6. If lumber is worth $ 22.50 per M, what are 650 feet worth? 7. If a pump delivers 500 gallons per minute, what is the delivery per second ? per hour ? 8. If an automobile will run 25 miles on a gallon of gasoline, what is the amount of gasoline consumed on a 1000-mile trip? 9. If trees are planted 64 to the acre, how many trees are there in 40 acres ? in a square mile? 10. If there are 3 quarter notes or their equivalent, per measure, how many quarter notes or equivalent are there in a piece of music of 32 bars ? 11. If the yield of potatoes is 400 bushels to the acre, what is the yield from 21 acres ? from į acre ? 12. If a certain milk sample yields 21 pounds of butter fat per gallon, how many pounds of butter fat will 100 gallons of such milk yield ? 13. If a stenographer writes at the rate of 225 words per minute, how long at the same rate would it take her to write 2000 words? 14. If a spindle tapers one sixteenth of an inch to the inch, what is the amount of taper per foot? 15. If a mechanically operated hack-saw makes 90 strokes per minute, what is the number per hour ? 16. If ammunition is served at the rate of 40 rounds per man, what is the number of rounds supplied to 250 men ? How many men will 10,000 rounds serve? 17. At $3.25 per day, how many days' work will $ 100 pay for ? 18. If a change of elevation of about 295 meters makes a difference of 1° Centigrade in the boiling point, what is the boiling point at a place whose elevation is 2 miles ? 1} miles? 19. If the fall in the barometer is .1 inch for every 100 feet of elevation, what is the fall due to an increase in elevation of 2600 feet? of 6200 feet? 20. If the yield of alfalfa is one and one-half tons per acre for each cutting, and the crop is cut seven times each year, what is the yield per year on 12 acres ? A knowledge of ratio is essential to a proper understanding of the mathematics of all industrial and mechanical pursuits. For example, mechanical advantage is defined in terms of ratio ; specific gravity is defined in terms of ratio, and such expressions as “gear ratio” and " coefficient of expansion " are of frequent occurrence. The term specific gravity is used to denote the ratio between the weight of a body and the weight of an equal volume of water. If a cubic foot of lead weighs 11.4 times as much as a cubic foot of water, its specific gravity is said to be 11.4. Thus, 11.4 is the number that expresses the ratio which the weight of a given volume of lead bears to the weight of an equal volume of water. EXAMPLE 1. If a cubic foot of water weighs 62.3 pounds and the specific gravity of brass is 8.6, what is the weight of a cubic foot of the latter ? 62.3 Weight of 1 cubic foot of water 8.6 Specific gravity of brass EXAMPLE 2. What is the weight of a cubic foot of cork if the specific gravity of cork is 0.24? 62.3 Weight of 1 cubic foot of water .24 Specific gravity of cork Exercise 1. If the specific gravity of wrought iron is 7.8 and water weighs 62.3 pounds per cubic foot, find the weight of a cubic foot of wrought iron. 2. If the specific gravity of lead is 11.4, what is the weight of 5 cubic feet of it? 3. If platinum has a specific gravity of 21.4, find the weight of a cubic inch of it. 4. If earth has a specific gravity of 1.6, what is its weight per load (cubic yard)? 5. If the specific gravity of wet sand is 2, what weight of it will a box 4 feet wide, 1 foot deep, and 8 feet long hold ? 6. If concrete has a specific gravity of 2.5, what is the weight of concrete per cubic yard ? 7. If the weight of a sphere is a little more than half as much as a cube of the same dimensions, find the weight of 10 lignum vitae nine-pin balls 6 inches in diameter. (Specific gravity lignum vitae 1.3.) 8. If the specific gravity of ice is .9, what is the weight of a block of it 4 feet wide, 4 feet thick, and 8 feet long ? 9. Olive oil has a specific gravity of about .9. What is the weight of 5 gallons of it? 10. What is the weight of 1000 gallons of petroleum if its specific gravity is .8? a Proportion A proportion is a statement of equality between two equal ratios. Thus = is a proportion. It is equivalent to saying that the ratio of 3 to 4 is equal to the ratio of 6 to 8. The numbers 3, 4, 6, and 8 are called the terms of the proportion. When two ratios are equal the four terms are said to be in proportion. The numbers 4 and 6 are called the means; the numbers 3 and 8 are called the extremes. The proportion = g can also be written in the form 3:4= 6:8. It is read : “ 3 is to 4 as 6 is to 8.” In the above proportion it is obvious that 3 x 8 equals 4 x 6. In the proportion 2:5 = 4:10 it is also obvious that 2 x 10 equals 5 x 4. This relation will be observed . to hold true in every proportion. Rule. — In every proportion, the product of the extremes is equal to the product of the means. = : The word proportion is sometimes used as synonymous with ratio, as for example in the expressions, "the proportion of copper to zinc in brass,” ,” “the proportion of business men who succeed,” etc. The process of finding from three given numbers a fourth one which bears the same ratio to the third that the second does to the first is sometimes called The Rule of Three. It can be stated as follows: Rule. - Multiply the means together and divide by the given extreme. The ratio of 4 to 3 equals the ratio of 8 to what EXAMPLE 1. number? Now, since the product of the extremes equals the product of the 4 x = 24 means and x = 6, the required numbe = EXAMPLE 2. What is the height of a tree which casts a shadow 80 feet long at the same time a man 6 feet tall casts a shadow 5 feet long? The ratio of the man's height to the length of his shadow is Ģ, or 6:5. The ratio of the tree's height to the length of its shadow is the same. That is, the height of the tree : 80 = 6:5. Then 5 x height of tree 6 X 80, |