REDUCTION OF MONOMIALS AND POLYNOMIALS.

§ 171. A monomial quantity, or simply a monomial, is a quantity expressed by a single name of measuring units, (§ 154). Thus 5 dollars is a monomial; 10 shillings is a monomial. $172. A polynomial quantity, or simply a polynomial, is a quantity expressed by two or more names of measuring units.

Thus 5 dollars 25 cents is a polynomial; 3 pounds, 10 shillings, and 6 pence, is a polynomial.

A polynomial is composed of two or more monomials, which may thence be called the terms of the polynomial.

Thus in the first example given, the terms are 5 dol. and 25 c.; and in the second, 3£, 10 s. 6 d.

Note.-Monomial quantities have by some been called denominate numbers, and polynomials have usually been called compound numbers.

§ 173. Reduction descending consists in finding the value of a given quantity in measuring units of a lower order, (§ 155). The quantity is then said to be reduced to a lower name or denomination.

RULE X X XIII.

§ 174. To reduce a Quantity to a LOWER DENOMINATION.

1. Multiply a monomial of a higher denomination, or the highest term of a polynomial, by that number of the next lower denomination which makes a unit of the higher the product will be in the lower denomination.

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

3. In reducing a MONOMIAL FRACTION to lower denominations, the integers in the successive products may be reserved, and afterwards arranged as the terms of a polynomial.

EXAMPLE.

To reduce 5£, 14s. 9d. to pence.

£5 14 s. 9 d.
20

1 14 s.

12

1377 d.

We multiply 5 £ by 20, because 20 s. make 1 £; the product is shillings-to which adding the 14 s., we have 114 s.

We next multiply the 114 s. by 12, because 12 d. make 1 s.; the product is pence-to which adding the 9d., we have 1377 d. Thus we find 5 £ 14 s. 9 d. to be equal to 1377 d.

EXERCISES.

1. Reduce 4 lb. 7 oz. 13 dwt. to dwt.

Recollect that 12 oz. make 1 lb., and 20 dwt. make 1 oz.

2. Reduce 7 lb. 10 dut. 2 gr. to gr..
3. Reduce 3 T. 2 cwt. 3 qr. to qr.
4. Reduce 9 cut. 1 gr. 13 oz. to oz..
5. Reduce 14 3, 2
12 gr.
to gr.
6. Reduce 8 ib, 1 3, 15 gr. to gr.
7. Reduce 15 bu. 2 pk. 7 qt. to qt..
8. Reduce 9 bu. 5 qt. 1 pt. to pt..
9. Reduce 3 pi. 1 hhd. 40 gal. to gal.
10. Reduce 4 tuns, 5 hhd. 3 qt. to qt..
11. Reduce 13 m. 7 fur. 25 r. to r.
12. Reduce 10 L. 16 fur. 15 p. to p.
13. Reduce 20 yd. 3 qr. 2 na. to na.
14. Reduce 31 yd. 3 na. 2 in. to in..
15. Reduce 14 A. 1 R. 20 P. to P.

Ans. 1113 dwt.

Ans. 43957 sq. yd.

Ans.
Ans.

Ans.

287 cu. ft.

186724 cu. in. 210785 min. 670805 min.

274688 oz.

437271 gr.

7356 gills. 30882 yd.

16. Reduce 9 A. 13 P. 4 sq. yd. to sq. yd. 17. Reduce 10 cu. yd. 17 cu. ft. to cu. ft. 18. Reduce 4 cu. yd. 100 cu. in. to cu. in. 19. Reduce 20 wk. 5 da. 33 hr. 5 min. to min. Ans. 20. Reduce 1 yr. 100 da. 20 hr. 5 min. to min. 21. Reduce 7 T. 13 cwt. 1 qr. 4 lb. to oz. Ans. 22. Reduce 75 th, 10 3,7 3, 29, 11 gr. to gr. Ans. 23. Reduce 3 hhd. 40 gal. 3 qt. 1 pt. to gills. Ans. 24. Reduce 5 L. 2 m. 4 fur. 15 r. to yards. Ans. 25. Reduce 11A. 2 R. 25 P. 25 sq.yd. to sq.yd. Ans. 564414 sq. yd.

Monomial Fractions Reduced to Integers.

26. Reduce £ to integers in shillings, &c.

£ multiplied by 20 produces 40 s.=55 shillings, (§ 174). Reserving the integer 5 s., and reducing the fraction 4 s. to pence,

s. multiplied by 12 produces 6 d.=84 pence. Reserving the integer 8 d., and reducing the fraction d. to farthings,

d. multiplied by 4 produces 16 qr.=2 farthings. Arranging the integers reserved as the terms of a polynomial, we find £=5 s. 8 d. 23 qr.

27. Reduce .23 £ to integers in shillings, &c.

.23 £ multiplied by 20 produces 4.60 shillings, (§ 174). Reserving the integer 4 s., and reducing the fraction .60 s. to pence,

.60 s. multiplied by 12 produces 7.20 pence. Reserving the integer 7 d., and reducing the fraction .20 d. to 20 d. multiplied by 4 produces .80 farthings. Arranging the integers reserved as the terms of a polynomial, we find .23 £=4 s. 7 d. 0.80 qr.

qr.,

The integers found in reducing are arranged with the quantity in the lowest denomination, whether that quantity be an integer or otherwise.

28. Reduce lb. to integers in oz., &c.

Ans. 5 oz. 6 dwt. 16 gr.

29. Reduce .17 lb. to integers in oz., &c.

Ans. 2 oz. O dut. 19.2 gr.

30. Reduce gr. to integers in lb., &c.

31. Reduce .19 T. to integers in cut.,

Ans. 18 lb. 10 oz. 10 dr.
&c.

Ans. 3 cwt. 3 qr. 5.6 lb.
Ans. 2 3,0 3,8 gr.
Ans. 2 3,2 9, 8 gr.
Ans. 4 qt. 1 pt. 1 gi.

32. Reduce 3 to integers in 3, &c.
33. Reduce .35 3 to integers in 3, &c.
34. Reduce pk. to integers in qt., &c.
35. Reduce pi. to integers in hhd., &c.

Ans. 1 hhd. 12 gal. 2 qt.

36. Reduce .31 bu. to integers in pk. &c.

Ans. 1 pk. 1 qt. 1.84 pt.

37. Reduce .6 tun, to integers in pi., &c.

Ans. 1 pi. 25 gal. 1.6 pt.

38. Reduce 87 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 1 yd. to integers in gr., &c.

Ans. 3 qr. 2 na. 11⁄21⁄2 in.

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

Ans. 3 gr. 3 na. 1.71 in.
Ans. 2 R. 26 P. 201 yd.

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

§ 173. 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 2 qr. and divide it by 4, because 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 3 s. and divide by 20, because 20 s. make 1£, a unit of the next higher denomination.

Thus we find 10 s. 6d. 2 gr. to be equal to

£.

The same reductions performed decimally, will be presented thus;

2 gr.÷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.

The fraction will then be 88 £=3§3 £. 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. 188 lb.

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

Ans. .890' lb.

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

Ans. cwt.

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

Ans. .466' T.

5. Reduce 27, 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.