Explanation. Here we cannot take }" from 1"' until we change 1" to eighths and add the 1 or to it which we have borrowed from the whole number 8. This makes the minuend read 710" and we can now proceed to subtract. The result is 23", which tells how much wider the one board is than the other.

If the minuend happens to be a whole number without a fraction we can borrow 1 and change it into a fraction as we did in the previous example. Example. What is the difference between 5]" and 8''?

8'' =73"

57"
21"

Ans. 26. Multiplication of Fractions and Mixed Numbers. We have already learned that multiplication is a short method of addition. In multiplying fractions and whole numbers, fractions and fractions, fractions and mixed numbers, and mixed numbers together, certain precautions must be taken which lead to speed and accuracy.

27. The Product of a Whole Number and a Fraction. To multiply a whole number by a fraction we multiply the number by the numerator and divide this product by the denominator, or if it is possible without a remainder, simply divide the denominator by the number.

Multiplying a whole number by a fraction is the same thing as multiplying the fraction by the whole number: thus, 5 X is the same thing as X5. Example: What is 5 times ? 5x5 = 25=31

Ans. Explanation. Multiplying the whole number 5 by the numerator of the fraction 5 gives the product 25, and

a

dividing this by the denominator 7 to reduce the improper fraction to a mixed number gives 34. Let us test this problem by addition: 5+5+5+5+5= 25=34, which is the same result as found before.

Example. What will be the combined thickness of 4 boards each }" thick?

7 7

=
31

Ans.
2
2

Explanation. This is a case in which the whole number will be contained exactly into the denominator, without a remainder. Performing this division, 4 goes into 8 two times. Cross out the 4 and the 8 and write the quotient of this division near the 8. Since 7 is left above the line and 2 is left below the line, the result of the multiplication is , which, when simplified, equals 3). It is suggested that the student test this result by multiplying the whole number by the numerator and dividing this product by the denominator. Which method is easier?

28. The Product of Two Fractions. To multiply one fraction by another, we multiply the numerators together for the new numerator and multiply the denominators together for the new denominator. Example. What is the product of 1 and 3? ixf= 32

Ans.

9

Explanation. Multiplying the numerators, 3X3 equals 9, which is the numerator of the product. Multiplying the denominators, 4X8 equals 32, which is the denominator of the product. The result is, therefore, 32.

29. “ Of " means “ Times." The word of is often seen in problems in fractions. For example, the statement sometimes appears like this: What is i of 5''? In such a case we work the problem by multiplying, so we say that of means times. You can see that this is so by taking a piece of wood 5" long and cutting it into four equal parts and then taking three of these parts. These three parts will be of 5", and by actual measurement, they will be 31" long. Thus we know that of 5" =3". Let us now see what times 5" is. X5'' = * =3}", which is the same value that we found

4

before. Therefore, we see that the word of when used thug signifies multiplication.

30. Multiplying a Group of Whole Numbers and Fractions. If we wish to multiply any number of whole numbers and fractions together, we multiply the whole numbers and the numerators together for the numerator of the product and multiply the denominators together for the denominator of the product. Reduce the result to its simplest form.

Example. What is the product of the following: 2XiX5X? 1 3 2X1X5X3 30 15

Ans. 8 4X8 32 16

2xįx5x

Explanation: The product of the whole numbers and the numerators is 30 and the product of the denominators is 32. This gives 3%, which, when reduced to its lowest terms, equals 15.

31. Multiplying Mixed Numbers. To multiply a mixed number by a fraction or to multiply two mixed numbers together, reduce the mixed numbers to improper fractions and proceed as in multiplying fractions, reducing the product to its lowest terms. Never try to multiply mixed numbers without first reducing them to improper fractions.

Example. Multiply 53 by 72.

16

Explanation. First reducing the mixed numbers to improper fractions, 53 = 48 and 73 = 1,5. We then multiply these just as we did when working with fractions. This gives 64.5, which may be reduced to the mixed number, 40%.

32. Factors. The factors of a quantity are the numbers which multiplied together will make that quantity. The factors of 4 are 2 and 2 because 2x2=4. If we multiply 2X2 X3 we get 12 and, therefore, the factors of 12 are 2 and 2 and 3. Notice that every factor of a quantity will exactly divide the quantity; that is, will divide it without a remainder.

33. Cancellation. Now that we know what a factor is we can work many prob

3'0" lems in the multiplication of fractions and whole numbers by the short method of cancellation. You remember from a previous lesson that we can divide both the numerator and de

A

534 nominator of a fraction by the same number without changing its value. Cancellation consists in separating the numerator and the denominator of the fraction into their factors and dividing by (canceling out) their common factors. If a number of fractions and whole numbers or mixed numbers are to be multiplied together, the method of canceling can be used to great advantage. Το show the advantage of the

Fig. 16.-Door. method of cancellation we will work the same example out both ways.

Example. Multiply the following fractions together:

7,0,4

5'456"