11. If alcohol and olive oil, specific gravity .91, were put into the same vessel, which would go to the bottom? 12. A cylindrical piece of cork 1 in. in diameter and 2 in. long weighs how many times as much as an equal volume of water? 13. How many cubic inches of water weigh a pound? Is it approximately true for a pint of water, that a pint is a pound the world around "? 14. Express the specific gravities in the table as per cents. 15. What is the weight of a cubic centimeter of water in metric units? Of a cubic meter of water? Of a liter? (See the metric tables.) 16. What is the weight of a liter of alcohol? Of a liter of sea water? Of a cubic meter of coal? Of a cubic meter of steel? 17. What is the volume of 15 grams of water? Of 15 grams of cork? 18. What is the volume of 250 kg. of each of the substances in the table of specific gravities? 19. A kilogram of platinum is how many times as large as a kilogram of cast iron? 20. A load of bituminous coal weighs 4200 lb. How many cubic feet of coal in the load? Use the result of exercise 9. 21. A stick of walnut timber contains 12.3 cu. ft. How much does it weigh? 22. The ice man hauls 60 cu. ft. of ice at a load. How much does the load weigh? 23. A piece of steel has a volume of 320 cu. in. How much does it weigh? 37. Dividing a number into parts having a given ratio. A B FIG. 20 What is the ratio of AB to BC? Into how many equal parts is AC divided? AB is what part of AC? BC is what part of AC? It is seen that the line AC is divided into two parts which have the ratio of 2 : 3. If it is desired to divide the line MN (Figure 21) into two parts having the ratio of 3: 4, into how many equal parts must the line MN be divided? How many of these equal parts in the first part of MN? In the second part? M FIG. 21 EXAMPLE. Divide 84¢ into two parts which have the ratio of 3: 4. SOLUTION. Since the two parts of 84¢ are to have the ratio of 3 to 4, we divide the 84¢ into 7 equal parts. For the first of the two parts we take 3 of the equal parts and for the second we take 4 of the equal parts. of 84¢=36¢, one of the required parts. 1. Divide $65 into two parts having the ratio of 5: 8. 2. Two boys have 30 marbles which they wish to divide in the ratio of 2:3. How many should each receive? 3. A man and a boy, working together, receive $30 for a piece of work. They agree to divide that amount in the ratio of 5 to 3. How much does each receive? The man gets how many times as much as the boy? 4. The shares received by two heirs to an estate are in the ratio of 2 to 5. Their combined shares are $16,000. Find the share of each. 5. The seventh and the eighth grades have a picnic. together. There are 47 pupils in the seventh grade and 62 in the eighth. The total expenses are $26.48. How much should each grade pay? 6. A mixture of water alcohol to 3 parts of water. of the mixture? and alcohol contains 1 part of How much of each in 12 ounces 7. Concrete blocks are made of 1 part of cement to 3 parts of sand. How much of each in 240 cu. ft. of concrete blocks? 8. A concrete mixture for large engine foundations contains 1 part cement, 2 parts sand, and 4 parts crushed stone. How much of each in a foundation 12 ft. wide, 20 ft. long, and 4 ft. deep? 9. Two men engage in business together. The first invests $5000 and the second $7000. What is the ratio of the investment of each to the whole investment? The profits for the first year are $1500. They are divided in the ratio of the investments. How much does each man receive? 10. A drainage ditch costs $3875. The expense is borne by four farmers according to the number of acres of land that is drained for each. The ditch drains 200 acres for A, 180 acres for B, 400 for C, and 80 for D. How much should each pay? 11. Two men ship their hogs to market in the same car. The freight and other charges are $119.74. A's hogs weigh 3540 pounds and B's weigh 6085 pounds. What is the share of each in the expenses? 12. Two boys work at a job and agree to share the pay according to the number of hours each works. One works 12 hours and the other 15 hours. What part of the pay should each receive? 13. A certain recipe for milk sherbet is as follows: Milk, 4 parts; sugar, 11⁄2 parts; lemon juice, part. How much of each should be used to make 2 gallons of the sherbet? 38. Proportion. The statement of the equality of two ratios is called a proportion. Thus, the statement 19 is a proportion. = may also be written a: b=c: d. In either case the proportion may be read "a is to b as c is to d," or "the ratio of a to b equals the ratio of c to d." In the proportion a: bc: d, a, b, c, and d, are called the terms of the proportion; a and d are called the extremes and b and c are the means. The four numbers which form a proportion are said to be proportional. Principle. In any proportion the product of the means equals the product of the extremes. If 2=2, both members of the equation may be multiplied b d' by bd. Then ad=bc. This principle gives a convenient way to find one term of a proportion when the other three terms are given. EXAMPLE 2. Find x if 9:20=x: 12. SOLUTION. 20x = 108. The product of the means equals the Exercise 40 Solve each of the following proportions for the letter in 39. Proportional lines. Four lines are said to be proportional when the ratio of the first to the second equals the ratio of the third to the fourth. Draw a triangle ABC with AB=6 What is the ratio of AH to HB? Of CK to KB? Compare these ratios. If your construction and measurements were accurate this statement can be made as a proportion. Make it. Locate point P on AB so that AB is divided in the ratio of 1 : 7. How long is AP? PB? Through P draw PQ parallel to AC. AP BQ Measure BQ and QC. Compare and making the PB QC' statement as a proportion. Draw a triangle XYZ and divide one side into segments having the ratio 3: 5. Through the point of division draw a line parallel to another side of the triangle. This line |