38. Reduce 37 m. to integers in fur., &c.

880

Ans. 3 fur. 20 r. 4 yd 39. Reduce .4 L. to integers in m., &c. Ans. 1 m. 1 fur. 24 r. 40. Reduce yd. to integers in gr., &c.

Ans. 3 qr. 2 na. 11⁄2 in.

41. Reduce .985 yd. to integers in qr. &c. Ans.

42. Reduce A. to integers in R., &c.

gr. 3 na. 1.71 in. Ans. 2 R. 26 P. 20§ yd.

43. Reduce .83 A. to integers in R., &c.

Ans. 3 R. 12 P. 24.2 yd.

44. Reduce cu. yd. to integers in cu. ft., &c.

Ans. 15 cu. ft. 1296 cu. in.

45. Reduce .3 cu. yd. to integers in cu. ft., &c.

Ans. 8 cu. ft. 172.8 cu. in.

46. Reduce degree to integers in min., &c.

Ans. 21 min. 25 sec.

47. Reduce .37 deg. to integers in min., &c.

48 Reduce wk. to integers in da., &c.

Ans. 22 min. 12 sec.

Ans. 4 da. 21 hr. 36 min.

49. Reduce .85 wk. to integers in da. &c.

Ans. 5 da. 22 hr. 48 min.

REDUCTION ASCENDING.

§ 175. Reduction ascending consists in finding the value of a given quantity in measuring units of a higher order. The quantity is then said to be reduced to a higher name or denomination.

RULE XXXIV.

§ 176. To reduce a Quantity to a HIGHER DENOMINATION.

1. Divide a monominal of a lower denomination, or the lowest term of a polynomial, by the number of that denomination which makes a unit of the next higher denomination: the quotient will be in the higher denomination.

2. This quotient may, in like manner, be reduced to a still higher denomination, and so on, observing that each higher term in a polynomial must be added to the quotient in the same denomination with itself.

3. In reducing a MONOMIAL INTEGER to higher denominations, each remainder may be reserved in the same denomination with the dividend whence it is derived, and the last quotient and the several remainders be afterwards arranged as the terms of a polynomial.

EXAMPLE.

To reduce 10 s. 6 d. 2 qr. to the denomination of £.

We take the lowest term

qr. and divide it by 4, ba ase 4 qr. make 1 d., a unit of the next higher denomination

2 gr. d. d.

We then add the 6d. to the d., and divide by 12, because 12 d. make 1 s., a unit of the next higher denomination.

We next add the 10 s. to the s. and divide by 20, because 20 s. make 1£, a unit of the next higher denomination.

10 20

253
80

1012 s.= £=383 £.

Thus we find 10 s. 6d. 2 qr. to be equal to 253£.

480

The same reductions performed decimally, will be presented thus;

2 qr.÷4.5 d.; 6.5 d.÷12.541' s.; 10.541' s.-20.527' £. Another method of Reducing a Polynomial to a Fraction of a Higher Denomination.

$177. A Polynomial may also be reduced to a vulgar fraction of a higher denomination, by reducing the given quantity to its lowest denomination, for a numerator, and reducing a unit of the higher denomination to the same lowest denomination, for a denominator.

Thus to reduce 10 s. 6 d. 2 qr. to the fraction of a £. 10 s. 6 d. 2 qr.=506 qr.; and 1 £=960 qr.

960

480

The fraction will then be 508 £=253 £. And this fraction reduced to a decimal (§ 153) gives .527' £, the same as in the preceding example.

EXERCISES.

1. Reduce 8 oz. 15 dwt. 18 gr. to a fraction of a lb.

Ans. 703 lb.

2. Reduce 10 oz. 13 dwt. 20 gr. to a decimal of a lb.

960

Ans. .890' lb.

3. Reduce 2'qr. 14 lb. 12 oz. to a fraction of a cut.

Ans. 23 cut.

4. Reduce 9 cut. 1 gr. 10 lb. to a decimal of a T.

Ans. .466' T.

160

5. Reduce 23, 29, 17 gr. to a fraction of an 3. Ans. % 3. 6. Reduce 3 hhd. 5 gal. 3 qt. to a decimal of a tun.

Ans. .772' tun.

7. Reduce 4 yd. 2 ft. 9 in. to a fraction of a r.
8. Reduce 6 fur. 30 p. 4 yd. to a decimal of a m.

Ans. r.

Ans. .846' m.

9. Reduce 2 qr. 3 na. 2 in. to a fraction of a yd. Ans. 107 yd. 10. Reduce 1 qr. 2 na. 11⁄2 in. to a decimal of a yd.

Ans. .416' yd.

11. Reduce 8 sq. ft. 100 sq. in. to a decimal of a sq. yd.

Ans. .966' sq. yd.

12. Reduce 3 R, 20 P. 9 sq. yd. to a decimal of an A.

Ans. .876' A.

13. Reduce 20 cu. ft. 1000 cu. in. to a decimal of a cu. yd.

Ans. .762' cu. yd.
Ans. 27 deg.

14. Reduce 40′ 30′′ to a fraction of a deg. 15. Reduce 15 min. 15 sec. to a decimal of an hr. Ans. .254' hr. 16. Reduce 3 hr. 4 min. 20 sec. to a decimal of a da.

17. Reduce 5 s. 10 d. 2 qr. to a fraction of a £. 18. Reduce 10 oz. 13 dwt. 4 gr. to a decimal of

19. Reduce 17 lb. 14 oz. 10 dr. to a decimal of a cut.

Ans. .159' cwt.

576

20. Reduce 5 3, 29, 10 gr. to a fraction of a lb. Ans. 35 lb. 21. Reduce 1 pk. 3 qt. 1 pt. to a decimal of a bu. Ans. .359' bu. 22. Reduce 35 gal. 1 pt. 1 gi. to a decimal of a hhd.

23. Reduce 25 r. 3 yd. 2 ft. to a decimal of a m. 24. Reduce 50 sq. yd. 5 sq. ft. to a decimal of an

25. Reduce 200 da. 14 hr. to a fraction of a yr. 26. Reduce 175 da. 23 hr. to a decimal of a yr.

Ans. .558' hhd.
Ans. .080' m.

A.

Ans. .010' A. Ans. yr. Ans. .482'yr.

Monomial Integers Reduced to Polynomials. 27. Reduce 3531 qr. to a polynomial in £, s. &c.

4) 353 1 qr.

12) 882 d. 3 qr.

20) 73 s. 6 d.

3£ 13 s. 6d. 3 qr.

4380

Dividing 3531 qr. by 4, we find 882 d., with the remainder 3 qr.; dividing the 882 d. by 12, we find 73 s., with the remainder 6 d.; dividing the 73 s. by 20, we find 3 £, with the remainder 13 s.

Arranging the last quotient 3 £ and the several remainders as the terms of a polynomial, we find 3531 gr. equal to 3£ 13s. 6 d. 3 qr.

28. Reduce 530 d. to a polynomial in £, &c. Ans. 2 £ 4s. 2 d. 29. Reduce 874 dwt. to a polynomial in lb., &c.

Ans. 3 lb. 7 oz. 14 dwt.

36. Reduce 1000 dr. to a polynomial in lb., &c.

Ans. 3 lb. 14 oz. 8 dr.

31. Reduce 785 lb. to a polynomial in cut., &c.

Ans. 7 cwt. 0 qr. 1lb.
Ans. 3 ib, 2 3.

32. Reduce 870 to a polynomial in , &c. 33. Reduce 748 3 to a polynomial in ib, &c. Ans. 7, 93, 43. 34. Reduce 62 pt. to a polynomial in pk., &c. Ans. 3 pk. 7 qt. 35. Reduce 730 qt. to a polynomial in bu., &c.

Ans. 22 bu. 3 pk. 2 qt.

36. Reduce 890 bbl. to a polynomial in tuns, &c.

Ans. 111 tuns, 1 hhd.

37. Reduce 75 hhd. to a polynomial in tuns, &c.

Ans. 18 tuns, 1 pi. 1 hhd.
Ans. 12 r. 2 ft.

38. Reduce 200 ft. to a polynomial in r., &c. 39. Reduce 540 yd. to a polynomial in fur., &c.

Ans. 2 fur. 18 r. 1 yd. 40. Reduce 1000 r. to a polynomial in L., &c. Ans. 1 L. 1 fur. 41. Reduce 375 na. to a polynomial in yd., &c.

Ans. 23 yd. 1 qr. 3 na.

42. Reduce 4750 sq. in. to a polynomial in sq. yd., &c.

Ans. 3 sq. yd. 5 ft. 142 in.

43. Reduce 7562 sq. yd. to a polynomial in A., &c.

Ans. 1 A. 2 R. 10 P.

44. Reduce 9374 cu. in. to a polynomial in cu. ft., &c.

Ans. 5 cu. ft. 734 in. 45. Reduce 4034" to a polynomial in deg., &c. Ans. 1° 7′ 14′′. 46. Reduce 371' to a polynomial in deg., &c. 47. Reduce 3875 sec. to a polynomial in hr., &c.

Ans. 6° 11'.

Ans. 1 hr. 4 min. 35 sec.

48. Reduce 4375 min. to a polynomial in da., &c.

Ans. 3 da. 55 min Ans. 20 wk. 4 da. 14 hr.

49. Reduce 3470 hr. to a polynomial in wk., &c.

50. Reduce 4831 d. to a polynomial in £, &c.

51. Reduce 3743 d. to a polynomial in £, &c.

Ans. 20 £ 2 s. 7 d. Ans. 15£ 11 s. 11 d. Ans. 2 cwt. 3 qr. 27 lb. Ans. 12, 11 3, 5 3.

52. Reduce 335 lb. to a polynomial in cwt., &c.

53. Reduce 3735 ⇒ to a polynomial in †, &c.

54. Reduce 17630" to a polynomial in deg., &c.

Ans. 4° 53′ 50′′.

A Polynomial Reduced to the Denomination of either of its

Terms.

$173. A Polynomial may be reduced to the denomination of either of its terms, by reducing the other terms to that denomination, and adding together the several parts.

EXAMPLE.

To reduce 7£ 10 s. 8d. 2 gr. to shillings.

By Rule XXXIII, 7£=140 s.; and by Rule XXXIV, 8 d. 2 qr.=.708's.;

then 7£ 10 s. 8 d. 2 gr.=140 s.+10 s.+.708's.=150.708's.

Ans. 55.910' cut.
Ans. 103.437' pk.
Ans. 1469.25 gal.
Ans. 1510.545 r.
Ans. 579.165' P.
Ans. `6.639' tuns.

55. Reduce 9 lb. 8 oz. 15 dut. 5 gr. to dwt. Ans. 2335.208' dwt. 56. Reduce 2 T. 15 cwt. 3 qr. 18 lb. to cut. 57. Reduce 25 bu. 3 pk. 3 qt. 1 pt. to pk. 58. Reduce 5 tuns, 3 hhd. 20 gal. 1 qt. to gal. 59. Reduce 4 m. 5 fur. 30 r. 3 yd. to r. 60. Reduce 3 A. 2 R. 19 P. 5 sq. yd. to P. 61. Reduce 6 tuns, 2 hhd. 35 gal. 1 pt. to tuns. 62. Reduce 12 A. 3 R. 21 P. 25 sq. yd. to A. 63. Reduce 10 T. 15 cut. 1 gr. 25 lb. to cut. 64. Reduce 25 L. 2 m. 7 fur. 30 p. to fur.

Ans. 12.886' A. Ans. 215,473 cwt. Ans. 623.75 fur.

Cubic Measure Reduced to Gallons, Bushels, &c.

§ 179. Cubic measure may be reduced to gallons in Beer or Wine Measure, or to bushels in Dry Measure, by dividing the number of cubic inches by the number of cubic inches in a gallon or bushel, respectively. (§ 167).

EXAMPLE.

65. How many bushels of wheat would be contained in a box, the capacity of which is 100 cu. ft. 500 cu. in.?

100 cu. ft. 500 cu. in.=173300 cubic inches;

and 2150.4 cu. in. make 1 bushel; then 1733002150.4= 80.589' bushels.

66. How many bushels of salt could be put into a receptacle which measures 2160 cu. ft. 1000 cu. in.? Ans. 1736.179' bu. 67. How many barrels of water will be contained in a cistern whose capacity is 25000 cu. ft. 1500 cu. in.?

68. How many barrels of wine will whose capacity is found to be 730 cu. ft.

Ans. 4255.466' bar. be contained in a vat 49 cu. in.?

Ans. 173.364' bar.