194. MISCELLANEOUS REVIEW. 1. A piece of land in the form of an equilateral triangle measures on one side 46 rods. What is the distance around it? 2. The perimeter of a piece of land that is an exact square is 2463 feet. How far across on one side? 3. The length of a certain rectangular field is three times its breadth; its perimeter is 360rd. What is its breadth? its length? NOTE. If its breadth is x feet then its length is 3x feet, and its perimeter is 2x+6x feet. Then, 2x+6x= 360. 4. If of the value of a farm is $2154, what is of the value of the farm? NOTE.-If of a certain number is 24, what is the number? What is of the number? 5. I spent of my money and had $3.60 remaining. (a) How much did I spend? (b) What I had remaining, equals what part of what I spent? 6. Change ator is 30. to an equivalent fraction whose denomin 7. Change to an equivalent fraction whose numerator is 30. 8. Change tor is bc. a2 b to an equivalent fraction whose denomina 9. Multiply by 3 and multiply the product by 25. 10. Multiply by 13 and multiply the product by 25. 11. If of an acre of land is worth $36, how much are 37 acres worth at the same rate? 12. The rent of a house for 2 yr. 4 mo. was $840. What was the rate per year? From the foregoing operations the following rules for dividing by a fraction are obtained: RULE I. —Reduce the dividend and the divisor to like fractional units, then divide the numerator of the dividend by the numerator of the divisor. RULE II.—“ Invert the divisor and proceed as in multiplication." Observe that the inverted divisor shows the number of times the divisor is contained in 1: then in 6 it is contained 6 times as many times; in 4, 4 times as many; in, as many; in 3, 3 as many, etc. NOTE. For another explanation of the inversion of the divisor, see Book II., page 232. 18 thirds 2 thirds = 9. +1=3. 3 thirds ÷ 2 thirds = = 1} = 2. 35 fortieths 24 fortieths = 11. §1. 5 fifths ÷ 3 fifths = = 13 = §. = (a) Find the sum of the nine quotients. * 375 + 3 8. 196 ÷ 3 9. 196 + / (b) Find the sum of the nine quotients. † 24 1414 8. 11+ 8 9. 11-19 *Why is the sum of the nine quotients equal to (46 +375 +196) × 5? + Why is the sum of the nine quotients equal to (++11) × 4? Fractions. 196. TO REDUCE COMPLEX FRACTIONS TO SIMPLE FRACTIONS. Observe that every reduction of a complex fraction may be regarded as a problem in division. (a) Find the sum of the four fractions. Observe that a complex fraction may be reduced to a simple fraction by multiplying its numerator and denominator by some number that will in each case give an integral product. When this number can be easily discovered by inspection this is a convenient × 2 1 method of reduction: thus 14 7 X 2 |