156. To divide a fraction by an integer, as ÷5.

the result is ; but when, or a quantity 8 times less than 7 is to be divided by 5, then the quotient must be 8 times less than

Therefore, to divide a fraction by an integer, multiply the integer by the denominator, and divide the numerator by the product.

Thus÷8%; }÷10=12%.

157. 3rd, to divide an integer by a mixed quantity, as 6÷33.

Having reduced the mixed quantity into an improper fraction, the explanation is like case 1st.

158. 4th, to divide a mixed quantity by an integer, as 43÷7.

When the mixed quantity is transformed into an improper fraction, we have 14÷7, which brings it under case 2nd.

Ex. Divide 5 by 9. We have 5÷9=40÷9=49.

361⁄2 by 12?

363÷12=223÷12=293=35%.

159. 5th, to divide a fraction by a fraction, as 7÷3.

If 7 were divided by 5, the quotient would be 7; then 7, a quantity 9 times less than 7, when divided by 5, gives a quotient 9 times less than, or; but if be divided by, a quantity 8 times less than 5, the result must be 8 times larger than, or $8=143.

=

Thus, 753; therefore, ÷5; and, therefore, ÷

Therefore, to divide a fraction by another, multiply the numerator of the dividend by the denominator of the divisor, and

divide the product, by the product of the numerator of the divisor by the denominator of the dividend.

160. 6th, to divide a mixed quantity by a fraction, as 33

After the mixed quantity has been reduced to an improper fraction, we proceed as in case 5th.

161. 7th, to divide a fraction by a mixed quantity, as ÷7.

Evidently, ÷ 7 = 1÷ 3 =

This case is like case 5th,

after the mixed quantity has been transformed into an improper fraction.

162. 8th, to divide a mixed quantity by another mixed quantity as 235%.

We know that 23÷54÷35 = 98=33.

Here we are led again into case 5th, after having reduced both mixed quantities to improper fractions.

163. RECAPITULATORY EXERCISES.

1. Find the quotient of 177; 11÷12; 36÷7; 15÷÷9; ÷1; 47÷1; 11÷87; 154÷4.

2. Required, the quotient of (+)÷4; (23 − 3)÷14; († of 34) ÷54; 43÷(24 of 4).

4. What is the result of (××16)÷(#× 11 × 15)?

6. Find the value of 4+÷31÷14.

8. Simplify the expression (4+1-3)× 163÷48.

9. Reduce }} 14+6%

; (34 - 1×4) ÷ (38 +31 × 8) to their

simplest expressions.

10. Determine the value of ((188+17+4381)-(341+261)}78.

13. 36 is the product of two numbers, one of which is 10g. Required, the other.

14. If of a railway share cost £30, how much will one share cost?

15. In 5 hours, a wheel makes 11500 revolutions. many revolutions are made per hour?

How

16. The of of a sum of money is 361. What is the sum? 17. A man gave away of his money and had £2 left? How

much had he?

18. The area of a rectangular field is 892 yards; its breadth is 32 yards. Required, its length.

19. A can perform a piece of work in 5 days, B in 7 days, and C in 9 days. In how many days will the three do it together?

20. It is required to place five persons, A, B, C, D, and E, according to their statures; A is 64 feet high, B is of A's, C is of B's, D is of C's, and E is g of D's height.

21. Three contractors propose to dig a canal; the first could do it in of a year, the second in of a year, and the third in of a year. In what time could the three contractors do it?

22, How many yards of linen, yard broad, would be required. to line 48 yards of cloth, & broad.

23. Two men, 225 miles apart, start at the same time, to meet each other; A goes 5 miles per hour, and B goes 6 miles per hour. In how many hours will they meet?

24. The difference between and of a number is 20. What is the number?

25. A ship's crew has 15 days' provisions, but circumstances oblige them to be at sea 20 days. To what must each man's daily allowance be reduced?

26. A post is coloured so that blue, and the 12 remaining feet post is required?

of it is white,

are red.

black,

The length of the

27. A cistern has two feeding pipes; the first fills it in 2 hours, the other in 3 hours; the water is let out through a tap, that empties it in 1 hours. Suppose the cistern empty, and the two pipes and the tap act at the same time, how long will it take to fill the cistern.

PART IV.

DECIMAL FRACTIONS.

164. In the common system of numeration, it has been shown that a figure has a value ten times smaller than if it occupied the place on its left side; the same law being extended to the right side of the units (after which let a point be placed), we shall find that the first figure next to the units expresses tenths, the second hundredths, the third thousandths, &c.

Thus, 4.4 signify 4 wholes and 343.507 is 10+1000 or 100%.

507

34.07 is the same as

175. Then, there are two ways of expressing a fraction, (when the denominator is 10, 100, 1000, &c.,) either as a common fraction, or by writing the numerator after the units' place, taking care to set a point between.

166. Quantities of this kind, which are a part of unity, and whose denominator is 1, followed by one or more ciphers, which denominator, however, is not set down, are called Decimal Fractions.

167. Therefore, in decimals, the figures after the point express the numerator; and the denominator, which is not set down, is always 1, followed by as many ciphers as there are figures after the point.

168. It will be well to exercise the pupils upon the following scheme, and make them find the value of the figures, taken singly or combined together, in each line, the middle column being that of the units.