315. The ratio of the squares of two numbers is called the Duplicate ratio; the ratio of their cubes, the Triplicate ratio; the ratio of their square roots, the Subduplicate ratio; the ratio of their cube roots, the Subtriplicate ratio of the numbers.

The duplicate ratio of a to b is a2 b2; the triplicate ratio, a3: b3; the subduplicate ratio, Va: √b; the subtriplicate ratio, Va: Vb.

316. PRINCIPLE.

Multiplying or dividing both terms of a ratio

by the same number does not change the ratio.

EXAMPLES

1. What is the ratio of 8m to 4m? of 4 m to 8 m?

2. Express the ratio of 6:9 in its lowest terms; 12x: 16 y; am: bm; 20 ab: 10 bc; (m + n) : (m2 — n2).

3. Which is the greater ratio, 2:3 or 3: 4? 4:9 or 2:5?

4. What is the ratio of to ? to? to 3?

SUGGESTION.

When fractions have a common denominator, they have the ratio of their numerators.

5. Reduce a b and x:y to ratios having the same consequent. 6. When the antecedent is 6x and the ratio is }, what is the consequent?

317. It is evident from § 316, that the ratio of two rational fractions may be expressed by the ratio of two integers.

b

b

: may be reduced to the form xny: xny,

n y

m

n

y

But the ratio of two numbers, when one is rational and the other irrational or when they are dissimilar surds, cannot be expressed by the ratio of two integers.

Thus, the ratio v2: 3 cannot be expressed by any two integers.