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DIVISION OF FRACTIONS.
EXAMPLES FOR THE BOARD.
Divide by : Divide 24, (4) by 16, (.)
The answer to the first question, dividing numerator by numerator, and denominator by denominator, is found to be The second example does not admit of so ready a division. But if we reduce both fractions to a common denominator, the question is resolved into the division of by , which gives 1992, or 198. [See Mental Arithmetic, Sect. XXIV.] We may obtain the quotient in another manner, as follows:
contains 1, times. It contains , 9 times as often as 1; that is, 9 times 18, or 192 times. It contains , la as often as ģ; that is, il of 192, 12 times. Now, if we had inverted the divisor, and multiplied by 14, the result would have been the same. Therefore, when one fraction cannot be directly divided by another, we may either reduce them both to a common denominator, and divide their numerators, or invert the divisor, and proceed as in multiplication.
1. Divide? by 2,(i); by }; by .
7. What is the quotient of 14; by 24; by 15; by 21?
8. What is the quotient of 3 by i ; by 11 ; by
9. If 5} barrels of flour cost $22}, what is the price per barrel ?
10. If a labourer receives $9.47} for 8} days' work, what are his daily wages ?
11. Divide 1313 by 9,3; by 217; by 42; ; by 16.
REDUCTION OF FRACTIONS.
EXAMPLE FOR THE BOARD.
Reduce i to its lowest terms. 3)}; 2)| Dividing any number by 1 does not alter its
value. Therefore, if we can find any number that will divide both the numerator and denominator of a frac tion, without a remainder, we may perform the division, and the resulting fraction will have the same value. In this exam. ple, we find that 7 will divide both 42 and 56. As o equal 1, , which is the quotient of by or 1, is equivalent to se. Dividing & again by ź, we obtain , as the lowest terms of the
The discovery of common divisors may often be facilitated, by attending to the following rules, viz. :
2 will divide any number, whose right-hand figure is either 0, 2, 4, 6, or 8.
3 will divide any number, if the sum of its figures is divisible by 3.
4 will divide any number, if its two right-hand figures are divisible by 4.
5 will divide any number, whose right-hand figure is either 0 or 5.
9 will divide any number, if the sum of its figures is divisible by 9.
10 will divide any number, whose right-hand figure is 0.
11 will divide any number, if the sum of its odd digits, (the 1st, 3d, 5th, &c.,) differs from the sum of its even digits, (the 2d, 4th, 6th, &c.,) by 0 or 11.
1. Reduce each of the following fractions to its lowest terms. fi poi ti ti ti to; }}; if ; 1}.
2. Reduce each of the following fractions to its lowest terms. fó; i ; 31; 11ới 600 ; 29160 ;
2 56: 63 1024,999.
TO REDUCE DECIMALS TO FRACTIONS.
Write the decimal for a numerator, and the denomination tenth, hundredth, fc. for a denominator, and reduce this fraction to its lowest terms. Thus, .5 is ii or ; .25 is 105 or ..
1. Reduce each of the following decimals to a fraction. .3; .07; .009; .216 ; .00309; .0007803; .91604; .0007.
TO REDUCE FRACTIONS OF A HIGHER DENOMINATION, TO WHOLE NUMBERS OF A LOWER, AND THE REVERSE.
This may be done in the same way as reduction of whole numbers, by multiplying, or dividing, as the case may require.
EXAMPLES FOR THE BOARD.
Reduce g of a mile to furlongs, &c. Reduce 24min. 31sec. to the fraction of a day. 1. Reduce of a bushel to pecks, &c. 2. Reduce / of a day to hours, &c. 3. Reduce of a gallon to pints, fluidounces, &c. 4. Reduce 5s. Od. 3qr. to the fraction of a £. 5. Reduce lqr. 2na. to the fraction of a yard. 6. Reduce 9d. lqr. to the fraction of a shilling. 7. Reduce .934£ to 8. d., foc. 8. Reduce 5cwt. 3qr. to the fraction of a ton. 9. Reduce .076 miles to furlongs, &c. 10. Reduce 1R. 30r, to the fraction of an acre. 11. Reduce .89m to the fraction of a gallon. 12. Reduce 5.7min. to the fraction of a year.
13. Reduce .9889T. to cwt., qr., fc. To drams and the fraction of a dram.
In the following chapter, a mental question is first given, to illustrate the succeeding example for the slate. Let the pupil learn to first find the answer for one, and afterwards for many.
1. If 4 barrels of flour cost $20.00, what will be 'the cost of 1 barrel ? Of 7 barrels ?
2. If 3.5 barrels of flour cost $17.75, what will be the cost of 1 barrel ? Of 9.25 barrels ?
3. If 5 bushels of wheat cost $5.00, what will 1 bushel cost ? 3 bushels ?
4. If 6.25 bushels of wheat cost $7.125, what will 1 bushel cost? 9.5 bushels ?
5. If 9 horses eat 54qts. of oats in a day, how much will 1 horse eat? 5 horses?
6. If 19 horses eat 4bu. 2pk. of oats in a day, how much will 1 horse eat? 13 horses?
7. What part of a shilling is 1 penny? 2 pence? 7 pence?
8. What part of a pound (240 pence) is 1 penny ? 78. 6d. (90 pence.)?
9. What part of a month is 1 day? 2 days? 16 days?
10. What part of a year is 1 day? 2mo. 19dy., (79 days.)?
11. What part of .8 is .1 ? 2? .3? 12. What part of 3.5 (3.50), is .01 ? .25? 9.25? 13. What part of 19 is 1? 13?
14. What part of 11s. 4d. Iqr. (545qr.) is 48. 3d. (204qr.)?
15. What part of 63gal. (2016gi.) is Igal. ipt. 2gi. (38gi.)?
16. What part of 3pk. 7qt. is 1pk. Ipt.?
17. If a man walks 3 miles an hour, how far will he walk in 5 hours ?
18. If a man walk 34 miles an hour, how far will he walk in 11 hours ?
19. If 3 nails of cloth cost 6 cents, what will 1 nail cost ? lqr. ? lyd.?
20. If 3 nails of cloth cost 7} cents, what will 1 nail cost? lyd.? 4$yd.?
21. Two railway trains start from the same point, in opposite directions; one travels 10 miles an hour, and the other 15 miles an hour. How far apart will they be in 1 hour? In 4 hours ?
22. Two railway trains start from the same point, in opposite directions; one travels 12į miles an hour, and the other 14} miles an hour. How far apart will they be in 73 hours ?
23. Anthony bought 4 oranges at 3 cents apiece ; what did he give for them all? How many apples at 2 cents apiece would pay for them?
24. A man sold 5} tons of hay, at $11.50 per ton, and received his pay in sheep at $3.83; apiece; how many did he receive ?
25. If a bushel of grain will last 5 horses a week, how long will it last 1 horse ? 2 horses?
26. If 12 bushels of grain will last 13 horses a month, how long will it last 9 horses ?
27. How long will it take 1 man to eat a barrel of provisions, that 10 men will eat in 6 days? How long will it take 2 men to eat the same barrel ? 3 men? 6 men ? 20 men ? 30 men ?
28. How long will it take 13 men to eat a barrel of provisions that 9 men will eat in 7$ days ?