Now, by cancelling out the common factors before multiplying, we can obtain the same product more quickly and with less work, as follows:

Explanation: First the 3 in the numerator and the 3 in the denominator are cancelled. In other words, a common factor, 3, is divided into the 3 in the numerator and the 3 in the denominator; 3 ÷ 3 = 1, so we cross out the 3's and place 1's in their places. Then a common factor, 2, is canceled from the 2 in the numerator and the 4 in the denominator, leaving a 1 in place of the 2 and a 2 in place of the 4. No more factors can be cancelled, so we multiply together the remaining figures above the line for the numerator of the answer, and below the line for the denominator, giving us the answer,, as before. Note that, whenever a factor is cancelled from the numerator, the same factor must be cancelled from the denominator.

Examples:

1. Suppose we have several fractions to multiply together, as

By cancellation, we could obtain the product in its lowest terms immediately, without reduction, as follows:

Explanation: First a 3 in the numerator is cancelled with a 3 in the denominator, leaving a 1 in place of each 3 thus cancelled. Next a common factor 2 is cancelled from the 2 and the 4; in other words, they are both divided by 2. We see that 7 is a common factor of 14 and 21, so we cancel out a 7 by dividing each of these terms by 7. This leaves 2 in place of the 14 and 3 in place of the 21. There are no more common factors; therefore, we multiply together the remaining numbers and find the answer to be 1.

Explanation: First cancel 250 out of 500 and also out of 250, then 9 out of 36 and 63; then 10 out of 20 and 50, and finally 2 out of 2 and 2, and 7 out of 7 and 42.

26. Division-The Reverse of Multiplication.-Division is just the opposite of multiplication, and this fact gives us the cue to a very simple method of dividing fractions.

To divide one fraction by another, invert the divisor and then multiply. To invert means to turn upside down. Invert and we get ; invert and we get.

Suppose we have a fraction to divide by a whole number; as

27. Compound Fractions. Sometimes we see a fraction which has a fraction for the numerator and another fraction for the denominator. This is called a Compound Fraction. If we remember that a fraction indicates the division of the numerator by the denominator, we will see that a compound fraction can be simplified by performing this division.

28. How to Analyze Practical Problems.-The chief trouble that students have in working practical problems is in analyzing the problems to find out just what operations they should use to solve them. Problems in multiplication or division of fractions will fall in one of the three following cases:

1. Given a whole; to find a part (multiply).

2. Given a part; to find the whole (divide).

3. To find what part one number is of another (divide).

Example of Case 1:

The total weight of a shipment of steel bars is 3425 lb.

3

1

this consists of 4-in. round bars and the balance is 2-in. round. What weight

is there of each size?

137

7 3425

X

=

959 lb. of -in. bars.

25

1

Explanation: In this example we have the whole (3425 lb.); to find a

959 2466 lb. of 4-in. bars. part (). If the whole is 3425, then

=

Example of Case 2:

1

The base of a dynamo weighs 270 lb.; the base is of the total weight; find the total weight.

A molder who is on piece work sets up 91 flasks, but the castings
What fractional part of his work does he get

from 7 of them are defective.

53. A gallon is about of a cubic foot. If a cubic foot of water weighs 62 lb., how much does a gallon weigh?

54. A machine part requires two "finishing" operations on a milling machine, and the cost of these two operations on every piece is of the total cost of manufacturing. If the total cost of each is 60

cts., what will be the cost of the finishing operations on 10,000 of these parts?

55. If a certain sized steel bar weighs 23 lb. to the foot, how long must a piece be to weigh 8 lb.?

56. How many steel pins to be finished 13 in. long can be cut from an 8-ft. rod if we allow 1 in. to be wasted in cutting off and finishing each pin?

57. A steam engine during a test developed 148 horsepower. If the steam consumed by the engine in an hour was 2277 lb., how much steam did it consume per horsepower hour?

58. A firm manufacturing pumps finds that it costs 73 cts. a pound to make its castings. A near-by foundry and machine company offers to make the castings at a price of 5 cts. per pound. However, the actual cost per pound when the castings are bought from this firm is found to be 13 times the net cost of 5 cts. Is it cheaper for the pump company to buy its castings or to make them in its own foundry?

59. It was estimated that the cableway of Fig. 7 could convey 24 loads of excavated earth per hour, each load containing about 24 cu. yd. If the crew works 93 hr. daily, how many days will it take to remove an excavation of about 12,000 cu. yd.

60. I want to cut 300 pieces of steel each 112 in. long for wagon tires. I have in stock a sufficient number of bars of the same size but they are 120 in. long; and I also have a sufficient number 235 in. long. Which length should I use in order to waste the least material? Calculate the total number of inches of stock that would be wasted in each case.